Editing Imaginary unit (section)

==Definition==
{| class="wikitable" style="float: right; margin-left: 1em; text-align: center;"
! The powers of {{mvar|i}}<br/> are cyclic:
|-
|<math>\ \vdots</math>
|-
|<math>\ i^{-4} = \phantom-1\phantom{i}</math>
|-
|<math>\ i^{-3} = \phantom-i\phantom1</math>
|-
|<math>\ i^{-2} = -1\phantom{i}</math>
|-
|<math>\ i^{-1} = -i\phantom1</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{0}\ = \phantom-1\phantom{i}</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{1}\  = \phantom-i\phantom1</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{2}\  = -1\phantom{i}</math>
|-
|style="background:#e1edfd;" | <math>\ \ i^{3}\  = -i\phantom1</math>
|-
|<math>\ \ i^{4}\  = \phantom-1\phantom{i}</math>
|-
|<math>\ \ i^{5}\  = \phantom-i\phantom1</math>
|-
|<math>\ \ i^{6}\  = -1\phantom{i}</math>
|-
|<math>\ \ i^{7}\  = -i\phantom1</math>
|-
|<math>\ \vdots</math>
|}

The imaginary unit {{mvar|i}} is defined solely by the property that its square is −1:
<math display=block>i^2 = -1.</math>

With {{mvar|i}} defined this way, it follows directly from [[algebra]] that {{mvar|i}} and {{math|−''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 {{mvar|i}} as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of {{math|''i''{{isup|2}}}} with {{math|−1}}). Higher integral powers of {{mvar|i}} are thus
<math display=block>\begin{alignat}{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{alignat}</math>
and so on, cycling through the four values {{math|1}}, {{mvar|i}}, {{math|−1}}, and {{math|−''i''}}. As with any non-zero real number, {{math|1=''i''{{isup|0}} = 1.}}

As a complex number, {{mvar|i}} can be represented in [[Rectangular coordinate system|rectangular form]] as {{math|0 + 1''i''}}, with a zero real component and a unit imaginary component. In [[polar form]], {{mvar|i}} can be represented as {{math|1 × ''e''{{isup|''πi'' /2}}}} (or just {{math|''e''{{isup|''πi'' /2}}}}), with an [[absolute value]] (or magnitude) of 1 and an [[argument (complex analysis)|argument]] (or angle) of <math>\tfrac\pi2</math> [[radian]]s. (Adding any integer multiple of {{math|2''π''}} to this angle works as well.)  In the [[complex plane]], which is a special interpretation of a [[Cartesian plane]], {{mvar|i}} is the point located one unit from the origin along the [[imaginary axis]] (which is [[perpendicular]] to the [[real axis]]).

==={{math|''i''}} vs. {{math|−''i''}}===
{{anchor|i and -i}}
Being a [[quadratic polynomial]] with no [[multiple root]], the defining equation {{math|1=''x''{{isup|2}} = −1}} has {{em|two}} distinct solutions, which are equally valid and which happen to be [[additive inverse|additive]] and [[multiplicative inverse]]s 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 {{math|+''i''}} (or simply {{mvar|i}}) and the other is labelled {{math|−''i''}}, though it is inherently ambiguous which is which.

The only differences between {{math|+''i''}} and {{math|−''i''}} arise from this labelling. For example, by convention {{math|+''i''}} is said to have an [[Argument (complex analysis)|argument]] of <math>+\tfrac\pi2</math> and {{math|−''i''}} is said to have an argument of <math>-\tfrac\pi2,</math> related to the convention of labelling orientations in the [[Cartesian plane]] relative to the positive {{mvar|x}}-axis with positive angles turning [[anticlockwise]] in the direction of the positive {{mvar|y}}-axis. Also, despite the signs written with them, neither {{math|+''i''}} nor {{math|−''i''}} is inherently positive or negative in the sense that real numbers are.<ref>{{cite book |first1=Apostolos K. |last1=Doxiadēs |first2=Barry |last2=Mazur |year=2012 |title=Circles Disturbed: The interplay of mathematics and narrative |page= [https://books.google.com/books?id=X9Uoug4lNWkC&pg=PA225 225] |edition=illustrated |publisher=Princeton University Press |isbn=978-0-691-14904-2 |via=Google Books |url=https://books.google.com/books?id=X9Uoug4lNWkC}}</ref>

A more formal expression of this indistinguishability of {{math|+''i''}} and {{math|−''i''}} is that, although the complex [[field (algebra)|field]] is [[unique (mathematics)|unique]] (as an extension of the real numbers) [[up to]] [[isomorphism]], it is {{em|not}} unique up to a {{em|unique}} isomorphism. That is, there are two [[automorphism|field automorphisms]] of the complex numbers <math>\C</math> that keep each real number fixed, namely the identity and [[complex conjugation]]. For more on this general phenomenon, see [[Galois group]].

