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<title>Negative number</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Negative number</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>negative number</b> is the <a href="Opposite_(mathematics)" class="mw-redirect" title="Opposite (mathematics)">opposite</a> of a positive <a href="Real_number" title="Real number">real number</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Equivalently, a negative number is a real number that is <a href="Inequality_(mathematics)" title="Inequality (mathematics)">less than</a> <a href="0" title="0">zero</a>. Negative numbers are often used to represent the <a href="Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a> of a loss or deficiency. A <a href="Debt" title="Debt">debt</a> that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as <i>positive</i> and <i>negative</i>. Negative numbers are used to describe values on a scale that goes below zero, such as the <a href="Celsius" title="Celsius">Celsius</a> and <a href="Fahrenheit" title="Fahrenheit">Fahrenheit</a> scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −<span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>(−3) = 3 because the opposite of an opposite is the original value.

</p><p>Negative numbers are usually written with a <a href="Plus_and_minus_signs" title="Plus and minus signs">minus sign</a> in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced and read as "minus three" or "negative three". Conversely, a number that is greater than zero is called <i>positive</i>; zero is usually (<a href="Signed_zero" title="Signed zero">but not always</a>) thought of as neither positive nor <a href="Negative_zero" class="mw-redirect" title="Negative zero">negative</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its <a href="Sign_(mathematics)" title="Sign (mathematics)">sign</a>.

</p><p>Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as <a href="Natural_number" title="Natural number">natural numbers</a> (i.e., 0, 1, 2, 3, ...), while the positive and negative whole numbers (together with zero) are referred to as <a href="Integer" title="Integer">integers</a>. (Some definitions of the natural numbers exclude zero.)

</p><p>In <a href="Bookkeeping" title="Bookkeeping">bookkeeping</a>, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.

</p><p>Negative numbers were used in the <i><a href="Nine_Chapters_on_the_Mathematical_Art" class="mw-redirect" title="Nine Chapters on the Mathematical Art">Nine Chapters on the Mathematical Art</a></i>, which in its present form dates from the period of the Chinese <a href="Han_dynasty" title="Han dynasty">Han dynasty</a> (202 BC – AD 220), but may well contain much older material.<sup id="cite_ref-struik33_3-0" class="reference"><a href="#cite_note-struik33-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="Liu_Hui" title="Liu Hui">Liu Hui</a> (c. 3rd century) established rules for adding and subtracting negative numbers.<sup id="cite_ref-Hodgkin_4-0" class="reference"><a href="#cite_note-Hodgkin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> By the 7th century, Indian mathematicians such as <a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a> were describing the use of negative numbers. <a href="Islamic_mathematicians" class="mw-redirect" title="Islamic mathematicians">Islamic mathematicians</a> further developed the rules of subtracting and multiplying negative numbers and solved problems with negative <a href="Coefficients" class="mw-redirect" title="Coefficients">coefficients</a>.<sup id="cite_ref-Rashed_5-0" class="reference"><a href="#cite_note-Rashed-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Prior to the concept of negative numbers, mathematicians such as <a href="Diophantus" title="Diophantus">Diophantus</a> considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Western mathematicians like <a href="Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a> held that negative numbers were invalid, but still used them in calculations.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2></div>

<div class="mw-heading mw-heading3"><h3 id="The_number_line">The number line</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Number_line" title="Number line">Number line</a></div>

<p>The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a <b>number line</b>:

</p>

<p>Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are lesser. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left.

</p><p>Note that a negative number with greater magnitude is considered less. For example, even though (positive) <span class="texhtml">8</span> is greater than (positive) <span class="texhtml">5</span>, written

</p>

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</style><div class="block-indent" style="padding-left: 1.5em;"><span class="texhtml">8 > 5</span></div>

<p>negative <span class="texhtml">8</span> is considered to be less than negative <span class="texhtml">5</span>:

</p>

<div class="mw-heading mw-heading3"><h3 id="Signed_numbers">Signed numbers</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Sign_(mathematics)" title="Sign (mathematics)">Sign (mathematics)</a></div>

<p>In the context of negative numbers, a number that is greater than zero is referred to as <b>positive</b>. Thus every <a href="Real_number" title="Real number">real number</a> other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a <a href="Plus_sign" class="mw-redirect" title="Plus sign">plus sign</a> in front, e.g. <span class="texhtml">+3</span> denotes a positive three.

</p><p>Because zero is neither positive nor negative, the term <b>nonnegative</b> is sometimes used to refer to a number that is either positive or zero, while <b>nonpositive</b> is used to refer to a number that is either negative or zero. Zero is a neutral number.

</p>

<div class="mw-heading mw-heading3"><h3 id="As_the_result_of_subtraction">As the result of subtraction</h3></div>

<p>Negative numbers can be thought of as resulting from the <a href="Subtraction" title="Subtraction">subtraction</a> of a larger number from a smaller. For example, negative three is the result of subtracting three from zero:

</p>

<p>In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example,

</p>

<p>since <span class="texhtml">8 − 5 = 3</span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Everyday_uses_of_negative_numbers">Everyday uses of negative numbers</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Sport">Sport</h3></div>

</div>

Negative golf scores relative to par.

</div>

<ul><li><a href="Goal_difference" title="Goal difference">Goal difference</a> in <a href="Association_football" title="Association football">association football</a> and <a href="Hockey" title="Hockey">hockey</a>; points difference in <a href="Rugby_football" title="Rugby football">rugby football</a>; <a href="Net_run_rate" title="Net run rate">net run rate</a> in <a href="Cricket" title="Cricket">cricket</a>; <a href="Golf" title="Golf">golf</a> scores relative to <a href="Golf#Scoring_and_handicapping" title="Golf">par</a>.</li>

<li><a href="Plus%E2%80%93minus_(sports)" title="Plus–minus (sports)">Plus-minus</a> differential in <a href="Ice_hockey" title="Ice hockey">ice hockey</a>: the difference in total goals scored for the team (+) and against the team (−) when a particular player is on the ice is the player's +/− rating. Players can have a negative (+/−) rating.</li>

