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Imaginary unit

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</style><div role="note" class="hatnote navigation-not-searchable">"i (number)" redirects here. For internet numbers, see <a href="I-number" title="I-number">i-number</a>.</div>

The imaginary unit or unit imaginary number (i) is a <a href="Mathematical_constant" title="Mathematical constant">mathematical constant</a> that is a solution to the <a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a> x2 + 1 = 0. Although there is no <a href="Real_number" title="Real number">real number</a> with this property, i can be used to extend the real numbers to what are called <a href="Complex_number" title="Complex number">complex numbers</a>, using <a href="Addition" title="Addition">addition</a> and <a href="Multiplication" title="Multiplication">multiplication</a>. A simple example of the use of i in a complex number is 2 + 3i.

<a href="Imaginary_number" title="Imaginary number">Imaginary numbers</a> are an important mathematical concept; they extend the real number system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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</math><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"> to the complex number system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ,}">

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

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</math><img src="./c6ff6a3dc2982018ff20f1d2c927afc74a217be6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} ,}" loading="lazy"> in which at least one <a href="Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> for every nonconstant <a href="Polynomial" title="Polynomial">polynomial</a> exists (see <a href="Algebraic_closure" title="Algebraic closure">Algebraic closure</a> and <a href="Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">Fundamental theorem of algebra</a>). Here, the term imaginary is used because there is no <a href="Real_number" title="Real number">real number</a> having a negative <a href="Square_(algebra)" title="Square (algebra)">square</a>.

There are two complex square roots of −1: i and −i, just as there are two complex <a href="Square_root" title="Square root">square roots</a> of every real number other than <a href="Zero" class="mw-redirect" title="Zero">zero</a> (which has one <a href="Multiple_root" class="mw-redirect" title="Multiple root">double square root</a>).

In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in <a href="Electrical_engineering" title="Electrical engineering">electrical engineering</a> and <a href="Control_systems_engineering" class="mw-redirect" title="Control systems engineering">control systems engineering</a>, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote <a href="Electric_current" title="Electric current">electric current</a>.<a href="#cite_note-1">[1]</a>

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

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

Square roots of negative numbers are called imaginary because in <a href="History_of_mathematics#Renaissance" title="History of mathematics">early-modern mathematics</a>, only what are now called <a href="Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even <a href="Negative_numbers" class="mw-redirect" title="Negative numbers">negative numbers</a> were treated with skepticism – so the square root of a negative number was previously considered undefined or nonsensical. The name imaginary is generally credited to <a href="Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, and <a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a> used the term as early as 1670.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> The i notation was introduced by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>.<a href="#cite_note-Boyer-4">[4]</a>

A unit is an undivided whole, and unity or the unit number is the number <a href="1" title="1">one</a> (1).

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

<th>The powers of i are cyclic:

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<annotation encoding="application/x-tex">{\displaystyle \ i^{-4}={\phantom {-}}1{\phantom {i}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ i^{-3}={\phantom {-}}i{\phantom {1}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ i^{-2}=-1{\phantom {i}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ i^{-1}=-i{\phantom {1}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}}</annotation>

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

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The imaginary unit i is defined solely by the property that its square is −1:

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<annotation encoding="application/x-tex">{\displaystyle i^{2}=-1.}</annotation>

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With i defined this way, it follows directly from <a href="Algebra" title="Algebra">algebra</a> that i and −i are both square roots of −1.

Although the construction is called imaginary, and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by treating i as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of i2 with −1). Higher integral powers of i are thus

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}i^{3}&=i^{2}i&&=(-1)i&&=-i,\\[3mu]i^{4}&=i^{3}i&&=\;\!(-i)i&&=\ \,1,\\[3mu]i^{5}&=i^{4}i&&=\ \,(1)i&&=\ \ i,\end{alignedat}}}</annotation>

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and so on, cycling through the four values 1, i, −1, and −i. As with any non-zero real number, i0 = 1.

As a complex number, i can be represented in <a href="Rectangular_coordinate_system" class="mw-redirect" title="Rectangular coordinate system">rectangular form</a> as 0 + 1i, with a zero real component and a unit imaginary component. In <a href="Polar_form" class="mw-redirect" title="Polar form">polar form</a>, i can be represented as 1 × eπi /2 (or just eπi /2), with an <a href="Absolute_value" title="Absolute value">absolute value</a> (or magnitude) of 1 and an <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> (or angle) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\pi }{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\pi }{2}}}</annotation>

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</math><img src="./b4e31a202557dfbf326b44ebcc914ba3ab08fff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.778ex; height:3.176ex;" alt="{\displaystyle {\tfrac {\pi }{2}}}" loading="lazy"> <a href="Radian" title="Radian">radians</a>. (Adding any integer multiple of 2π to this angle works as well.) In the <a href="Complex_plane" title="Complex plane">complex plane</a>, which is a special interpretation of a <a href="Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a>, i is the point located one unit from the origin along the <a href="Imaginary_axis" class="mw-redirect" title="Imaginary axis">imaginary axis</a> (which is <a href="Perpendicular" title="Perpendicular">perpendicular</a> to the <a href="Real_axis" class="mw-redirect" title="Real axis">real axis</a>).