===Matrices===
Using the concepts of [[matrix (mathematics)|matrices]]  and [[matrix multiplication]], complex numbers can be represented in linear algebra. The real unit {{math|1}} and imaginary unit {{mvar|i}} can be represented by any pair of matrices {{mvar|I}} and {{mvar|J}} satisfying {{math|1=''I''{{isup|2}} = ''I'',}} {{math|1=''IJ'' = ''JI'' = ''J'',}} and {{math|1=''J''{{isup|2}} = −''I''.}} Then a complex number {{math|''a'' + ''bi''}} can be represented by the matrix {{math|''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 {{math|1}} and {{mvar|i}} by the {{math|2&hairsp;×&hairsp;2}} [[identity matrix]] {{mvar|I}} and the matrix {{mvar|J}},

<math display=block>
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad
J = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.</math>

Then an arbitrary complex number {{math|''a'' + ''bi''}} can be represented by:

<math display=block>aI + bJ = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.</math>

More generally, any real-valued {{math|2&hairsp;×&hairsp;2}} matrix with a [[trace (linear algebra)|trace]] of zero and a [[determinant]] of one squares to {{math|−''I''}}, so could be chosen for {{mvar|J}}. Larger matrices could also be used; for example, {{math|1}} could be represented by the {{math|4&hairsp;×&hairsp;4}} identity matrix and {{mvar|i}} could be represented by any of the [[Dirac matrices]] for spatial dimensions.

===Root of {{math|''x''<sup>2</sup> + 1}}===
[[Polynomial]]s (weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose [[coefficient]]s are real numbers form a [[ring (mathematics)|ring]], denoted <math>\R[x],</math> an algebraic structure with addition and multiplication and sharing many properties with the ring of [[integer]]s.

The polynomial <math>x^2 + 1</math> has no real-number [[root of a polynomial|roots]], but the set of all real-coefficient polynomials divisible by <math>x^2 + 1</math> forms an [[ideal (ring theory)|ideal]], and so there is a [[Polynomial ring#Quotient ring|quotient ring]] <math>\reals[x] / \langle x^2 + 1\rangle.</math> This quotient ring is [[isomorphism|isomorphic]] to the complex numbers, and the variable <math>x</math> expresses the imaginary unit.

=== Graphic representation ===
{{main|Complex plane}}
The complex numbers can be represented graphically by drawing the real [[number line]] as the horizontal axis and the imaginary numbers as the vertical axis of a [[Cartesian coordinate system|Cartesian plane]] called the ''[[complex plane]]''. In this representation, the numbers {{math|1}} and {{mvar|i}} are at the same distance from {{math|0}}, with a right angle between them. Addition by a complex number corresponds to [[translation (geometry)|translation]] in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every [[similarity (geometry)|similarity]] transformation of the plane can be represented by a complex-linear function <math>z \mapsto az + b.</math>

=== Geometric algebra ===

In the [[geometric algebra]] of the [[Euclidean plane]], the geometric product or quotient of two arbitrary [[Euclidean vector|vectors]] is a sum of a scalar (real number) part and a [[bivector]] 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 {{math|1=1 = ''u''/''u''}}, and when multiplied by any vector leaves it unchanged (the [[Identity function|identity transformation]]). The quotient of any two perpendicular vectors of the same magnitude, {{math|1=''J'' = ''u''/''v''}}, which when multiplied rotates the divisor a quarter turn into the dividend, {{math|1=''Jv'' = ''u''}}, is a unit bivector which squares to {{math|&minus;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 [[isomorphic]] to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects.<ref>The interpretation of the imaginary unit as the ratio of two perpendicular vectors was proposed by [[Hermann Grassmann]] in the foreword to his ''Ausdehnungslehre'' of 1844; later [[William Kingdon Clifford|William Clifford]] realized that this ratio could be interpreted as a bivector. {{pb}} {{cite book |last=Hestenes |first=David |author-link=David Hestenes |year=1996 |chapter=Grassmann’s Vision |editor-last=Schubring |editor-first=G. |title=Hermann Günther Graßmann (1809–1877) |series=Boston Studies in the Philosophy of Science |volume=187 |pages=243–254 |publisher=Springer |doi=10.1007/978-94-015-8753-2_20 |isbn=978-90-481-4758-8 |chapter-url=https://davidhestenes.net/geocalc/pdf/GrassmannsVision.pdf }}</ref>

More generally, in the geometric algebra of any higher-dimensional [[Euclidean space]], a unit bivector of any arbitrary planar orientation squares to {{math|−1}}, so can be taken to represent the imaginary unit {{mvar|i}}.