<li><a href="Run_differential" title="Run differential">Run differential</a> in <a href="Baseball" title="Baseball">baseball</a>: the run differential is negative if the team allows more runs than they scored.</li>

<li>Clubs may be deducted points for breaches of the laws, and thus have a negative points total until they have earned at least that many points that season.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li>

<li>Lap (or sector) times in <a href="Formula_1" class="mw-redirect" title="Formula 1">Formula 1</a> may be given as the difference compared to a previous lap (or sector) (such as the previous record, or the lap just completed by a driver in front), and will be positive if slower and negative if faster.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li>

<li>In some <a href="Athletics_(sport)" class="mw-redirect" title="Athletics (sport)">athletics</a> events, such as <a href="Sprint_(running)" title="Sprint (running)">sprint races</a>, the <a href="110_metres_hurdles" title="110 metres hurdles">hurdles</a>, the <a href="Triple_jump" title="Triple jump">triple jump</a> and the <a href="Long_jump" title="Long jump">long jump</a>, the <a href="Wind_assistance" title="Wind assistance">wind assistance</a> is measured and recorded,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and is positive for a <a href="Tailwind" class="mw-redirect" title="Tailwind">tailwind</a> and negative for a headwind.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Science">Science</h3></div>

<ul><li><a href="Temperature" title="Temperature">Temperatures</a> which are colder than 0 °C or 0 °F.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li>

<li><a href="Latitude" title="Latitude">Latitudes</a> south of the equator and <a href="Longitude" title="Longitude">longitudes</a> west of the <a href="Prime_meridian" title="Prime meridian">prime meridian</a>.</li>

<li><a href="Topography" title="Topography">Topographical</a> features of the earth's surface are given a <a href="Height" title="Height">height</a> above <a href="Sea_level" title="Sea level">sea level</a>, which can be negative (e.g. the surface elevation of the <a href="Dead_Sea" title="Dead Sea">Dead Sea</a> or <a href="Death_Valley" title="Death Valley">Death Valley</a>, or the elevation of the <a href="Thames_Tideway_Tunnel" title="Thames Tideway Tunnel">Thames Tideway Tunnel</a>).</li>

<li><a href="Electrical_circuits" class="mw-redirect" title="Electrical circuits">Electrical circuits</a>. When a battery is connected in reverse polarity, the voltage applied is said to be the opposite of its rated voltage. For example, a 6-volt battery connected in reverse applies a voltage of −6 volts.</li>

<li><a href="Ions" class="mw-redirect" title="Ions">Ions</a> have a positive or negative electrical charge.</li>

<li><a href="Wave_impedance" title="Wave impedance">Impedance</a> of an AM broadcast tower used in multi-tower <a href="Directional_antenna" title="Directional antenna">directional antenna</a> arrays, which can be positive or negative.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Finance">Finance</h3></div>

<ul><li><a href="Financial_statement" title="Financial statement">Financial statements</a> can include negative balances, indicated either by a minus sign or by enclosing the balance in parentheses.<sup id="cite_ref-CarysforthNeild2002_16-0" class="reference"><a href="#cite_note-CarysforthNeild2002-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Examples include bank account <a href="Overdraft" title="Overdraft">overdrafts</a> and business losses (negative <a href="Earnings" title="Earnings">earnings</a>).</li>

<li>The annual percentage growth in a country's <a href="Gross_domestic_product" title="Gross domestic product">GDP</a> might be negative, which is one indicator of being in a <a href="Recession" title="Recession">recession</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li>

<li>Occasionally, a rate of <a href="Inflation" title="Inflation">inflation</a> may be negative (<a href="Deflation" title="Deflation">deflation</a>), indicating a fall in average prices.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li>

<li>The daily change in a <a href="Share_(finance)" title="Share (finance)">share</a> price or <a href="Stock_market_index" title="Stock market index">stock market index</a>, such as the <a href="FTSE_100_Index" title="FTSE 100 Index">FTSE 100</a> or the <a href="Dow_Jones_Industrial_Average" title="Dow Jones Industrial Average">Dow Jones</a>.</li>

<li>A negative number in financing is synonymous with "debt" and "deficit" which are also known as "being in the red".</li>

<li><a href="Interest_rates" class="mw-redirect" title="Interest rates">Interest rates</a> can be negative,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> when the lender is charged to deposit their money.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Other">Other</h3></div>

<ul><li>The numbering of <a href="Storey" title="Storey">stories</a> in a building below the ground floor.</li>

<li>When playing an <a href="Audio_signal" title="Audio signal">audio</a> file on a <a href="Portable_media_player" title="Portable media player">portable media player</a>, such as an <a href="IPod" title="IPod">iPod</a>, the screen display may show the time remaining as a negative number, which increases up to zero time remaining at the same rate as the time already played increases from zero.</li>

<li>Television <a href="Game_shows" class="mw-redirect" title="Game shows">game shows</a>:

<ul><li>Participants on <i><a href="QI" title="QI">QI</a></i> often finish with a negative points score.</li>

<li>Teams on <i><a href="University_Challenge" title="University Challenge">University Challenge</a></i> have a negative score if their first answers are incorrect and interrupt the question.</li>

<li><i><a href="Jeopardy!" title="Jeopardy!">Jeopardy!</a></i> has a negative money score – contestants play for an amount of money and any incorrect answer that costs them more than what they have now can result in a negative score.</li>

<li>In <i><a href="The_Price_Is_Right_(U.S._game_show)" class="mw-redirect" title="The Price Is Right (U.S. game show)">The Price Is Right</a>'</i>s pricing game Buy or Sell, if an amount of money is lost that is more than the amount currently in the bank, it incurs a negative score.</li></ul></li>

<li>The change in support for a political party between elections, known as <a href="Swing_(politics)" title="Swing (politics)">swing</a>.</li>

<li>A politician's <a href="United_States_presidential_approval_rating" title="United States presidential approval rating">approval rating</a>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li>

<li>In <a href="Video_games" class="mw-redirect" title="Video games">video games</a>, a negative number indicates loss of life, damage, a score penalty, or consumption of a resource, depending on the genre of the simulation.</li>

