Editing Complex number (section)

==Definition and basic operations==

[[File:Complex_numbers_intheplane.svg|right|thumb|Various complex numbers depicted in the complex plane.]]

A complex number is an expression of the form {{math|1=''a'' + ''bi''}}, where {{mvar|a}} and {{mvar|b}} are real numbers, and {{math|''i''}} is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, {{math|2 + 3''i''}} is a complex number.<ref>{{cite book|title=College algebra |url=https://archive.org/details/collegealgebrawi00axle |url-access=limited |last=Axler |first=Sheldon |page=[https://archive.org/details/collegealgebrawi00axle/page/n285 262]|publisher=Wiley|year=2010|isbn=9780470470770 }}</ref>

For a complex number {{math|''a'' + ''bi''}}, the real number {{mvar|a}} is called its ''real part'', and the real number {{mvar|b}} (not the complex number {{math|''bi''}}) is its ''imaginary part''.<ref>{{cite book |last1=Spiegel |first1=M.R. |title=Complex Variables |last2=Lipschutz |first2=S. |last3=Schiller |first3=J.J. |last4=Spellman |first4=D. |date=14 April 2009 |publisher=McGraw Hill |isbn=978-0-07-161569-3 |edition=2nd |series=Schaum's Outline Series}}</ref><ref>{{harvnb|Aufmann|Barker|Nation|loc=p. 66, Chapter P|2007}}</ref> The real part of a complex number {{mvar|z}} is denoted {{math|Re(''z'')}}, <math>\mathcal{Re}(z)</math>, or <math>\mathfrak{R}(z)</math>; the imaginary part is {{math|Im(''z'')}}, <math>\mathcal{Im}(z)</math>, or <math>\mathfrak{I}(z)</math>: for example, <math display="inline">  \operatorname{Re}(2 + 3i) = 2  </math>, <math>   \operatorname{Im}(2 + 3i) = 3  </math>.

A complex number {{mvar|z}} can be identified with the [[ordered pair]] of real numbers <math>(\Re (z),\Im (z))</math>, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the ''[[complex plane]]'' or ''[[Argand diagram]].''<ref>{{cite book |last=Pedoe |first=Dan |author-link=Daniel Pedoe |title=Geometry: A comprehensive course |publisher=Dover |year=1988 |isbn=978-0-486-65812-4}}</ref><ref name=":2">{{Cite web |last=Weisstein |first=Eric W. |title=Complex Number |url=https://mathworld.wolfram.com/ComplexNumber.html |access-date=2020-08-12 |website=mathworld.wolfram.com}}</ref>{{efn| {{harvnb|Solomentsev|2001}}: "The plane <math>\R^2</math> whose points are identified with the elements of <math>\Complex</math> is called the complex plane&nbsp;... The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the 'Argand diagram', came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel".}} The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards. 

[[File:Complex number illustration.svg|thumb|right|A complex number {{mvar|z}}, as a point (black) and its [[vector (geometric)|position vector]] (blue).]]
A real number {{mvar|a}} can be regarded as a complex number {{math|''a'' + 0''i''}}, whose imaginary part is 0. A purely imaginary number {{math|''bi''}} is a complex number {{math|0 + ''bi''}}, whose real part is zero. It is common to write {{math|1=''a'' + 0''i'' = ''a''}}, {{math|1=0 + ''bi'' = ''bi''}}, and {{math|1=''a'' + (−''b'')''i'' = ''a'' − ''bi''}}; for example, {{math|1=3 + (−4)''i'' = 3 − 4''i''}}.

The [[Set (mathematics)|set]] of all complex numbers is denoted by <math>\Complex</math> ([[blackboard bold]]) or {{math|'''C'''}} (upright bold).

In some disciplines such as electromagnetism and electrical engineering, {{mvar|j}} is used instead of {{mvar|i}}, as {{mvar|i}} frequently represents electric current,<ref name="Campbell_1911" /><ref name="Brown-Churchill_1996" /> and complex numbers are written as {{math|''a'' + ''bj''}} or {{math|''a'' + ''jb''}}.

===Addition and subtraction===
[[File:Vector Addition.svg|right|thumb|Addition of two complex numbers can be done geometrically by constructing a parallelogram.]]