Being a <a href="Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomial</a> with no <a href="Multiple_root" class="mw-redirect" title="Multiple root">multiple root</a>, the defining equation x2 = −1 has two distinct solutions, which are equally valid and which happen to be <a href="Additive_inverse" title="Additive inverse">additive</a> and <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverses</a> of each other. Although the two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled +i (or simply i) and the other is labelled −i, though it is inherently ambiguous which is which.

The only differences between +i and −i arise from this labelling. For example, by convention +i is said to have an <a href="Argument_(complex_analysis)" title="Argument (complex analysis)">argument</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +{\tfrac {\pi }{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {\pi }{2}},}</annotation>

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</math><img src="./9646bd065f3aecf19e0e4f6bb9a7a33a187da62b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.233ex; height:3.176ex;" alt="{\displaystyle -{\tfrac {\pi }{2}},}" loading="lazy"> related to the convention of labelling orientations in the <a href="Cartesian_plane" class="mw-redirect" title="Cartesian plane">Cartesian plane</a> relative to the positive x-axis with positive angles turning <a href="Anticlockwise" class="mw-redirect" title="Anticlockwise">anticlockwise</a> in the direction of the positive y-axis. Also, despite the signs written with them, neither +i nor −i is inherently positive or negative in the sense that real numbers are.<a href="#cite_note-5">[5]</a>

A more formal expression of this indistinguishability of +i and −i is that, although the complex <a href="Field_(algebra)" class="mw-redirect" title="Field (algebra)">field</a> is <a href="Unique_(mathematics)" class="mw-redirect" title="Unique (mathematics)">unique</a> (as an extension of the real numbers) <a href="Up_to" title="Up to">up to</a> <a href="Isomorphism" title="Isomorphism">isomorphism</a>, it is not unique up to a unique isomorphism. That is, there are two <a href="Automorphism" title="Automorphism">field automorphisms</a> of the complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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

Using the concepts of <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> and <a href="Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>, complex numbers can be represented in linear algebra. The real unit 1 and imaginary unit i can be represented by any pair of matrices I and J satisfying I2 = I, IJ = JI = J, and J2 = −I. Then a complex number a + bi can be represented by the matrix aI + bJ, and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.

The most common choice is to represent 1 and i by the 2 × 2 <a href="Identity_matrix" title="Identity matrix">identity matrix</a> I and the matrix J,

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<annotation encoding="application/x-tex">{\displaystyle I={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.}</annotation>

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Then an arbitrary complex number a + bi can be represented by:

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<annotation encoding="application/x-tex">{\displaystyle aI+bJ={\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.}</annotation>

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More generally, any real-valued 2 × 2 matrix with a <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of zero and a <a href="Determinant" title="Determinant">determinant</a> of one squares to −I, so could be chosen for J. Larger matrices could also be used; for example, 1 could be represented by the 4 × 4 identity matrix and i could be represented by any of the <a href="Dirac_matrices" class="mw-redirect" title="Dirac matrices">Dirac matrices</a> for spatial dimensions.

<a href="Polynomial" title="Polynomial">Polynomials</a> (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose <a href="Coefficient" title="Coefficient">coefficients</a> are real numbers form a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [x],}">

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

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</math><img src="./8a990f60289eb6d302674d3d54b8e52f1c9955ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.948ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [x],}" loading="lazy"> an algebraic structure with addition and multiplication and sharing many properties with the ring of <a href="Integer" title="Integer">integers</a>.

The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1}">

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<annotation encoding="application/x-tex">{\displaystyle x^{2}+1}</annotation>

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</math><img src="./92a3a8d23f9f8123651e496dcf8490990c65cf9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}+1}" loading="lazy"> has no real-number <a href="Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">roots</a>, but the set of all real-coefficient polynomials divisible by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+1}</annotation>

</semantics>

</math><img src="./92a3a8d23f9f8123651e496dcf8490990c65cf9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}+1}" loading="lazy"> forms an <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a>, and so there is a <a href="Polynomial_ring#Quotient_ring" title="Polynomial ring">quotient ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .}</annotation>

</semantics>

</math><img src="./c90eaee192547028462852200009bcb20cf10261.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.307ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} [x]/\langle x^{2}+1\rangle .}" loading="lazy"> This quotient ring is <a href="Isomorphism" title="Isomorphism">isomorphic</a> to the complex numbers, and the variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> expresses the imaginary unit.