<li>Employees with <a href="Flextime" title="Flextime">flexible working hours</a> may have a negative balance on their <a href="Timesheet" title="Timesheet">timesheet</a> if they have worked fewer total hours than contracted to that point. Employees may be able to take more than their annual holiday allowance in a year, and carry forward a negative balance to the next year.</li>

<li><a href="Transposition_(music)" title="Transposition (music)">Transposing</a> notes on an <a href="Electronic_keyboard" title="Electronic keyboard">electronic keyboard</a> are shown on the display with positive numbers for increases and negative numbers for decreases, e.g. "−1" for one <a href="Semitone" title="Semitone">semitone</a> down.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Arithmetic_involving_negative_numbers">Arithmetic involving negative numbers</h2></div>

<p>The <a href="Plus_and_minus_signs" title="Plus and minus signs">minus sign</a> "−" signifies the <a href="Operator_(mathematics)" title="Operator (mathematics)">operator</a> for both the binary (two-<a href="Operand" title="Operand">operand</a>) <a href="Operation_(mathematics)" title="Operation (mathematics)">operation</a> of <a href="Subtraction" title="Subtraction">subtraction</a> (as in <span class="texhtml"><i>y</i> − <i>z</i></span>) and the unary (one-operand) operation of <a href="Additive_inverse" title="Additive inverse">negation</a> (as in <span class="texhtml">−<i>x</i></span>, or twice in <span class="texhtml">−(−<i>x</i>)</span>). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in <span class="texhtml">−5</span>).

</p><p>The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand.

</p><p>For example, the expression <span class="texhtml">7 + −5</span> may be clearer if written <span class="texhtml">7 + (−5)</span> (even though they mean exactly the same thing formally). The <a href="Subtraction" title="Subtraction">subtraction</a> expression <span class="texhtml">7 − 5</span> is a different expression that doesn't represent the same operations, but it evaluates to the same result.

</p><p>Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>

</p>

<div class="block-indent" style="padding-left: 1.5em;"><span class="texhtml"><sup>−</sup>2 + <sup>−</sup>5</span>  gives <span class="texhtml"><sup>−</sup>7</span>.</div>

<div class="mw-heading mw-heading3"><h3 id="Addition">Addition</h3></div>

<p>Addition of two negative numbers is very similar to addition of two positive numbers. For example,

</p>

<p>The idea is that two debts can be combined into a single debt of greater magnitude.

</p><p>When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example:

</p>

<p>In the first example, a credit of <span class="texhtml">8</span> is combined with a debt of <span class="texhtml">3</span>, which yields a total credit of <span class="texhtml">5</span>. If the negative number has greater magnitude, then the result is negative:

</p>

<p>Here the credit is less than the debt, so the net result is a debt.

</p>

<div class="mw-heading mw-heading3"><h3 id="Subtraction">Subtraction</h3></div>

<p>As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer:

</p>

<p>In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus

</p>

<p>and

</p>

<p>On the other hand, subtracting a negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that <i>losing</i> a debt is the same thing as <i>gaining</i> a credit.) Thus

</p>

<p>and

</p>

<div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3></div>

<p>When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The <a href="Sign_(mathematics)" title="Sign (mathematics)">sign</a> of the product is determined by the following rules:

</p>

<ul><li>The product of one positive number and one negative number is negative.</li>

<li>The product of two negative numbers is positive.</li></ul>

<p>Thus

</p>

<p>and

</p>

<p>The reason behind the first example is simple: adding three <span class="texhtml">−2</span>'s together yields <span class="texhtml">−6</span>:

</p>

<p>The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:

</p>

<div class="block-indent" style="padding-left: 1.5em;"><span class="texhtml"> (−2</span> debts <span class="texhtml">) × (−3</span> each<span class="texhtml">) = +6</span> credit.</div>

<p>The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the <a href="Distributive_law" class="mw-redirect" title="Distributive law">distributive law</a>. In this case, we know that

</p>

<p>Since <span class="texhtml">2 × (−3) = −6</span>, the product <span class="texhtml">(−2) × (−3)</span> must equal <span class="texhtml">6</span>.

</p><p>These rules lead to another (equivalent) rule—the sign of any product <i>a</i> × <i>b</i> depends on the sign of <i>a</i> as follows:

</p>

<ul><li>if <i>a</i> is positive, then the sign of <i>a</i> × <i>b</i> is the same as the sign of <i>b</i>, and</li>

<li>if <i>a</i> is negative, then the sign of <i>a</i> × <i>b</i> is the opposite of the sign of <i>b</i>.</li></ul>

<p>The justification for why the product of two negative numbers is a positive number can be observed in the analysis of <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Division">Division</h3></div>

<p>The sign rules for <a href="Division_(mathematics)" title="Division (mathematics)">division</a> are the same as for multiplication. For example,

</p>

<p>and

</p>

<p>If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative.

</p>

<div class="mw-heading mw-heading2"><h2 id="Negation">Negation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Additive_inverse" title="Additive inverse">Additive inverse</a></div>

<p>The negative version of a positive number is referred to as its <a href="Additive_inverse" title="Additive inverse">negation</a>. For example, <span class="texhtml">−3</span> is the negation of the positive number <span class="texhtml">3</span>. The <a href="Addition" title="Addition">sum</a> of a number and its negation is equal to zero:

</p>

<p>That is, the negation of a positive number is the <a href="Additive_inverse" title="Additive inverse">additive inverse</a> of the number.