Two complex numbers <math>a =x+yi</math> and <math>b =u+vi</math> are [[addition|added]] by separately adding their real and imaginary parts. That is to say:

<math display=block>a + b =(x+yi) + (u+vi) = (x+u) + (y+v)i.</math>
Similarly, [[subtraction]] can be performed as
<math display=block>a - b =(x+yi) - (u+vi) = (x-u) + (y-v)i.</math>

The addition can be geometrically visualized as follows: the sum of two complex numbers {{mvar|a}} and {{mvar|b}}, interpreted as points in the complex plane, is the point obtained by building a [[parallelogram]] from the three vertices {{mvar|O}}, and the points of the arrows labeled {{mvar|a}} and {{mvar|b}} (provided that they are not on a line). Equivalently, calling these points {{mvar|A}}, {{mvar|B}}, respectively and the fourth point of the parallelogram {{mvar|X}} the [[triangle]]s {{mvar|OAB}} and {{mvar|XBA}} are [[Congruence (geometry)|congruent]].

===Multiplication{{anchor|Multiplication|Square}}===
The product of two complex numbers is computed as follows:
:<math>(a+bi) \cdot (c+di) = ac - bd + (ad+bc)i.</math>
For example, <math>(3+2i)(4-i) = 3 \cdot 4 - (2 \cdot (-1)) + (3 \cdot (-1) + 2 \cdot 4)i = 14 +5i.</math> 
In particular, this includes as a special case the fundamental formula
:<math>i^2 = i \cdot i = -1.</math>
This formula distinguishes the complex number ''i'' from any real number, since the square of any (negative or positive) real number is always a non-negative real number.

With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the [[distributive property]], the [[commutative property|commutative properties]] (of addition and multiplication) hold. Therefore, the complex numbers form an algebraic structure known as a [[field (mathematics)|''field'']], the same way as the rational or real numbers do.{{sfn|Apostol|1981|pp=15–16}}

===Complex conjugate, absolute value, argument and division===

[[File:Complex conjugate picture.svg|right|thumb|upright=0.8|Geometric representation of {{mvar|z}} and its conjugate {{mvar|{{overline|z}}}} in the complex plane.]]
The ''[[complex conjugate]]'' of the complex number {{math|1=''z'' = ''x'' + ''yi''}} is defined as
<math>\overline z = x-yi.</math><ref>{{harvnb|Apostol|1981|pp=15–16}}</ref> It is also denoted by some authors by <math>z^*</math>. Geometrically, {{mvar|{{overline|z}}}} is the [[reflection symmetry|"reflection"]] of {{mvar|z}} about the real axis. Conjugating twice gives the original complex number: <math>\overline{\overline{z}}=z.</math> A complex number is real if and only if it equals its own conjugate. The [[unary operation]] of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division. 

[[File:Complex number illustration modarg.svg|right|thumb|Argument {{mvar|φ}} and modulus {{mvar|r}} locate a point in the complex plane.]]
For any complex number {{math|1=''z'' = ''x'' + ''yi''}} , the product 
:<math>z \cdot \overline z = (x+iy)(x-iy) = x^2 + y^2</math>
is a ''non-negative real'' number. This allows to define the ''[[absolute value]]'' (or ''modulus'' or ''magnitude'') of ''z'' to be the square root{{sfn|Apostol|1981|p=18}}
<math display="block">|z|=\sqrt{x^2+y^2}.</math>
By [[Pythagoras' theorem]], <math>|z|</math> is the distance from the origin to the point representing the complex number ''z'' in the complex plane. In particular, the [[unit circle|circle of radius one]] around the origin consists precisely of the numbers ''z'' such that <math>|z| = 1 </math>. If <math>  z = x = x + 0i  </math> is a real number, then <math>  |z|= |x| </math>: its absolute value as a complex number and as a real number are equal. 

Using the conjugate, the [[multiplicative inverse|reciprocal]] of a nonzero complex number <math>z = x + yi</math> can be computed to be 