<div class="mw-heading mw-heading3"><h3 id="Graphic_representation">Graphic representation</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Complex_plane" title="Complex plane">Complex plane</a></div>

The complex numbers can be represented graphically by drawing the real <a href="Number_line" title="Number line">number line</a> as the horizontal axis and the imaginary numbers as the vertical axis of a <a href="Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian plane</a> called the <a href="Complex_plane" title="Complex plane">complex plane</a>. In this representation, the numbers 1 and i are at the same distance from 0, with a right angle between them. Addition by a complex number corresponds to <a href="Translation_(geometry)" title="Translation (geometry)">translation</a> in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every <a href="Similarity_(geometry)" title="Similarity (geometry)">similarity</a> transformation of the plane can be represented by a complex-linear function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\mapsto az+b.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\mapsto az+b.}</annotation>

</semantics>

</math><img src="./37b1c613f6185677fe5204bf6e3925bb93f589ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.505ex; height:2.343ex;" alt="{\displaystyle z\mapsto az+b.}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Geometric_algebra">Geometric algebra</h3></div>

In the <a href="Geometric_algebra" title="Geometric algebra">geometric algebra</a> of the <a href="Euclidean_plane" title="Euclidean plane">Euclidean plane</a>, the geometric product or quotient of two arbitrary <a href="Euclidean_vector" title="Euclidean vector">vectors</a> is a sum of a scalar (real number) part and a <a href="Bivector" title="Bivector">bivector</a> part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector is a negative scalar.

The quotient of a vector with itself is the scalar 1 = u/u, and when multiplied by any vector leaves it unchanged (the <a href="Identity_function" title="Identity function">identity transformation</a>). The quotient of any two perpendicular vectors of the same magnitude, J = u/v, which when multiplied rotates the divisor a quarter turn into the dividend, Jv = u, is a unit bivector which squares to −1, and can thus be taken as a representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums is <a href="Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects.<a href="#cite_note-6">[6]</a>

More generally, in the geometric algebra of any higher-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, a unit bivector of any arbitrary planar orientation squares to −1, so can be taken to represent the imaginary unit i.

<div class="mw-heading mw-heading2"><h2 id="Proper_use">Proper use</h2></div>

The imaginary unit was historically written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {-1}},}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\sqrt {-1}},}</annotation>

</semantics>

</math><img src="./bec99ead8ec9d1e87896212e7d9a95b1f8371988.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.553ex; height:2.843ex;" alt="{\textstyle {\sqrt {-1}},}" loading="lazy"> and still is in some modern works. However, great care needs to be taken when manipulating formulas involving <a href="Nth_root" title="Nth root">radicals</a>. The radical sign notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {x}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\sqrt {x}}}</annotation>

</semantics>

</math><img src="./02f01419d50f8331ed8f948d3b0dce7d5bd75950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.266ex; height:2.843ex;" alt="{\textstyle {\sqrt {x}}}" loading="lazy"> is reserved either for the principal (positive) square root of a positive real number or for the <a href="Square_root#Principal_square_root_of_a_complex_number" title="Square root">principal square root of a complex number</a>. Attempting to apply the calculation rules of square roots of positive real numbers to manipulate square roots of complex numbers can produce false results:<a href="#cite_note-7">[7]</a>

<msqrt>

</msqrt>

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<msqrt>

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<mover>

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</mover>

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<msqrt>

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<msqrt>

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<mtext>(incorrect).</mtext>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}\mathrel {\stackrel {\mathrm {fallacy} }{=}} {\textstyle {\sqrt {(-1)\cdot (-1)}}}={\sqrt {1}}=1\qquad {\text{(incorrect).}}}</annotation>

</semantics>

</math>

Generally, the calculation rules

<msqrt>

</mphantom>

</mpadded>

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</msqrt>

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<msqrt>

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<msqrt>

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<annotation encoding="application/x-tex">{\textstyle {\sqrt {x{\vphantom {ty}}}}\cdot \!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x\cdot y{\vphantom {ty}}}}}</annotation>

</semantics>

</math><img src="./0e92426bb7eaa8da544ca56ed9a2246df22973e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -1.995ex; width:18.843ex; height:2.843ex;" alt="{\textstyle {\sqrt {x{\vphantom {ty}}}}\cdot \!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x\cdot y{\vphantom {ty}}}}}" loading="lazy">

and

<msqrt>

</mphantom>

</mpadded>

</mrow>

</msqrt>

</mrow>

</mrow>

<msqrt>

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<msqrt>

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<annotation encoding="application/x-tex">{\textstyle {\sqrt {x{\vphantom {ty}}}}{\big /}\!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x/y}}}</annotation>

</semantics>

</math><img src="./803d9bf9e7ba2bb45d700a729a33ec4882c51b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.383ex; height:3.343ex;" alt="{\textstyle {\sqrt {x{\vphantom {ty}}}}{\big /}\!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x/y}}}" loading="lazy">

are guaranteed to be valid only when x and y are both positive real numbers.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a>

When x or y is real but negative, these problems can be avoided by writing and manipulating expressions like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle i{\sqrt {7}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle i{\sqrt {7}}}</annotation>

</semantics>

</math><img src="./fa5508cd971f570f2b4357d77a7ed67c82175a57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.901ex; height:3.009ex;" alt="{\textstyle i{\sqrt {7}}}" loading="lazy">, rather than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\sqrt {-7}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\sqrt {-7}}}</annotation>

</semantics>

</math><img src="./f4b6c234373104826f5748b928578b8cb35ce3a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.906ex; height:3.009ex;" alt="{\textstyle {\sqrt {-7}}}" loading="lazy">. For a more thorough discussion, see the articles <a href="Square_root" title="Square root">Square root</a> and <a href="Branch_point" title="Branch point">Branch point</a>.