</p><p>Using <a href="Algebra" title="Algebra">algebra</a>, we may write this principle as an <a href="Algebraic_identity" class="mw-redirect" title="Algebraic identity">algebraic identity</a>:

</p>

<p>This identity holds for any positive number <span class="texhtml"><i>x</i></span>. It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers. Specifically:

</p>

<ul><li>The negation of 0 is 0, and</li>

<li>The negation of a negative number is the corresponding positive number.</li></ul>

<p>For example, the negation of <span class="texhtml">−3</span> is <span class="texhtml">+3</span>. In general,

</p>

<p>The <a href="Absolute_value" title="Absolute value">absolute value</a> of a number is the non-negative number with the same magnitude. For example, the absolute value of <span class="texhtml">−3</span> and the absolute value of <span class="texhtml">3</span> are both equal to <span class="texhtml">3</span>, and the absolute value of <span class="texhtml">0</span> is <span class="texhtml">0</span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Formal_construction_of_negative_integers">Formal construction of negative integers</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Integer#Construction" title="Integer">Integer § Construction</a></div>

<p>In a similar manner to <a href="Rational_number" title="Rational number">rational numbers</a>, we can extend the <a href="Natural_number" title="Natural number">natural numbers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

</math></span><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"></span> to the integers <b><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span></b> by defining integers as an <a href="Ordered_pair" title="Ordered pair">ordered pair</a> of natural numbers (<i>a</i>, <i>b</i>). We can extend addition and multiplication to these pairs with the following rules:

</p>

<p>We define an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> ~ upon these pairs with the following rule:

</p>

<p>This equivalence relation is compatible with the addition and multiplication defined above, and we may define <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span> to be the <a href="Quotient_set" class="mw-redirect" title="Quotient set">quotient set</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ^{2}/\sim }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ^{2}/\sim }</annotation>

</semantics>

</math></span><img src="./11a05ef4d29d7a24def5c703a5ec210bdaf9f941.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.348ex; height:3.176ex;" alt="{\displaystyle \mathbb {N} ^{2}/\sim }" loading="lazy"></span>, i.e. we identify two pairs (<i>a</i>, <i>b</i>) and (<i>c</i>, <i>d</i>) if they are equivalent in the above sense. Note that <b><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span></b>, equipped with these operations of addition and multiplication, is a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, and is in fact, the prototypical example of a ring.

</p><p>We can also define a <a href="Total_order" title="Total order">total order</a> on <b><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</p>

<p>This will lead to an <i>additive zero</i> of the form (<i>a</i>, <i>a</i>), an <i><a href="Additive_inverse" title="Additive inverse">additive inverse</a></i> of (<i>a</i>, <i>b</i>) of the form (<i>b</i>, <i>a</i>), a multiplicative unit of the form (<i>a</i> + 1, <i>a</i>), and a definition of <a href="Subtraction" title="Subtraction">subtraction</a>

</p>

<p>This construction is a special case of the <a href="Grothendieck_group#Explicit_constructions" title="Grothendieck group">Grothendieck construction</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Uniqueness">Uniqueness</h3></div>

<p>The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero.

</p><p>Let <i>x</i> be a number and let <i>y</i> be its additive inverse. Suppose <i>y′</i> is another additive inverse of <i>x</i>. By definition,

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x+y'=0,\quad {\text{and}}\quad x+y=0.}</annotation>

</semantics>

</math></span></span>

</p><p>And so, <i>x</i> + <i>y′</i> = <i>x</i> + <i>y</i>. Using the law of cancellation for addition, it is seen that <i>y′</i> = <i>y</i>. Thus <i>y</i> is equal to any other additive inverse of <i>x</i>. That is, <i>y</i> is the unique additive inverse of <i>x</i>.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>

</p>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Complex_number#History" title="Complex number">Complex number § History</a></div>

<p>For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false".

</p><p>In <a href="Hellenistic_Egypt" class="mw-redirect" title="Hellenistic Egypt">Hellenistic Egypt</a>, the <a href="Greek_mathematics" class="mw-redirect" title="Greek mathematics">Greek</a> mathematician <a href="Diophantus" title="Diophantus">Diophantus</a> in the 3rd century AD referred to an equation that was equivalent to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x+20=4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4x+20=4}</annotation>

</semantics>

</math></span><img src="./3a3370feaf08c7734572dd9ea9895c100953e479.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.918ex; height:2.343ex;" alt="{\displaystyle 4x+20=4}" loading="lazy"></span> (which has a negative solution) in <i><a href="Arithmetica" class="mw-redirect" title="Arithmetica">Arithmetica</a></i>, saying that the equation was absurd.<sup id="cite_ref-Needham_volume_3_p90_24-0" class="reference"><a href="#cite_note-Needham_volume_3_p90-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots, while they could take no account of others.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>

</p><p>Negative numbers appear for the first time in history in the <i><a href="Nine_Chapters_on_the_Mathematical_Art" class="mw-redirect" title="Nine Chapters on the Mathematical Art">Nine Chapters on the Mathematical Art</a></i> (九章算術, <i>Jiǔ zhāng suàn-shù</i>), which in its present form dates from the <a href="Han_dynasty" title="Han dynasty">Han period</a>, but may well contain much older material.<sup id="cite_ref-struik33_3-1" class="reference"><a href="#cite_note-struik33-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The mathematician <a href="Liu_Hui" title="Liu Hui">Liu Hui</a> (c. 3rd century) established rules for the addition and subtraction of negative numbers. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese <a href="Natural_philosophy" title="Natural philosophy">natural philosophy</a> made it easier for the Chinese to accept the idea of negative numbers.<sup id="cite_ref-Hodgkin_4-1" class="reference"><a href="#cite_note-Hodgkin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The Chinese were able to solve simultaneous equations involving negative numbers. The <i>Nine Chapters</i> used red <a href="Counting_rods" title="Counting rods">counting rods</a> to denote positive <a href="Coefficient" title="Coefficient">coefficients</a> and black rods for negative.<sup id="cite_ref-Hodgkin_4-2" class="reference"><a href="#cite_note-Hodgkin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Needham_volume_3_pp90-91_26-0" class="reference"><a href="#cite_note-Needham_volume_3_pp90-91-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values. Liu Hui writes:

</p>

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</style><blockquote class="templatequote"><p>Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative. Red counting rods are positive, black counting rods are negative.<sup id="cite_ref-Hodgkin_4-3" class="reference"><a href="#cite_note-Hodgkin-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></p></blockquote>

<p>The ancient Indian <i><a href="Bakhshali_Manuscript" class="mw-redirect" title="Bakhshali Manuscript">Bakhshali Manuscript</a></i> carried out calculations with negative numbers, using "+" as a negative sign.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries,<sup id="cite_ref-HayashiEncy_29-0" class="reference"><a href="#cite_note-HayashiEncy-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup>