<math display=block>
\frac{1}{z}
= \frac{\bar{z}}{z\bar{z}}
= \frac{\bar{z}}{|z|^2}
= \frac{x - yi}{x^2 + y^2}
= \frac{x}{x^2 + y^2} - \frac{y}{x^2 + y^2}i.</math>
More generally, the division of an arbitrary complex number <math>w = u + vi</math> by a non-zero complex number <math>z = x + yi</math> equals
<math display=block>
\frac{w}{z}
= \frac{w\bar{z}}{|z|^2}
= \frac{(u + vi)(x - iy)}{x^2 + y^2}
= \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2}i.
</math>
This process is sometimes called "[[rationalisation (mathematics)|rationalization]]" of the denominator (although the denominator in the final expression may be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator.<ref>{{cite book |title=Numerical Linear Algebra with Applications: Using MATLAB and Octave |author1=William Ford |edition=reprinted |publisher=Academic Press |year=2014 |isbn=978-0-12-394784-0 |page=570 |url=https://books.google.com/books?id=OODs2mkOOqAC}} [https://books.google.com/books?id=OODs2mkOOqAC&pg=PA570 Extract of page 570]</ref><ref>{{cite book |title=Precalculus with Calculus Previews: Expanded Volume |author1=Dennis Zill |author2=Jacqueline Dewar |edition=revised |publisher=Jones & Bartlett Learning |year=2011 |isbn=978-0-7637-6631-3 |page=37 |url=https://books.google.com/books?id=TLgjLBeY55YC}} [https://books.google.com/books?id=TLgjLBeY55YC&pg=PA37 Extract of page 37]</ref>

The ''[[argument (complex analysis)|argument]]'' of {{mvar|z}} (sometimes called the "phase" {{mvar|φ}})<ref name=":2" /> is the angle of the [[radius]] {{mvar|Oz}} with the positive real axis, and is written as {{math|arg ''z''}}, expressed in [[radian]]s in this article. The angle is defined only up to adding integer multiples of <math>  2\pi </math>, since a rotation by <math>2\pi</math> (or 360°) around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval <math>  (-\pi,\pi]  </math>, which is referred to as the [[principal value]].<ref>Other authors, including {{harvnb|Ebbinghaus|Hermes|Hirzebruch|Koecher|Mainzer|Neukirch|Prestel|Remmert|1991|loc=§6.1}}, chose the argument to be in the interval <math>[0, 2\pi)</math>.</ref>
The argument can be computed from the rectangular form {{mvar|x + yi}} by means of the [[arctan]] (inverse tangent) function.<ref>{{cite book
|title=Complex Variables: Theory And Applications
|edition=2nd
|chapter=Chapter 1
|first1=H.S.
|last1=Kasana
|publisher=PHI Learning Pvt. Ltd
|year=2005
|isbn=978-81-203-2641-5
|page=14
|chapter-url=https://books.google.com/books?id=rFhiJqkrALIC&pg=PA14}}</ref>

===Polar form{{anchor|Polar form}}===
{{Main|Polar coordinate system}}
{{Redirect|Polar form|the higher-dimensional analogue|Polar decomposition}}

[[File:Complex multi.svg|right|thumb|Multiplication of {{math|2 + ''i''}} (blue triangle) and {{math|3 + ''i''}} (red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the terms ''φ''<sub>1</sub>+''φ''<sub>2</sub> in the equation) and stretched by the length of the [[hypotenuse]] of the blue triangle (the multiplication of both radiuses, as per term ''r''<sub>1</sub>''r''<sub>2</sub> in the equation).]]

For any complex number ''z'', with absolute value <math>r = |z|</math> and argument <math>\varphi</math>, the equation 
:<math>z=r(\cos\varphi +i\sin\varphi) </math>
holds. This identity is referred to as the polar form of ''z''. It is sometimes abbreviated as <math display="inline"> z = r \operatorname\mathrm{cis} \varphi </math>. 
In electronics, one represents a [[Phasor (sine waves)|phasor]] with amplitude {{mvar|r}} and phase {{mvar|φ}} in [[angle notation]]:<ref>
{{cite book |last1=Nilsson |first1=James William |title=Electric circuits |last2=Riedel |first2=Susan A. |publisher=Prentice Hall |year=2008 |isbn=978-0-13-198925-2 |edition=8th |page=338 |chapter=Chapter 9 |chapter-url=https://books.google.com/books?id=sxmM8RFL99wC&pg=PA338}}
</ref><math display="block">z = r \angle \varphi . </math>