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

As a complex number, the imaginary unit follows all of the rules of <a href="Complex_number#Relations_and_operations" title="Complex number">complex arithmetic</a>.

<div class="mw-heading mw-heading3"><h3 id="Imaginary_integers_and_imaginary_numbers">Imaginary integers and imaginary numbers</h3></div>

When the imaginary unit is repeatedly added or subtracted, the result is some <a href="Integer" title="Integer">integer</a> times the imaginary unit, an imaginary integer; any such numbers can be added and the result is also an imaginary integer:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ai+bi=(a+b)i.}</annotation>

</semantics>

</math>

Thus, the imaginary unit is the generator of a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> under addition, specifically an infinite <a href="Cyclic_group" title="Cyclic group">cyclic group</a>.

The imaginary unit can also be multiplied by any arbitrary <a href="Real_number" title="Real number">real number</a> to form an <a href="Imaginary_number" title="Imaginary number">imaginary number</a>. These numbers can be pictured on a <a href="Number_line" title="Number line">number line</a>, the imaginary axis, which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally.

<div class="mw-heading mw-heading3"><h3 id="Gaussian_integers">Gaussian integers</h3></div>

Integer sums of the real unit 1 and the imaginary unit i form a <a href="Square_lattice" title="Square lattice">square lattice</a> in the complex plane called the <a href="Gaussian_integers" class="mw-redirect" title="Gaussian integers">Gaussian integers</a>. The sum, difference, or product of Gaussian integers is also a Gaussian integer:

<mtr>

<mtd>

</mtd>

<mtd>

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<mtr>

<mtd>

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<mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a+bi)+(c+di)&=(a+c)+(b+d)i,\\[5mu](a+bi)(c+di)&=(ac-bd)+(ad+bc)i.\end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Quarter-turn_rotation">Quarter-turn rotation</h3></div>

When multiplied by the imaginary unit i, any arbitrary complex number in the complex plane is rotated by a quarter turn (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\pi }">

<mfrac>

</mfrac>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\pi }</annotation>

</semantics>

</math><img src="./cfe27b6e44239efb1973b09f9eae178854860d5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.99ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\pi }" loading="lazy"> radians or 90°) <a href="Anticlockwise" class="mw-redirect" title="Anticlockwise">anticlockwise</a>. When multiplied by −i, any arbitrary complex number is rotated by a quarter turn clockwise. In polar form:

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle i\,re^{\varphi i}=re^{(\varphi +\pi /2)i},\quad -i\,re^{\varphi i}=re^{(\varphi -\pi /2)i}.}</annotation>

</semantics>

</math>

In rectangular form,

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i(a+bi)=-b+ai,\quad -i(a+bi)=b-ai.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Integer_powers">Integer powers</h3></div>

The powers of i repeat in a cycle expressible with the following pattern, where n is any integer:

<msup>

</mrow>

</msup>

<msup>

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</msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle i^{4n}=1,\quad i^{4n+1}=i,\quad i^{4n+2}=-1,\quad i^{4n+3}=-i.}</annotation>

</semantics>

</math>

Thus, under multiplication, i is a generator of a <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of order 4, a discrete subgroup of the continuous <a href="Circle_group" title="Circle group">circle group</a> of the unit complex numbers under multiplication.

Written as a special case of <a href="Euler's_formula" title="Euler's formula">Euler's formula</a> for an integer n,

<msup>

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</msup>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle i^{n}={\exp }{\bigl (}{\tfrac {1}{2}}\pi i{\bigr )}^{n}={\exp }{\bigl (}{\tfrac {1}{2}}n\pi i{\bigr )}={\cos }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}+{i\sin }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}.}</annotation>

</semantics>

</math>

With a careful choice of <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cuts</a> and <a href="Principal_value" title="Principal value">principal values</a>, this last equation can also apply to arbitrary complex values of n, including cases like n = i.

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

Just like all nonzero complex numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle i=e^{\pi i/2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle i=e^{\pi i/2}}</annotation>

</semantics>

</math><img src="./bc02f756f6e0da6e202a3631859487652fb48fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.37ex; height:2.676ex;" alt="{\textstyle i=e^{\pi i/2}}" loading="lazy"> has two distinct <a href="Square_root" title="Square root">square roots</a> which are <a href="Additive_inverse" title="Additive inverse">additive inverses</a>. In polar form, they are

<mtr>

<mtd>

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<mtd>

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<msup>

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<mtd>

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<mtr>

<mtd>

<msqrt>

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<mtd>

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<mfrac>

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<mtd>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}{\sqrt {i}}&={\exp }{\bigl (}{\tfrac {1}{2}}{\pi i}{\bigr )}^{1/2}&&{}={\exp }{\bigl (}{\tfrac {1}{4}}\pi i{\bigr )},\\-{\sqrt {i}}&={\exp }{\bigl (}{\tfrac {1}{4}}{\pi i}-\pi i{\bigr )}&&{}={\exp }{\bigl (}{-{\tfrac {3}{4}}\pi i}{\bigr )}.\end{alignedat}}}</annotation>