</p><p>During the 7th century AD, negative numbers were used in India to represent debts. The <a href="Indian_mathematics" title="Indian mathematics">Indian mathematician</a> <a href="Brahmagupta" title="Brahmagupta">Brahmagupta</a>, in <i><a href="Brahmasphutasiddhanta" class="mw-redirect" title="Brahmasphutasiddhanta">Brahma-Sphuta-Siddhanta</a></i> (written c. AD 630), discussed the use of negative numbers to produce a general form <a href="Quadratic_formula" title="Quadratic formula">quadratic formula</a> similar to the one in use today.<sup id="cite_ref-Needham_volume_3_p90_24-1" class="reference"><a href="#cite_note-Needham_volume_3_p90-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p><p>In the 9th century, <a href="Islamic_mathematicians" class="mw-redirect" title="Islamic mathematicians">Islamic mathematicians</a> were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid.<sup id="cite_ref-Rashed_5-1" class="reference"><a href="#cite_note-Rashed-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="Al-Khwarizmi" title="Al-Khwarizmi">Al-Khwarizmi</a> in his <i><a href="The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" class="mw-redirect" title="The Compendious Book on Calculation by Completion and Balancing">Al-jabr wa'l-muqabala</a></i> (from which the word "algebra" derives) did not use negative numbers or negative coefficients.<sup id="cite_ref-Rashed_5-2" class="reference"><a href="#cite_note-Rashed-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> But within fifty years, <a href="Abu_Kamil" title="Abu Kamil">Abu Kamil</a> illustrated the rules of signs for expanding the multiplication <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\pm b)(c\pm d)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (a\pm b)(c\pm d)}</annotation>

</semantics>

</math></span><img src="./55fabcbec688d4fcf5d447872affba9c76021dfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.749ex; height:2.843ex;" alt="{\displaystyle (a\pm b)(c\pm d)}" loading="lazy"></span>,<sup id="cite_ref-Ismail_31-0" class="reference"><a href="#cite_note-Ismail-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> and <a href="Al-Karaji" title="Al-Karaji">al-Karaji</a> wrote in his <i>al-Fakhrī</i> that "negative quantities must be counted as terms".<sup id="cite_ref-Rashed_5-3" class="reference"><a href="#cite_note-Rashed-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In the 10th century, <a href="Ab%C5%AB_al-Waf%C4%81'_al-B%C5%ABzj%C4%81n%C4%AB" class="mw-redirect" title="Abū al-Wafā' al-Būzjānī">Abū al-Wafā' al-Būzjānī</a> considered debts as negative numbers in <i><a href="A_Book_on_What_Is_Necessary_from_the_Science_of_Arithmetic_for_Scribes_and_Businessmen" class="mw-redirect" title="A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen">A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen</a></i>.<sup id="cite_ref-Ismail_31-1" class="reference"><a href="#cite_note-Ismail-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup>

</p><p>By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve <a href="Polynomial_division" class="mw-redirect" title="Polynomial division">polynomial divisions</a>.<sup id="cite_ref-Rashed_5-4" class="reference"><a href="#cite_note-Rashed-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> As <a href="Al-Samaw'al" class="mw-redirect" title="Al-Samaw'al">al-Samaw'al</a> writes:

</p>

<blockquote><p>the product of a negative number—<i>al-nāqiṣ</i> (loss)—by a positive number—<i>al-zāʾid</i> (gain)—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (<i>martaba khāliyya</i>), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.<sup id="cite_ref-Rashed_5-5" class="reference"><a href="#cite_note-Rashed-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></p></blockquote>

<p>In the 12th century in India, <a href="Bh%C4%81skara_II" title="Bhāskara II">Bhāskara II</a> gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots."

</p><p><a href="Leonardo_of_Pisa" class="mw-redirect" title="Leonardo of Pisa">Fibonacci</a> allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of <i><a href="Liber_Abaci" title="Liber Abaci">Liber Abaci</a></i>, 1202) and later as losses (in <a href="Leonardo_of_Pisa" class="mw-redirect" title="Leonardo of Pisa"><i>Flos</i></a>, 1225).

</p><p>In the 15th century, <a href="Nicolas_Chuquet" title="Nicolas Chuquet">Nicolas Chuquet</a>, a Frenchman, used negative numbers as <a href="Exponentiation" title="Exponentiation">exponents</a><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> but referred to them as "absurd numbers".<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Michael_Stifel" title="Michael Stifel">Michael Stifel</a> dealt with negative numbers in his <a href="1544" title="1544">1544</a> AD <i><a href="Arithmetica_Integra" class="mw-redirect" title="Arithmetica Integra">Arithmetica Integra</a></i>, where he also called them <i>numeri absurdi</i> (absurd numbers).

</p><p>In 1545, <a href="Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>, in his <a href="Ars_Magna_(Gerolamo_Cardano)" class="mw-redirect" title="Ars Magna (Gerolamo Cardano)"><i>Ars Magna</i></a>, provided the first satisfactory treatment of negative numbers in Europe.<sup id="cite_ref-Needham_volume_3_p90_24-2" class="reference"><a href="#cite_note-Needham_volume_3_p90-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> He did not allow negative numbers in his consideration of <a href="Cubic_equation" title="Cubic equation">cubic equations</a>, so he had to treat, for example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}+ax=b}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{3}+ax=b}</annotation>

</semantics>

</math></span><img src="./6d95bcd867d92ac1fcaa1806650fb5ecf779d926.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.88ex; height:2.843ex;" alt="{\displaystyle x^{3}+ax=b}" loading="lazy"></span> separately from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}=ax+b}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{3}=ax+b}</annotation>

</semantics>

</math></span><img src="./3cd7a3e129091e122e60175aa909efd20816e7da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.88ex; height:2.843ex;" alt="{\displaystyle x^{3}=ax+b}" loading="lazy"></span> (with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b>0}</annotation>

</semantics>

</math></span><img src="./7b4689ff0639c824e92e7469fae91926382aa24f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.522ex; height:2.509ex;" alt="{\displaystyle a,b>0}" loading="lazy"></span> in both cases). In all, Cardano was driven to the study of thirteen types of cubic equations, each with all negative terms moved to the other side of the = sign to make them positive. (Cardano also dealt with <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>, but understandably liked them even less.)

