Definition |
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Set of all complex numbers
$ \C=\{a+bi|a,b\in\R\} $ , where $ i^2=-1 $ |
A complex number is any number of the form
- $ a+bi $
where $ a,b $ are real numbers and $ i $ is the imaginary unit. The set of complex numbers, denoted C or $ \C $ , is a field under the operations of addition, multiplication, and exponentiation defined as follows:
- $ (a_1+b_1i)+(a_2+b_2i)=(a_1+a_2)+(b_1+b_2)i $
- $ (a_1+b_1i)\cdot(a_2+b_2i)=(a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i $
- $ (a+bi)^n=a^n+na^{n-1}bi+\cdots+na(bi)^{n-1}+(bi)^n $
This means that these complex-number operations are commutative, associative and distributive in the same way as their real-number counterparts.
Note that the final expression above is the result of multiplying out the left hand side in the usual way (using the FOIL method) and simplifying using the fact that $ i^2=-1 $ (see Imaginary unit).
It is easily seen that the set of real numbers R is a subset of the set of complex numbers C, since every real number is equal to a complex number $ a + bi $ with the imaginary part equals 0:
- $ a=a+0i $
All purely imaginary numbers are also complex, since
- $ b\cdot i=0+bi $
The complex numbers can also be thought of as a two-dimensional vector space over the real numbers, with basis vectors 1 (one, the real unit) and $ i $ (the imaginary unit). In this case, a complex number may be written as:
- $ a\cdot1+b\cdot i $
Number lines and rectangular form
If the real numbers have a real number line, and the imaginary numbers have their own number line, these two number lines can be interpreted as being perpendicular to one another.
These perpendicular lines form axes in a Cartesian coordinate system where all complex numbers lie somewhere on the plane.
In this fashion, the real value of magnitude $ a $ forms the x-coordinate and the imaginary value of magnitude $ b $ forms the y-coordinate. This way, all complex numbers exist somewhere on the complex number plane at coordinate loci $ (a,b) $ , which equates to a simple numeric value of $ a+bi $ .
Polar form
Since all coordinates on a rectangular coordinate plane can be interpreted using the polar coordinate system, all complex numbers can also be interpreted in terms of a polar coordinate set $ (r,\theta) $ and using the trigonometric based function cis.
In this way:
- $ z=a+bi=r\cdot\text{cis}(\theta)=re^{\theta i}=r\angle\theta $
Where:
- $ a=r\cos(\theta) $
- $ b=r\sin(\theta) $
- $ \theta=\text{arg}(z)=\arctan\left(\tfrac{b}{a}\right) $
- $ r=\sqrt{a^2+b^2} $
Matrix representations
The complex number $ a+bi $ can also be represented as $ 2\times2 $ matrices of the form $ \left[\begin{array}{cc}a&-b\\b&a\end{array}\right] $ , where $ a,b $ are real numbers. The complex conjugate will simply be the transpose of the matrix.
As a vector space composed of the span of the real and imaginary units over the real numbers, the following subspace can be shown to be a field, and moreover, isomorphic to $ \C $ :
$ \text{span}\left\{\left[\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right],\left[\begin{array}{cc} 0 & -1\\ 1 & 0\end{array}\right]\right\} $
Where the isomorphism is given by $ \phi(a+bi)=a\left[\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right]+b\left[\begin{array}{cc} 0 & -1\\ 1 & 0\end{array}\right] $
Operations of complex numbers
Division
Given two complex numbers, $ z_1=a_1+b_1i $ and $ z_2=a_2+b_2i $ , they can be divided my multiplying the numerator and denominator by the complex conjugate of the denominator as follows:
- $ \frac{a_1+b_1i}{a_2+b_2i}=\frac{a_1+b_1i}{a_2+b_2i}\cdot\frac{a_2-b_2i}{a_2-b_2i}=\frac{a_1a_2+b_1b_2+(a_2b_1-a_1b_2)i}{a_2^2+b_2^2} $
Because division of a complex number by another complex number always results in another complex number, the complex field is also closed over division.
This formula also allows us to take the reciprocal of complex numbers.
$ z^{-1}=\frac{1+0i}{a_2+b_2i}=\frac{1+b_2i}{a_2^2+b_2^2} $
As a corollary,
- $ \frac{1}{i}=\frac{-i}{-i\cdot i}=-i $
Exponentiation
As previously mentioned, complex numbers can be raised to powers by the binomial theorem. However, it is much simpler in polar form. By De Moivre's formula,
- $ z^n=(re^{\theta i})^n=(r\cdot\text{cis}\theta)^n=r^n\text{cis}(n\theta) $
or, written another way,
- $ z^n=(r,\theta)^n=(r^n,n\theta) $
Logarithms
Since any complex number can be represented as $ re^{\theta i} $ , the logarithm of a complex number is simply
- $ \log_a(z)=\log_a(re^{\theta i})=\log_a(r)+i(\theta+2\pi n)\log_a(e) $
Since negative numbers can be represented by complex numbers with an argument of $ \pi $ , logarithms of negative numbers can be defined.