Complex Numbers

From Wikiversity

Jump to: navigation, search

Complex Numbers

Subject classification: this is a mathematics resource .

Educational level: this is a tertiary (university) resource.

Completion status: this resource has been started but most of the work is still to be done.

Complex numbers arise from dealing with the square root of negative numbers, such as the solutions to $x^2 = -1\,$ , which 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 $\mathbf{z}$ is the sum of a real part $a\,$ and an imaginary part $b\,$ ,
i.e. $\mathbf{z} = a + \sqrt{-1}b = a + ib$ .

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, the equvalent to the x axis on the cartesian plane (where $\operatorname{Im}(z)=0$ ), and the other an imaginary axis, the equvalent to the y axis on the cartesian plane (where $\operatorname{Re}(z)=0$ ).

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

[edit] Basics

Imagine the equation $x^2=-1\,$

What are the roots?

There is no number that multiplies by itself to give -1, so we make up a number called the imaginary number,i so that

$\,i=\sqrt{-1}$ ,

(letting i = the "positive" root-though in fact it is irrelevant as this is defined as such)

So in our second equation, $\,x=i$ or $\,x=-i$

And also any negative square root can now be found, for example:

$\,x=\sqrt{-64}$

$\,x=\sqrt{64}\times\sqrt{-1}$

$\,x=8\times\sqrt{-1}$

so $\,x=8i$

Any number that is a multiple of i is an imaginary number.

But, if $\,a=i$ , and $\,b=10$ , what does $\,a+b$ equal?

So we say $\,z = a+b$

and $\,z = 10+i$

so z is a complex number, that is a number that has a real part and an imaginary part.

[edit] Manipulation

In this section we will learn how to use basic maths on complex numbers.

[edit] Addition and Subtraction

Here we treat a complex number like any other piece of algebra. We add the real parts and the imaginary parts seperately, just like numbers and constants:
$(a+ib) ~+~ (c+id) = ((a+c) + i(b+d))$
and
$(a+ib)~-~(c+id) ~+~ (e+if) = ( ( a-c+e ) ~+~ i( b-d+f ) )$

[edit] Example

We know that $\,(3 + 7x) + (1 + x) = 4 + 8x$

In the same way $\,(3 + 7i) + (1 + i) = 4 + 8i$

The same is for subtraction , if $\,z=4+3i$

then $\,z-2i = 4+i$

[edit] Multiplication

The same procedure to multiply polynomials is applicable to multiplying complex numbers: term by term. Simply keep in mind that:
$i * i ~=~ i^2 ~=~ \sqrt{-1} ~ * ~ \sqrt{-1} ~=~-1$

So:
$(a + ib)~*~(c + id) ~~=~~ ac ~+~ iad ~+~ ibc ~+~ i^2bd$

Group the terms (remember: $i^2=-1\,$ )

$~~=~~ ac ~+~ iad ~+~ ibc ~+~ (-1)bd$

$~~=~~ ac ~+~ iad ~+~ ibc ~-~ bd$

$~~=~~ ac ~-~ bd ~+~ iad ~+~ ibc$

$~~=~~ (ac - bd)~+~i(ad + bc)$

[edit] Scalar Multiplication

Multiplying by a real number (w:scalar), again uses the same rules as algebra.

So that $\,3\times(1+i) = 3+3i$

[edit] Example

We know that $\,(2+3x)\times(1+x) = 2 + 5x + 3x^2$

So to $\,(2+3i)\times(1+i) = 2 + 5i + 3i^2$

Group the terms (remember: $i^2=-1\,$ )

$\,~~=~~ 2 + 5i + 3i^2$

$\,~~=~~ 2 + 5i - 3$

$\,~~=~~ -1 + 5i$

[edit] Division and the "Complex Conjugate"

We do not have a way to divide by a complex number, so we need to elminate any complex numbers we have in the denominator. First step is to use the above methods (addition, subtraction and multiplication to get the denominator into one complex number, $\,a+bi$ .

We know that $\,(2+i)(2-i) = 4+2i-2i-i^2 = 4-(-1)= 5$

So too in general $\,(a+bi)(a-bi) = a^2+abi-abi-(b^2)(i^2) = a^2 + b^2$

That is to divide, we can get rid of a complex number a+bi on the denominator by multiplying it by a-bi numerator and denominator.

If : $z=a+ib \,$ then : $\,a-ib=\overline(z)$ which is the complex conjugate.

So the rule is:

Multiply numerator and denominator by the complex conjugate of the denominator, and then result can be simplfied.

e.g. $\frac{\,4i}{\,1+i} = \frac{\,4i\times(1-i)}{\,(1+i)\times(1-i)} = \frac{\,4i - 4i^2}{\,1-i^2} =\frac{\,4+ 4i}{\,2}= \,2+2i$

[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,

$(r,\theta)\,$

Euler's formula shown graphically when r = 1

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,

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

When r = 1 this is known as w:Euler's formula.^[2]^[3]

$e^{i\theta} = \cos \theta + i\sin \theta \!$

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

[edit] Complex Plane

It is useful to be able to represent complex numbers geometrically, just as we are able to represent real numbers geometrically on the number line. Moreover, we don’t need any genuinely new ideas to do this.

We know that each pair of real numbers (x, y) corresponds uniquely to a point P(x, y) in the cartesian plane.
To represent complex numbers geometrically, we use the same representation: a complex number z = x+iy is represented by the point P(x, y) in the plane, as in the following diagram:
When the plane is used to represent complex numbers in this way, it is called the Complex Plane or the Argand Diagram.

[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] Basics

[edit] Manipulation

[edit] Addition and Subtraction

[edit] Example

[edit] Multiplication

[edit] Scalar Multiplication

[edit] Example

[edit] Division and the "Complex Conjugate"

[edit] Polar Notation

[edit] Complex Plane

[edit] References

[edit] See Also

Views

Personal tools

Search

Navigation

Community

Toolbox

In other languages