</p>

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</style><div class="div-col" style="column-width: 20em;">

<ul><li><a href="Signed_zero" title="Signed zero">Signed zero</a></li>

<li><a href="Additive_inverse" title="Additive inverse">Additive inverse</a></li>

<li><a href="History_of_zero" class="mw-redirect" title="History of zero">History of zero</a></li>

<li><a href="Integers" class="mw-redirect" title="Integers">Integers</a></li>

<li><a href="Positive_and_negative_parts" title="Positive and negative parts">Positive and negative parts</a></li>

<li><a href="Rational_numbers" class="mw-redirect" title="Rational numbers">Rational numbers</a></li>

<li><a href="Real_numbers" class="mw-redirect" title="Real numbers">Real numbers</a></li>

<li><a href="Sign_function" title="Sign function">Sign function</a></li>

<li><a href="Sign_(mathematics)" title="Sign (mathematics)">Sign (mathematics)</a></li>

<li><a href="Signed_number_representations" title="Signed number representations">Signed number representations</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">"Integers are the set of whole numbers and their opposites.", Richard W. Fisher, No-Nonsense Algebra, 2nd Edition, Math Essentials, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0999443330</bdi></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">The convention that zero is neither positive nor negative is not universal. For example, in the French convention, zero is considered to be <i>both</i> positive and negative. The French words <a href="https://fr.wikipedia.org/wiki/Nombre_positif" class="extiw external" title="fr:Nombre positif">positif</a> and <a href="https://fr.wikipedia.org/wiki/Nombre_n%C3%A9gatif" class="extiw external" title="fr:Nombre négatif">négatif</a> mean the same as English "positive or zero" and "negative or zero" respectively.</span>

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<li id="cite_note-struik33-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-struik33_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-struik33_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Struik, pages 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."</span>

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<li id="cite_note-Hodgkin-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hodgkin_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hodgkin_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hodgkin_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Hodgkin_4-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFHodgkin2005" class="citation book cs1">Hodgkin, Luke (2005). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000hodg"><i>A History of Mathematics: From Mesopotamia to Modernity</i></a></span>. Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema0000hodg/page/88">88</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-152383-0</bdi>. <q>Liu is explicit on this; at the point where the <i>Nine Chapters</i> give a detailed and helpful 'Sign Rule'</q></cite></span>

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<li id="cite_note-Rashed-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rashed_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rashed_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Rashed_5-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Rashed_5-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Rashed_5-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Rashed_5-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFRashed1994" class="citation book cs1">Rashed, R. (30 June 1994). <i>The Development of Arabic Mathematics: Between Arithmetic and Algebra</i>. Springer. pp. <span class="nowrap">36–</span>37. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780792325659</bdi>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="Diophantus" title="Diophantus">Diophantus</a>, <i><a href="Arithmetica" class="mw-redirect" title="Arithmetica">Arithmetica</a></i>.</span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFKline1972" class="citation book cs1">Kline, Morris (1972). <i>Mathematical Thought from Ancient to Modern Times</i>. Oxford University Press, New York. p. 252.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFSmith2001" class="citation web cs1"><a href="Martha_K._Smith" title="Martha K. Smith">Smith, Martha K.</a> (19 February 2001). <a rel="nofollow" class="external text" href="https://web.ma.utexas.edu/users/mks/326K/Negnos.html">"History of Negative Numbers"</a>. <a href="University_of_Texas" class="mw-redirect" title="University of Texas">University of Texas</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20250227120108/https://web.ma.utexas.edu/users/mks/326K/Negnos.html">Archived</a> from the original on 27 February 2025.</cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/sport/rugby-union/50457698">"Saracens salary cap breach: Premiership champions will not contest sanctions"</a>. <i>BBC Sport</i><span class="reference-accessdate">. Retrieved <span class="nowrap">18 November</span> 2019</span>. <q>Mark McCall's side have subsequently dropped from third to bottom of the Premiership with −22 points</q></cite></span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/sport/football/50356053">"Bolton Wanderers 1−0 Milton Keynes Dons"</a>. <i>BBC Sport</i><span class="reference-accessdate">. Retrieved <span class="nowrap">30 November</span> 2019</span>. <q>But in the third minute of stoppage time, the striker turned in Luke Murphy's cross from eight yards to earn a third straight League One win for Hill's side, who started the campaign on −12 points after going into administration in May.</q></cite></span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.formula1.com/en/championship/inside-f1/glossary.html">"Glossary"</a>. Formula1.com<span class="reference-accessdate">. Retrieved <span class="nowrap">30 November</span> 2019</span>. <q>Delta time: A term used to describe the time difference between two different laps or two different cars. For example, there is usually a negative delta between a driver's best practice lap time and his best qualifying lap time because he uses a low fuel load and new tyres.</q></cite></span>

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<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120805042254/http://london2012.bbc.co.uk/athletics/event/men-long-jump/index.html">"BBC Sport - Olympic Games - London 2012 - Men's Long Jump : Athletics - Results"</a>. 5 August 2012. Archived from <a rel="nofollow" class="external text" href="http://london2012.bbc.co.uk/athletics/event/men-long-jump/index.html">the original</a> on 5 August 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2018</span>.</cite></span>

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<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://elitefeet.com/how-wind-assistance-works-in-track-field">"How Wind Assistance Works in Track & Field"</a>. <i>elitefeet.com</i>. 3 July 2008<span class="reference-accessdate">. Retrieved <span class="nowrap">18 November</span> 2019</span>. <q>Wind assistance is normally expressed in meters per second, either positive or negative. A positive measurement means that the wind is helping the runners and a negative measurement means that the runners had to work against the wind. So, for example, winds of −2.2m/s and +1.9m/s are legal, while a wind of +2.1m/s is too much assistance and considered illegal. The terms "tailwind" and "headwind" are also frequently used. A tailwind pushes the runners forward (+) while a headwind pushes the runners backwards (−)</q></cite></span>