</semantics>

</math>

In rectangular form, they are<a href="#cite_note-12">[a]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{3}{\sqrt {i}}&={\frac {1+i}{\sqrt {2}}}&&{}={\phantom {-}}{\tfrac {\sqrt {2}}{2}}+{\tfrac {\sqrt {2}}{2}}i,\\[5mu]-{\sqrt {i}}&=-{\frac {1+i}{\sqrt {2}}}&&{}=-{\tfrac {\sqrt {2}}{2}}-{\tfrac {\sqrt {2}}{2}}i.\end{alignedat}}}</annotation>

</semantics>

</math>

Squaring either expression yields

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \left(\pm {\frac {1+i}{\sqrt {2}}}\right)^{2}={\frac {1+2i-1}{2}}={\frac {2i}{2}}=i.}</annotation>

</semantics>

</math>

The three <a href="Cube_root" title="Cube root">cube roots</a> of i are<a href="#cite_note-13">[12]</a>

<mroot>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{i}}={\exp }{\bigl (}{\tfrac {1}{6}}\pi i{\bigr )}={\tfrac {\sqrt {3}}{2}}+{\tfrac {1}{2}}i,\quad {\exp }{\bigl (}{\tfrac {5}{6}}\pi i{\bigr )}=-{\tfrac {\sqrt {3}}{2}}+{\tfrac {1}{2}}i,\quad {\exp }{\bigl (}{-{\tfrac {1}{2}}\pi i}{\bigr )}=-i.}</annotation>

</semantics>

</math>

For a general positive integer n, the <a href="Nth_root" title="Nth root">n-th roots</a> of i are, for k = 0, 1, ..., n − 1,

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<annotation encoding="application/x-tex">{\displaystyle \exp \left(2\pi i{\frac {k+{\frac {1}{4}}}{n}}\right)=\cos \left({\frac {4k+1}{2n}}\pi \right)+i\sin \left({\frac {4k+1}{2n}}\pi \right).}</annotation>

</semantics>

</math>

The value associated with k = 0 is the <a href="Principal_value" title="Principal value">principal</a> n-th root of i. The set of roots equals the corresponding set of <a href="Root_of_unity" title="Root of unity">roots of unity</a> rotated by the principal n-th root of i. These are the vertices of a <a href="Regular_polygon" title="Regular polygon">regular polygon</a> inscribed within the complex <a href="Unit_circle" title="Unit circle">unit circle</a>.

<div class="mw-heading mw-heading3"><h3 id="Exponential_and_logarithm">Exponential and logarithm</h3></div>

The <a href="Complex_exponential" class="mw-redirect" title="Complex exponential">complex exponential</a> function relates complex addition in the domain to <a href="Complex_multiplication" title="Complex multiplication">complex multiplication</a> in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with 1 representing multiplication by e, while imaginary values in the domain represent rotation in the codomain (multiplication by a unit complex number) with i representing a rotation by 1 radian. The complex exponential is thus a <a href="Periodic_function" title="Periodic function">periodic function</a> in the imaginary direction, with period 2πi and image 1 at points 2kπi for all integers k, a real multiple of the lattice of imaginary integers.

The complex exponential can be broken into <a href="Even_and_odd_functions" title="Even and odd functions">even and odd</a> components, the <a href="Hyperbolic_functions" title="Hyperbolic functions">hyperbolic functions</a> cosh and sinh or the <a href="Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> cos and sin:

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \exp z=\cosh z+\sinh z=\cos(-iz)+i\sin(-iz)}</annotation>

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</math>

<a href="Euler's_formula" title="Euler's formula">Euler's formula</a> decomposes the exponential of an imaginary number representing a rotation:

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<annotation encoding="application/x-tex">{\displaystyle \exp i\varphi =\cos \varphi +i\sin \varphi .}</annotation>

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</math>

This fact can be used to demonstrate, among other things, the apparently counterintuitive result that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{i}}">

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<annotation encoding="application/x-tex">{\displaystyle i^{i}}</annotation>

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</math><img src="./7d4035ae18d005ce695d4fc30cb23c0771ab4cd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.602ex; height:2.676ex;" alt="{\displaystyle i^{i}}" loading="lazy"> is a real number.<a href="#cite_note-14">[13]</a>

The quotient coth z = cosh z / sinh z, with appropriate scaling, can be represented as an infinite <a href="Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction decomposition</a> as the sum of <a href="Reciprocal_function" class="mw-redirect" title="Reciprocal function">reciprocal functions</a> translated by imaginary integers:<a href="#cite_note-15">[14]</a>

</mrow>

</munder>

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</munderover>

<mfrac>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \pi \coth \pi z=\lim _{n\to \infty }\sum _{k=-n}^{n}{\frac {1}{z+ki}}.}</annotation>

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</math>

Other functions based on the complex exponential are well-defined with imaginary inputs. For example, a number raised to the ni power is:

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<annotation encoding="application/x-tex">{\displaystyle x^{ni}=\cos(n\ln x)+i\sin(n\ln x).}</annotation>

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Because the exponential is periodic, its inverse the <a href="Complex_logarithm" title="Complex logarithm">complex logarithm</a> is a <a href="Multi-valued_function" class="mw-redirect" title="Multi-valued function">multi-valued function</a>, with each complex number in the domain corresponding to multiple values in the codomain, separated from each-other by any integer multiple of 2πi. One way of obtaining a single-valued function is to treat the codomain as a <a href="Cylinder" title="Cylinder">cylinder</a>, with complex values separated by any integer multiple of 2πi treated as the same value; another is to take the domain to be a <a href="Riemann_surface" title="Riemann surface">Riemann surface</a> consisting of multiple copies of the complex plane stitched together along the negative real axis as a <a href="Branch_cut" class="mw-redirect" title="Branch cut">branch cut</a>, with each branch in the domain corresponding to one infinite strip in the codomain.<a href="#cite_note-16">[15]</a> Functions depending on the complex logarithm therefore depend on careful choice of branch to define and evaluate clearly.

For example, if one chooses any branch where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln i={\tfrac {1}{2}}\pi i}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \ln i={\tfrac {1}{2}}\pi i}</annotation>

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</math><img src="./48d4a4afe147b7f42f1d029b2610bbf70ddc0743.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.02ex; height:3.509ex;" alt="{\displaystyle \ln i={\tfrac {1}{2}}\pi i}" loading="lazy"> then when x is a positive real number,

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<annotation encoding="application/x-tex">{\displaystyle \log _{i}x=-{\frac {2i\ln x}{\pi }}.}</annotation>

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

The <a href="Factorial" title="Factorial">factorial</a> of the imaginary unit i is most often given in terms of the <a href="Gamma_function" title="Gamma function">gamma function</a> evaluated at 1 + i:<a href="#cite_note-17">[16]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i!=\Gamma (1+i)=i\Gamma (i)\approx 0.4980-0.1549\,i.}</annotation>

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The magnitude and argument of this number are:<a href="#cite_note-18">[17]</a>

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<annotation encoding="application/x-tex">{\displaystyle |\Gamma (1+i)|={\sqrt {\frac {\pi }{\sinh \pi }}}\approx 0.5216,\quad \arg {\Gamma (1+i)}\approx -0.3016.}</annotation>

</semantics>

</math>

<ul><li><a href="Hyperbolic_unit" class="mw-redirect" title="Hyperbolic unit">Hyperbolic unit</a></li>

<li><a href="Right_versor" class="mw-redirect" title="Right versor">Right versor</a> in quaternions</li></ul>

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

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</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-12"><a href="#cite_ref-12">^</a> To find such a number, one can solve the equation (x + iy)2 = i where x and y are real parameters to be determined, or equivalently x2 + 2ixy − y2 = i. Because the real and imaginary parts are always separate, we regroup the terms, x2 − y2 + 2ixy = 0 + i. By <a href="Equating_coefficients" title="Equating coefficients">equating coefficients</a>, separating the real part and imaginary part, we have a system of two equations:

<mtr>

<mtd>

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</msup>

<msup>

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<mtr>

<mtd>

</mtd>

<mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x^{2}-y^{2}&=0\\[3mu]2xy&=1.\end{aligned}}}</annotation>

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Substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y={\tfrac {1}{2}}x^{-1}}">

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<msup>

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<annotation encoding="application/x-tex">{\textstyle y={\tfrac {1}{2}}x^{-1}}</annotation>

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</math><img src="./070d9c82c27e27e4357432bfee89047769045954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.575ex; height:3.509ex;" alt="{\textstyle y={\tfrac {1}{2}}x^{-1}}" loading="lazy"> into the first equation, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x^{2}-{\tfrac {1}{4}}x^{-2}=0}">

<msup>

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</msup>

<mfrac>

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<msup>

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<annotation encoding="application/x-tex">{\textstyle x^{2}-{\tfrac {1}{4}}x^{-2}=0}</annotation>

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</math><img src="./a94279be23653ece74fe96263acbe8569f616acc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.806ex; height:3.509ex;" alt="{\textstyle x^{2}-{\tfrac {1}{4}}x^{-2}=0}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \implies 4x^{4}=1.}">

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<annotation encoding="application/x-tex">{\textstyle \implies 4x^{4}=1.}</annotation>

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</math><img src="./04ede6a90f3f6c34929abf3e2f608077cc58b8c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.195ex; height:2.509ex;" alt="{\textstyle \implies 4x^{4}=1.}" loading="lazy"> Because x is a real number, this equation has two real solutions for x<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {1}{\sqrt {2}}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle x={\tfrac {1}{\sqrt {2}}}}</annotation>

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</math> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=-{\tfrac {1}{\sqrt {2}}}}">

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x=-{\tfrac {1}{\sqrt {2}}}}</annotation>

