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| − | ==The complex plane==
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| − | A complex number <math>z</math> can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the '''complex plane''' or '''Argand diagram'''. The point and hence the complex number <math>z</math> can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part <math>x=\text{Re}(z)</math> and the imaginary part <math>y=\text{Im}(z)</math> . The representation of a complex number by its Cartesian coordinates is called the ''Cartesian form'' or ''rectangular form'' or ''algebraic form'' of that complex number.
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| − | ===Polar form===
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| − | Alternatively, the complex number <math>z</math> can be specified by polar coordinates. The polar coordinates are <math>r=|z|\ge0</math> , called the '''absolute value''' or '''modulus''', and <math>\phi=\arg(z)</math> , called the '''argument''' of <math>z</math> . For <math>r=0</math> any value of <math>\varphi</math> describes the same number. To get a unique representation, a conventional choice is to set <math>\arg(0)=0</math> . For <math>r>0</math> the argument <math>\varphi</math> is unique modulo <math>2\pi</math> ; that is, if any two values of the complex argument differ by an exact integer multiple of <math>2\pi</math> , they are considered equivalent. To get a unique representation, a conventional choice is to limit <math>\varphi</math> to the interval <math>(-\pi,\pi]</math> i.e. <math>-\pi<\varphi\le\pi</math> . The representation of a complex number by its polar coordinates is called the ''polar form'' of the complex number.
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| − | ===Conversion from the polar form to the Cartesian form===
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| − | :<math>x=r\cos(\varphi)</math>
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| − | :<math>y=r\sin(\varphi)</math>
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| − | ===Conversion from the Cartesian form to the polar form===
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| − | :<math>r=\sqrt{x^2+y^2}</math>
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| − | :<math>\varphi=
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| − | \begin{cases}
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| − | \arctan\left(\frac{y}{x}\right)&\text{if }x>0\\
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| − | \arctan\left(\frac{y}{x}\right)+\pi&\text{if }x<0,y\ge0\\
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| − | \arctan\left(\frac{y}{x}\right)-\pi&\text{if }x<0,y<0\\
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| − | \frac{\pi}{2}&\text{if }x=0,y>0\\
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| − | -\frac{\pi}{2}&\text{if }x=0,y<0\\
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| − | \text{undefined}&\text{if }x=0,y=0
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| − | \end{cases}</math>
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| − | The previous formula requires rather laborious case differentiations. However, many programming languages provide a variant of the arctangent function. A formula that uses the arccos function requires fewer case differentiations:
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| − | :<math>\varphi=
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| − | \begin{cases}
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| − | \arccos\left(\frac{x}{r}\right)&\text{if }y\ge0,r\ne0\\
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| − | -\arccos\left(\frac{x}{r}\right)&\text{if }y<0\\
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| − | \text{undefined}&\text{if }r=0
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| − | \end{cases}</math>
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| − | ===Notation of the polar form===
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| − | The notation of the polar form as
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| − | :<math>z=r\big(\cos(\varphi)+i\sin(\varphi)\big)</math>
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| − | is called ''trigonometric form''. The notation <math>\text{cis}(\varphi)</math> is sometimes used as an abbreviation for <math>\cos(\varphi)+i\sin(\varphi)</math> . Using [[w:Euler's formula|Euler's formula]] it can also be written as
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| − | :<math>z=re^{i\varphi}</math>
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| − | which is called ''exponential form''.
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| − | ===Multiplication, division, exponentiation, and root extraction in the polar form===
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| − | Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.
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| − | Using [[w:List of trigonometric identities#Angle sum and difference identities|sum and difference identities]] its possible to obtain that
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| − | :<math>r_1e^{i\varphi_1}\cdot r_2e^{i\varphi_2}=r_1r_2e^{i(\varphi_1+\varphi_2)}</math>
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| − | and that
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| − | :<math>\frac{r_1e^{i\varphi_1}}{r_2e^{i\varphi_2}}=\frac{r_1}{r_2}\cdot e^{i(\varphi_1-\varphi_2)}</math>
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| − | Exponentiation with integer exponents; according to [[w:De Moivre's formula|de Moivre's formula]],
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| − | :<math>\big(re^{i\varphi}\big)^n=r^ne^{ni\varphi}</math>
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| − | Exponentiation with arbitrary complex exponents is discussed in the article on [[exponentiation]].
