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[edit] Introduction

Complex numbers arise from dealing with the square root of negative numbers, such as the solutions to $x 2 = - 1$ . Here the solutions are $x=\pm\sqrt{-1}$ . The foundation of complex number theory is the definition $i = \sqrt{-1}$ , where $i$ is referred to as an 'imaginary number'. A complex number $z$ is the sum of a real part $a$ and an imaginary part $b$ , i.e. $z = a + \sqrt{-1}b = a + ib$ .

If $b = 0$ then $z\in\mathbb{R}$ . That is, $z$ is a real number, and can be called 'pure real'. Conversely, if $a = 0$ then $z\in\mathbb{I}$ , and $z$ is 'pure imaginary'. We can refer specifically to the real and imaginary parts ( $a$ and $b$ ) of $z$ respectively as follows: $\operatorname{Re}(z) \equiv a$ and $\operatorname{Im}(z) \equiv b$ .

Whilst the real numbers are readily visualised by considering a straight "number line", complex numbers are best seen as being positions on a plane. This plane would have one axis considered to be the real axis (where $\operatorname{Im}(z)=0$ ), and the other an imaginary axis (where $\operatorname{Re}(z)=0$ ).

A typical example of complex numbers is the quadratic formula for finding the roots of a second order polynomial: $a x 2 + b x + c = 0$ . The roots are given by the well known formula: $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ^[1].

[edit] Polar Notation

We have already seen the cartesian representation of complex numbers, expressed in terms of real and imaginary parts. Alternatively, we can represent a complex number by a magnitude and direction, $\mathbf{r}$ and $\mathbf\theta$ . In this representation a complex number z can be represented as,

$\mathbf{z}=(r,\theta)$

where,

${r}=\sqrt{Re(z)^2+Im(z)^2}\ (=\lVert{z}\rVert)$

$\theta=\tan^{-1}\frac{Im(z)}{Re(z)}$

This also allows one to write,

$\mathbf{z}=r(\cos\theta+i\sin\theta)\Leftrightarrow\mathbf{z}=re^{i\theta}$

This is known as Euler's formula.^[2]^[3]

Note:To represent a complex number graphically, simply draw a vector on the X - Y plane with an X intercept $\mathbf r\cos\theta$ and a Y intercept $\mathbf r\sin\theta$ .

[edit] Adding/Subtracting

Adding and subtraction is simple, just add/subtract the reals and add/subtract the imaginaries.
$(a+ib) ~+~ (c+id) = ((a+c) + i(b+d))$
and
$(a+ib)~-~(c+id) ~+~ (e+if) = ( ( a-c+e ) ~+~ i( b-d+f ) )$

[edit] Multiplication

The same procedure to multiply polynomials is applicable to multiplying complex numbers: term by term. Simply keep in mind that: $i * i ~=~ \sqrt{-1} ~ * ~ \sqrt{-1} ~=~-1$
So:
$(a + ib)~ * ~(c + id) ~~=~~ ac ~+~ iad ~+~ ibc ~+~ iibd$
Group the terms (remember: i*i = -1)
$(ac - bd) ~+~ i(ad + bc)$

[edit] References

[edit] See Also

Complex Numbers.

[0] Quadratic Equation.

[1] Euler's formula.

[2] Taylor series.

[1]

[2]

[3]

Complex numbers

From Wikiversity

Contents

[edit] Introduction

[edit] Polar Notation

[edit] Adding/Subtracting

[edit] Multiplication

[edit] References

[edit] See Also

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