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</math><img src="./652f272aa91b193591e7f1bed4cfc7efe297b178.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.263ex; height:4.176ex;" alt="{\displaystyle x=-{\tfrac {1}{\sqrt {2}}}}" loading="lazy">. Substituting either of these results into the equation 2xy = 1 in turn, we will get the corresponding result for y. Thus, the square roots of i are the numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{\sqrt {2}}}+{\tfrac {1}{\sqrt {2}}}i}">

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mstyle>

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<mfrac>

<msqrt>

</msqrt>

</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{\sqrt {2}}}+{\tfrac {1}{\sqrt {2}}}i}</annotation>

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</math><img src="./55a64ecc5c1fd1ecdd2c8868e5334d08a61335f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.697ex; height:4.176ex;" alt="{\displaystyle {\tfrac {1}{\sqrt {2}}}+{\tfrac {1}{\sqrt {2}}}i}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{\sqrt {2}}}-{\tfrac {1}{\sqrt {2}}}i}">

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{\sqrt {2}}}-{\tfrac {1}{\sqrt {2}}}i}</annotation>

</semantics>

</math><img src="./4939de1cea5d8fdabea076df65ddb0ed26624d2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.505ex; height:4.176ex;" alt="{\displaystyle -{\tfrac {1}{\sqrt {2}}}-{\tfrac {1}{\sqrt {2}}}i}" loading="lazy">.<a href="#cite_note-11">[11]</a>

</li>

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

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

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFStubbings1945" class="citation book cs1">Stubbings, George Wilfred (1945). <a rel="nofollow" class="external text" href="https://archive.org/details/elementaryvector00stub/page/69/">Elementary vectors for electrical engineers</a>. London: I. Pitman. p. 69.</cite> <div class="paragraphbreak" style="margin-top:0.5em"></div> <cite id="CITEREFBoas2006" class="citation book cs1">Boas, Mary L. (2006). Mathematical Methods in the Physical Sciences (3rd ed.). New York [u.a.]: Wiley. p. 49. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-19826-9</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFSilver2017" class="citation journal cs1">Silver, Daniel S. (November–December 2017). <a rel="nofollow" class="external text" href="https://www.americanscientist.org/article/the-new-language-of-mathematics">"The New Language of Mathematics"</a>. <a href="American_Scientist" title="American Scientist">American Scientist</a>. 105 (6): 364–371. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1511%2F2017.105.6.364">10.1511/2017.105.6.364</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFReference-OED-imaginary_number" class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.oed.com/search/dictionary/?q=imaginary+number">"imaginary number"</a>. <a href="Oxford_English_Dictionary" title="Oxford English Dictionary">Oxford English Dictionary</a> (Online ed.). <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>.</cite> (Subscription or <a rel="nofollow" class="external text" href="https://www.oed.com/public/login/loggingin#withyourlibrary">participating institution membership</a> required.)

</li>

<li id="cite_note-Boyer-4"><a href="#cite_ref-Boyer_4-0">^</a> <cite id="CITEREFBoyerMerzbach1991" class="citation book cs1"><a href="Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl B.</a>; <a href="Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/439">A History of Mathematics</a>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/439">439–445</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-54397-8</bdi>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFDoxiadēsMazur2012" class="citation book cs1">Doxiadēs, Apostolos K.; Mazur, Barry (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X9Uoug4lNWkC">Circles Disturbed: The interplay of mathematics and narrative</a> (illustrated ed.). Princeton University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X9Uoug4lNWkC&pg=PA225">225</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-14904-2</bdi> – via Google Books.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> The interpretation of the imaginary unit as the ratio of two perpendicular vectors was proposed by <a href="Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a> in the foreword to his Ausdehnungslehre of 1844; later <a href="William_Kingdon_Clifford" title="William Kingdon Clifford">William Clifford</a> realized that this ratio could be interpreted as a bivector. <div class="paragraphbreak" style="margin-top:0.5em"></div> <cite id="CITEREFHestenes1996" class="citation book cs1"><a href="David_Hestenes" title="David Hestenes">Hestenes, David</a> (1996). <a rel="nofollow" class="external text" href="https://davidhestenes.net/geocalc/pdf/GrassmannsVision.pdf">"Grassmann's Vision"</a> (PDF). In Schubring, G. (ed.). Hermann Günther Graßmann (1809–1877). Boston Studies in the Philosophy of Science. Vol. 187. Springer. pp. 243–254. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-94-015-8753-2_20">10.1007/978-94-015-8753-2_20</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-90-481-4758-8</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFBunch2012" class="citation book cs1">Bunch, Bryan (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jUTCAgAAQBAJ">Mathematical Fallacies and Paradoxes</a> (illustrated ed.). Courier Corporation. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jUTCAgAAQBAJ&pg=PA31">31</a>-34. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-13793-3</bdi> – via Google Books.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFKramer2012" class="citation book cs1">Kramer, Arthur (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gdAJAAAAQBAJ">Math for Electricity & Electronics</a> (4th ed.). Cengage Learning. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gdAJAAAAQBAJ&pg=PA81">81</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-133-70753-0</bdi> – via Google Books.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFPicciottoWah1994" class="citation book cs1">Picciotto, Henri; Wah, Anita (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_cOhDl3J3ZMC">Algebra: Themes, tools, concepts</a> (Teachers' ed.). Henri Picciotto. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_cOhDl3J3ZMC&pg=PA424">424</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-56107-252-1</bdi> – via Google Books.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFNahin2010" class="citation book cs1">Nahin, Paul J. (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PflwJdPhBlEC">An Imaginary Tale: The story of "i" [the square root of minus one]</a>. Princeton University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12">12</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4008-3029-9</bdi> – via Google Books.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.math.utoronto.ca/mathnet/questionCorner/rootofi.html">"What is the square root of i ?"</a>. University of Toronto Mathematics Network. Retrieved 26 March 2007.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFZillShanahan2003" class="citation book cs1">Zill, Dennis G.; Shanahan, Patrick D. (2003). A first course in complex analysis with applications. Boston: Jones and Bartlett. pp. 24–25. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7637-1437-2</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/50495529">50495529</a>.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.hmc.edu/funfacts/i-to-the-i-is-a-real-number/">"i to the i is a Real Number – Math Fun Facts"</a>. math.hmc.edu. Retrieved 22 August 2024.</cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> Euler expressed the partial fraction decomposition of the trigonometric cotangent as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi \cot \pi z={\frac {1}{z}}+{\frac {1}{z-1}}+{\frac {1}{z+1}}+{\frac {1}{z-2}}+{\frac {1}{z+2}}+\cdots .}">