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<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><cite id="CITEREFForbes1975" class="citation book cs1">Forbes, Robert B. (6 January 1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VxPI4SdOaBcC&q=colder+than+0+%C2%B0C+or+0+%C2%B0F&pg=PA194"><i>Contributions to the Geology of the Bering Sea Basin and Adjacent Regions: Selected Papers from the Symposium on the Geology and Geophysics of the Bering Sea Region, on the Occasion of the Inauguration of the C. T. Elvey Building, University of Alaska, June 26-28, 1970, and from the 2d International Symposium on Arctic Geology Held in San Francisco, February 1-4, 1971</i></a>. Geological Society of America. p. 194. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780813721514</bdi>.</cite></span>

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<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilks2018" class="citation book cs1">Wilks, Daniel S. (6 January 2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IJuCVtQ0ySIC&q=colder+than+0+%C2%B0C+or+0+%C2%B0F&pg=PA17"><i>Statistical Methods in the Atmospheric Sciences</i></a>. Academic Press. p. 17. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780123850225</bdi>.</cite></span>

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<li id="cite_note-CarysforthNeild2002-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-CarysforthNeild2002_16-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFCarysforthNeild2002" class="citation cs2">Carysforth, Carol; Neild, Mike (2002), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vMTmjC7fgcYC&pg=PA375"><i>Double Award</i></a>, Heinemann, p. 375, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-435-44746-5</bdi></cite></span>

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<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.bbc.com/news/business-21193525">"UK economy shrank at end of 2012"</a>. <i>BBC News</i>. 25 January 2013<span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2018</span>.</cite></span>

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<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.independent.co.uk/news/business/news/first-negative-inflation-figure-since-1960-1671736.html">"First negative inflation figure since 1960"</a></span>. <i>The Independent</i>. 21 April 2009. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20220618/https://www.independent.co.uk/news/business/news/first-negative-inflation-figure-since-1960-1671736.html">Archived</a> from the original on 18 June 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2018</span>.</cite></span>

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<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.bbc.com/news/business-27717594">"ECB imposes negative interest rate"</a>. <a href="BBC_News" title="BBC News">BBC News</a>. 5 June 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2018</span>.</cite></span>

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<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFLynn" class="citation news cs1">Lynn, Matthew. <a rel="nofollow" class="external text" href="https://www.marketwatch.com/story/think-negative-interest-rates-cant-happen-here-think-again-2015-01-21">"Think negative interest rates can't happen here? Think again"</a>. <i>MarketWatch</i><span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2018</span>.</cite></span>

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<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.bbc.com/news/business-30528404">"Swiss interest rate to turn negative"</a>. <a href="BBC_News" title="BBC News">BBC News</a>. 18 December 2014<span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2018</span>.</cite></span>

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<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFWintour2014" class="citation news cs1">Wintour, Patrick (17 June 2014). <a rel="nofollow" class="external text" href="https://www.theguardian.com/politics/2014/jun/17/ed-miliband-nick-clegg-fall-lowest-popularity-guardian-icm">"Popularity of Miliband and Clegg falls to lowest levels recorded by ICM poll"</a>. <i>The Guardian</i><span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2018</span> – via www.theguardian.com.</cite></span>

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<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite id="CITEREFGrant_P._WigginsJay_McTighe2005" class="citation book cs1">Grant P. Wiggins; Jay McTighe (2005). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780131950849/page/210"><i>Understanding by design</i></a></span>. ACSD Publications. p. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780131950849/page/210">210</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-4166-0035-3</bdi>.</cite></span>

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<li id="cite_note-Needham_volume_3_p90-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Needham_volume_3_p90_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Needham_volume_3_p90_24-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Needham_volume_3_p90_24-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFNeedhamWang1995" class="citation book cs1">Needham, Joseph; Wang, Ling (1995) [1959]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jfQ9E0u4pLAC"><i>Science and Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth</i></a> (reprint ed.). Cambridge: Cambridge University Press. p. 90. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-05801-5</bdi>.</cite></span>

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<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><cite id="CITEREFHeath1897" class="citation book cs1">Heath, Thomas L. (1897). <a rel="nofollow" class="external text" href="https://archive.org/details/worksofarchimede029517mbp/page/n123/mode/2up"><i>The works of Archimedes</i></a>. Cambridge University Press. pp. cxxiii.</cite></span>

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<li id="cite_note-Needham_volume_3_pp90-91-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-Needham_volume_3_pp90-91_26-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFNeedhamWang1995" class="citation book cs1">Needham, Joseph; Wang, Ling (1995) [1959]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jfQ9E0u4pLAC"><i>Science and Civilisation in China: Volume 3; Mathematics and the Sciences of the Heavens and the Earth</i></a> (reprint ed.). Cambridge: Cambridge University Press. pp. <span class="nowrap">90–</span>91. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-05801-5</bdi>.</cite></span>

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<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Teresi, Dick. (2002). <i>Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas</i>. New York: Simon & Schuster. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-684-83718-8</bdi>. Page 65.</span>

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<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><cite id="CITEREFPearce2002" class="citation web cs1">Pearce, Ian (May 2002). <a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript.html">"The Bakhshali manuscript"</a>. The MacTutor History of Mathematics archive<span class="reference-accessdate">. Retrieved <span class="nowrap">24 July</span> 2007</span>.</cite></span>

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<li id="cite_note-HayashiEncy-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-HayashiEncy_29-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFHayashi2008" class="citation cs2">Hayashi, Takao (2008), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kt9DIY1g9HYC&pg=RA1-PA1">"Bakhshālī Manuscript"</a>, in <a href="Helaine_Selin" title="Helaine Selin">Helaine Selin</a> (ed.), <i>Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures</i>, vol. 1, Springer, p. B2, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781402045592</bdi></cite></span>

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<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Teresi, Dick. (2002). <i>Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas</i>. New York: Simon & Schuster. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-684-83718-8</bdi>. Page 65–66.</span>

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<li id="cite_note-Ismail-31"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ismail_31-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ismail_31-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBin_Ismail2008" class="citation cs2">Bin Ismail, Mat Rofa (2008), "Algebra in Islamic Mathematics", in <a href="Helaine_Selin" title="Helaine Selin">Helaine Selin</a> (ed.), <i>Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures</i>, vol. 1 (2nd ed.), Springer, p. 115, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781402045592</bdi></cite></span>