If two complex numbers are given in polar form, i.e., {{math|1=''z''<sub>1</sub> = ''r''<sub>1</sub>(cos ''φ''<sub>1</sub> + ''i'' sin ''φ''<sub>1</sub>)}} and {{math|1=''z''<sub>2</sub> = ''r''<sub>2</sub>(cos ''φ''<sub>2</sub> + ''i'' sin ''φ''<sub>2</sub>)}}, the product and division can be computed as 
<math display=block>z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).</math>
<math display=block>\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right), \text{if }z_2 \ne 0.</math>
(These are a consequence of the [[trigonometric identities]] for the sine and cosine function.)
In other words, the absolute values are ''multiplied'' and the arguments are ''added'' to yield the polar form of the product. The picture at the right illustrates the multiplication of
<math display=block>(2+i)(3+i)=5+5i. </math>
Because the real and imaginary part of {{math|5 + 5''i''}} are equal, the argument of that number is 45 degrees, or {{math|''π''/4}} (in [[radian]]). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are [[arctan]](1/3) and arctan(1/2), respectively. Thus, the formula
<math display=block>\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) </math>
holds. As the [[arctan]] function can be approximated highly efficiently, formulas like this – known as [[Machin-like formula]]s – are used for high-precision approximations of [[pi|{{pi}}]]:<ref>{{cite book |title=Modular Forms: A Classical And Computational Introduction |author1=Lloyd James Peter Kilford |edition= 2nd|publisher=World Scientific Publishing Company |year=2015 |isbn=978-1-78326-547-3 |page=112 |url=https://books.google.com/books?id=qDk8DQAAQBAJ}} [https://books.google.com/books?id=qDk8DQAAQBAJ&pg=PA112 Extract of page 112]</ref>
<math display=block>\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right) </math>

===Powers and roots===
{{see also|Square root#Square roots of negative and complex numbers|l1=Square roots of negative and complex numbers}}
The ''n''-th power of a complex number can be computed using [[de Moivre's formula]], which is obtained by repeatedly applying the above formula for the product:
<math display=block> z^{n}=\underbrace{z \cdot \dots \cdot z}_{n \text{ factors}} = (r(\cos \varphi + i\sin \varphi ))^n = r^n \, (\cos n\varphi + i \sin n \varphi).</math>
For example, the first few powers of the imaginary unit ''i'' are <math>i, i^2 = -1, i^3 = -i, i^4 = 1, i^5 = i, \dots</math>.

{{Visualisation complex number roots|1=upright=1.35}}
The {{mvar|n}} [[nth root|{{mvar|n}}th roots]] of a complex number {{mvar|z}} are given by
<math display=block>z^{1/n} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)</math>
for {{math|0 ≤ ''k'' ≤ ''n'' − 1}}. (Here <math>\sqrt[n]r</math> is the usual (positive) {{mvar|n}}th root of the positive real number {{mvar|r}}.) Because sine and cosine are periodic, other integer values of {{mvar|k}} do not give other values. For any <math>z \ne 0</math>, there are, in particular ''n'' distinct complex ''n''-th roots. For example, there are 4 fourth roots of 1, namely
:<math>z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i.</math>
In general there is ''no'' natural way of distinguishing one particular complex {{mvar|n}}th root of a complex number. (This is in contrast to the roots of a positive real number ''x'', which has a unique positive real ''n''-th root, which is therefore commonly referred to as ''the'' ''n''-th root of ''x''.) One refers to this situation by saying that the {{mvar|n}}th root is a [[multivalued function|{{mvar|n}}-valued function]] of {{mvar|z}}. 

===Fundamental theorem of algebra===
The [[fundamental theorem of algebra]], of [[Carl Friedrich Gauss]] and [[Jean le Rond d'Alembert]], states that for any complex numbers (called [[coefficient]]s) {{math|''a''<sub>0</sub>, ..., ''a''<sub>''n''</sub>}}, the equation
<math display=block>a_n z^n + \dotsb + a_1 z + a_0 = 0</math>
has at least one complex solution ''z'', provided that at least one of the higher coefficients {{math|''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>}} is nonzero.<ref name="Bourbaki 1998 loc=§VIII.1">{{harvnb|Bourbaki|1998|loc=§VIII.1}}</ref> This property does not hold for the [[rational number|field of rational numbers]] <math>\Q</math> (the polynomial {{math|''x''<sup>2</sup> − 2}} does not have a rational root, because {{math|√2}} is not a rational number) nor the real numbers <math>\R</math> (the polynomial {{math|''x''<sup>2</sup> + 4}} does not have a real root, because the square of {{mvar|x}} is positive for any real number {{mvar|x}}). 

Because of this fact, <math>\Complex</math> is called an [[algebraically closed field]]. It is a cornerstone of various applications of complex numbers, as is detailed further below. 
There are various proofs of this theorem, by either analytic methods such as [[Liouville's theorem (complex analysis)|Liouville's theorem]], or [[topology|topological]] ones such as the [[winding number]], or a proof combining [[Galois theory]] and the fact that any real polynomial of ''odd'' degree has at least one real root.