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

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<mfrac>

<mrow>

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</mfrac>

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<mrow>

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<annotation encoding="application/x-tex">{\textstyle \pi \cot \pi z={\frac {1}{z}}+{\frac {1}{z-1}}+{\frac {1}{z+1}}+{\frac {1}{z-2}}+{\frac {1}{z+2}}+\cdots .}</annotation>

</semantics>

</math><img src="./40732a64c32982cda9a8bfaa6176120a35cc2486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:45.166ex; height:3.676ex;" alt="{\textstyle \pi \cot \pi z={\frac {1}{z}}+{\frac {1}{z-1}}+{\frac {1}{z+1}}+{\frac {1}{z-2}}+{\frac {1}{z+2}}+\cdots .}" loading="lazy"> <div class="paragraphbreak" style="margin-top:0.5em"></div> <cite id="CITEREFVaradarajan2007" class="citation journal cs1">Varadarajan, V. S. (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-07-01175-5">"Euler and his Work on Infinite Series"</a>. Bulletin of the American Mathematical Society. New Series. 44 (4): 515–539. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-07-01175-5">10.1090/S0273-0979-07-01175-5</a>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFGbur2011" class="citation book cs1"><a href="Greg_Gbur" title="Greg Gbur">Gbur, Greg</a> (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. pp. 278–284. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-511-91510-9</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/704518582">704518582</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFIvanThornberKoubaConstales2013" class="citation journal cs1">Ivan, M.; Thornber, N.; Kouba, O.; Constales, D. (2013). "Arggh! Eye factorial . . . Arg(i!)". <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 120 (7): 662–665. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Famer.math.monthly.120.07.660">10.4169/amer.math.monthly.120.07.660</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:24405635">24405635</a>.</cite> <div class="paragraphbreak" style="margin-top:0.5em"></div> <a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). "Decimal expansion of the real part of i!", Sequence <a href="https://oeis.org/A212877" class="extiw external" title="oeis:A212877">A212877</a>; and "Decimal expansion of the negated imaginary part of i!", Sequence <a href="https://oeis.org/A212878" class="extiw external" title="oeis:A212878">A212878</a>. The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). "Decimal expansion of the absolute value of i!", Sequence <a href="https://oeis.org/A212879" class="extiw external" title="oeis:A212879">A212879</a>; and "Decimal expansion of the negated argument of i!", Sequence <a href="https://oeis.org/A212880" class="extiw external" title="oeis:A212880">A212880</a>. The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>. OEIS Foundation.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFNahin1998" class="citation book cs1">Nahin, Paul J. (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/imaginarytales00nahi">An Imaginary Tale: The story of i [the square root of minus one]</a>. Chichester: Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-02795-1</bdi> – via Archive.org.</cite></li></ul>

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

<ul><li><cite id="CITEREFEuler" class="citation web cs1"><a href="Leonhard_Euler" title="Leonhard Euler">Euler, Leonhard</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191216104926/https://www.maa.org/press/periodicals/convergence/eulers-investigations-on-the-roots-of-equations-factoring-rational-functions">"Imaginary Roots of Polynomials"</a>. Archived from <a rel="nofollow" class="external text" href="http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2245&bodyId=2439">the original</a> on 16 December 2019. Retrieved 29 November 2012.</cite> at <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070713083148/http://mathdl.maa.org/convergence/1/">"Convergence"</a>. mathdl.maa.org. Mathematical Association of America. Archived from <a rel="nofollow" class="external text" href="http://mathdl.maa.org/convergence/1/">the original</a> on 13 July 2007.</cite></li></ul></div><div><div class="zim-footer">

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