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<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><cite id="CITEREFFleggHayMoss1985" class="citation cs2">Flegg, Graham; Hay, C.; Moss, B. (1985), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_rO6lVwdbjcC&pg=PA354"><i>Nicolas Chuquet, Renaissance Mathematician: a study with extensive translations of Chuquet's mathematical manuscript completed in 1484</i></a>, D. Reidel Publishing Co., p. 354, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789027718723</bdi></cite>.</span>

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<li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><cite id="CITEREFJohnson1999" class="citation cs2">Johnson, Art (1999), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=STKX4qadFTkC&pg=PA56"><i>Famous Problems and Their Mathematicians</i></a>, Greenwood Publishing Group, p. 56, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781563084461</bdi></cite>.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Bibliography">Bibliography</h3></div>

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<ul><li>Bourbaki, Nicolas (1998). <i>Elements of the History of Mathematics</i>. Berlin, Heidelberg, and New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-64767-8</bdi>.</li>

<li>Struik, Dirk J. (1987). <i>A Concise History of Mathematics</i>. New York: Dover Publications.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Negative_number" class="extiw external" title="q:Special:Search/Negative number">Negative number</a></b></i>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maseres.html">Maseres' biographical information</a></li>

<li><a rel="nofollow" class="external text" href="https://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml">BBC Radio 4 series <i>In Our Time</i>, on "Negative Numbers", 9 March 2006</a></li>

<li><a rel="nofollow" class="external text" href="http://www.free-ed.net/sweethaven/Math/arithmetic/SignedValues01_EE.asp">Endless Examples & Exercises: <i>Operations with Signed Integers</i></a></li>

<li><a rel="nofollow" class="external text" href="http://mathforum.org/dr.math/faq/faq.negxneg.html">Math Forum: Ask Dr. Math FAQ: Negative Times a Negative</a></li></ul>

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</style><div id="Number_systems351" style="font-size:114%;margin:0 4em"><a href="Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Natural_number" title="Natural number">Natural numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

<li><a href="Integer" title="Integer">Integers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

<li><a href="Rational_number" title="Rational number">Rational numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math></span><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"></span>)</li>

<li><a href="Constructible_number" title="Constructible number">Constructible numbers</a></li>

<li><a href="Algebraic_number" title="Algebraic number">Algebraic numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation>

</semantics>

</math></span><img src="./3fb423c16a5f403edbaf66438b75e7a36e725af6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }" loading="lazy"></span>)</li>

<li><a href="Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li>

<li><a href="Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

</semantics>

</math></span><img src="./10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"></span>)</li>

<li><a href="Computable_number" title="Computable number">Computable numbers</a></li>

<li><a href="Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li>

<li><a href="Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li>

<li><a href="Gaussian_integer" title="Gaussian integer">Gaussian integers</a>

<ul><li><a href="Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li>

<li><a href="Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Composition_algebra" title="Composition algebra">Composition algebras</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Division_algebra" title="Division algebra">Division algebras</a>: <a href="Real_number" title="Real number">Real numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>)</li>

<li><a href="Complex_number" title="Complex number">Complex numbers</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math></span><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span>)</li>

<li><a href="Quaternion" title="Quaternion">Quaternions</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation>

</semantics>

</math></span><img src="./e050965453c42bcc6bd544546703c836bdafeac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }" loading="lazy"></span>)</li>

<li><a href="Octonion" title="Octonion">Octonions</a> (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {O} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation>

</semantics>

</math></span><img src="./c1ed2664a4fe515e6fbed25a7193ce663b82920c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }" loading="lazy"></span>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br>types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

<li><a href="Split-complex_number" title="Split-complex number">Split-complex numbers</a></li>

<li><a href="Split-quaternion" title="Split-quaternion">Split-quaternions</a></li>

<li><a href="Split-octonion" title="Split-octonion">Split-octonions</a><br> Over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

<li><a href="Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li>

<li><a href="Biquaternion" title="Biquaternion">Biquaternions</a></li>

<li><a href="Bioctonion" title="Bioctonion">Bioctonions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dual_number" title="Dual number">Dual numbers</a></li>

<li><a href="Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li>

<li><a href="Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li>

<li><a href="Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li>

<li><a href="Sedenion" title="Sedenion">Sedenions</a>  (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation>

</semantics>

</math></span><img src="./9f9d5874c5d7f68eba1cec9da9ccbe53903303bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }" loading="lazy"></span>)</li>

<li><a href="Trigintaduonion" title="Trigintaduonion">Trigintaduonions</a>  (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation>

</semantics>

</math></span><img src="./9c039979935c00b3b216cbb065999207872677f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {T} }" loading="lazy"></span>)</li>

<li><a href="Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li>

<li><a href="Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li>

<li><a href="Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a>

<ul><li><a href="Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li>

<li><a href="Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li>

<li><a href="Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Infinity" title="Infinity">Infinities</a> and <a href="Infinitesimal" title="Infinitesimal">infinitesimals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cardinal_number" title="Cardinal number">Cardinal numbers</a></li>

<li><a href="Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li>

<li><a href="Extended_real_number_line" title="Extended real number line">Extended real numbers</a>

<ul><li><a href="Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li>

<li><a href="Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li>

<li><a href="Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li>

<li><a href="Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal numbers</a></li>

<li><a href="Supernatural_number" title="Supernatural number">Supernatural numbers</a></li>

<li><a href="Surreal_number" title="Surreal number">Surreal numbers</a></li>

<li><a href="Superreal_number" title="Superreal number">Superreal numbers</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Irrational_number" title="Irrational number">Irrational numbers</a></li>

<li><a href="Fuzzy_number" title="Fuzzy number">Fuzzy numbers</a></li>

<li><a href="Transcendental_number" title="Transcendental number">Transcendental numbers</a></li>

<li><a href="P-adic_number" title="P-adic number"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li>

<li><a href="Profinite_integer" title="Profinite integer">Profinite integers</a></li>

<li><a href="Normal_number" title="Normal number">Normal numbers</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="Number#Main_classification" title="Number">Classification</a></li>

</div></td></tr></tbody></table></div>

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