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| − | The addition of two complex numbers is just the addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.
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| − | Multiplication by <math>i</math> corresponds to a counter-clockwise rotation by 90° or <math>\frac{\pi}{2}</math> radians. The geometric content of the equation <math>i^2=-1</math> is that a sequence of two 90° rotations results in a 180° (<math>\pi</math> radians) rotation. Even the fact <math>(-1)\cdot(-1)=1</math> from arithmetic can be understood geometrically as the combination of two 180° turns.
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| − | All the roots of any number, real or complex, may be found with a simple algorithm. The <math>n</math>-th roots are given by
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| − |
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| − | :<math>\sqrt[n]{re^{i\varphi}}=\sqrt[n]{r}\,e^{i\left(\frac{\varphi+2k\pi}{n}\right)}</math>
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| − | for <math>k=0,1,2,\ldots,n-1</math> , where <math>\sqrt[n]{r}</math> represents the principal <math>n</math>-th root of <math>r</math> .
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| − | ==Absolute value, conjugation and distance==
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| − | The '''absolute value''' (or ''modulus'' or ''magnitude'') of a complex number <math>z=re^{i\varphi}</math> is defined as <math>|z|=r</math> .
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| − | Algebraically, if <math>z=a+bi</math> then <math>|z|=\sqrt{a^2+b^2}</math> .<!--keep sentence-terminator within math element to make it align better with the formula-->
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| − | One can check readily that the absolute value has three important properties:
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| − | :<math>|z|=0</math> if and only if <math>z=0</math>
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| − | :<math>|z+w|\le|z|+|w|</math> (triangle inequality)
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| − | :<math>|z\cdot w|=|z|\cdot|w|</math>
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| − | for all complex numbers <math>z,w</math> . It then follows, for example, that <math>|1|=1</math> and <math>\left|\frac{z}{w}\right|=\frac{|z|}{|w|}</math> . By defining the '''distance''' function <math>d(z,w)=|z-w|</math> we turn the set of complex numbers into a [[w:metric space|metric space]] and we can therefore talk about limits and continuity.
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| − | The '''complex conjugate''' of the complex number <math>z=a+bi</math> is defined to be <math>a-bi</math> , written as <math>\bar z</math> or <math>z^*</math> . As seen in the figure, <math>\bar z</math> is the "reflection" of <math>z</math> about the real axis. The following can be checked:
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| − | :<math>\overline{z+w}=\bar z+\bar w</math>
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| − | :<math>\overline{z\cdot w}=\bar z\cdot\bar w</math>
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| − | :<math>\overline{\left(\frac{z}{w}\right)}=\frac{\bar z}{\bar w}</math>
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| − | :<math>\bar{\bar z}=z</math>
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| − | :<math>\bar z=z</math> if and only if <math>z</math> is real
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| − | :<math>|z|=|\bar z|</math>
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| − | :<math>|z|^2=z\cdot\bar z</math>
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| − | :<math>z^{-1}=\bar z\cdot|z|^{-2}</math> if <math>z</math> is non-zero.
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| − | The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.
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| − | That conjugation commutes with all the algebraic operations (and many functions; ''e.g.'' <math>\sin(\bar z)=\overline{\sin(z)}</math>) is rooted in the ambiguity in choice of <math>i</math> (−1 has two square roots). It is important to note, however, that the function <math>f(z)=\bar z</math> is not complex-differentiable.
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| − | ==Resources==
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| − | * [https://en.wikibooks.org/wiki/Calculus/Complex_numbers Complex Numbers], Wikibooks: Calculus
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| | * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Equations/Esparza%201093%20Notes%203.3B.pdf Trigonometric Equations]. Written notes created by Professor Esparza, UTSA. | | * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Trigonometric%20Equations/Esparza%201093%20Notes%203.3B.pdf Trigonometric Equations]. Written notes created by Professor Esparza, UTSA. |
