1.3 Complex Numbers

Complex Numbers

 
A Complex Number

A Complex Number is a combination of a
Real Number and an Imaginary Number

Real Numbers are numbers like:

1

12.38

−0.8625

3/4

√2

1998

Nearly any number you can think of is a Real Number!

Imaginary Numbers when squared give a negative result.

Normally this doesn't happen, because:

• when we square a positive number we get a positive result, and
• when we square a negative number we also get a positive result (because a negative times a negative gives a positive), for example −2 × −2 = +4

But just imagine such numbers exist, because we will need them.

The "unit" imaginary number (like 1 for Real Numbers) is i, which is the square root of −1

Because when we square i we get −1

i² = −1

Examples of Imaginary Numbers:

3i

1.04i

−2.8i

3i/4

(√2)i

1998i

And we keep that little "i" there to remind us we need to multiply by √−1

Complex Numbers

A Complex Number is a combination of a Real Number and an Imaginary Number:

Examples:

1 + i

39 + 3i

0.8 − 2.2i

−2 + πi

√2 + i/2

Can a Number be a Combination of Two Numbers?

Can we make up a number from two other numbers? Sure we can!

We do it with fractions all the time. The fraction ³/₈ is a number made up of a 3 and an 8. We know it means "3 of 8 equal parts".

Well, a Complex Number is just two numbers added together (a Real and an Imaginary Number).

Either Part Can Be Zero

So, a Complex Number has a real part and an imaginary part.

But either part can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers.

Complex Number	Real Part	Imaginary Part
3 + 2i	3	2
5	5	0
−6i	0	−6

Complicated?

Complex does not mean complicated.

It means the two types of numbers, real and imaginary, together form a complex, just like a building complex (buildings joined together).

A Visual Explanation

You know how the number line goes left-right?

Well let's have the imaginary numbers go up-down:

And we get the Complex Plane

And a complex number can now be shown as a point:

The complex number 3 + 4i

Adding

To add two complex numbers we add each part separately:

(a+bi) + (c+di) = (a+c) + (b+d)i

Example: add the complex numbers 3 + 2i and 1 + 7i

• add the real numbers, and
• add the imaginary numbers:

(3 + 2i) + (1 + 7i)
= 3 + 1 + (2 + 7)i
= (4 + 9i)

Let's try one visually:

Example: add the complex numbers 3 + 5i and 4 − 3i

(3 + 5i) + (4 − 3i)
= 3 + 4 + (5 − 3)i
= 7 + 2i

Multiplying

To multiply complex numbers:

Each part of the first complex number gets multiplied by
each part of the second complex number

Just use "FOIL", which stands for "Firsts, Outers, Inners, Lasts" 

	Firsts: a × c Outers: a × di Inners: bi × c Lasts: bi × di
(a+bi)(c+di) = ac + adi + bci + bdi2

Like this:

Example: (3 + 2i)(1 + 7i)

(3 + 2i)(1 + 7i)		= 3×1 + 3×7i + 2i×1+ 2i×7i
		= 3 + 21i + 2i + 14i2
		= 3 + 21i + 2i − 14	(because i2 = −1)
		= −11 + 23i

And this:

Example: (1 + i)²

(1 + i)2 = (1 + i)(1 + i)		= 1×1 + 1×i + 1×i + i2
		= 1 + 2i - 1	(because i2 = −1)
		= 0 + 2i

But There is a Quicker Way!

Use this rule:

(a+bi)(c+di) = (ac−bd) + (ad+bc)i

Example: (3 + 2i)(1 + 7i) = (3×1 − 2×7) + (3×7 + 2×1)i = −11 + 23i

Why Does That Rule Work?

It is just the "FOIL" method after a little work:

(a+bi)(c+di)	=	ac + adi + bci + bdi2		FOIL method
	=	ac + adi + bci − bd		(because i2 = −1)
	=	(ac − bd) + (ad + bc)i		(gathering like terms)

And there we have the (ac − bd) + (ad + bc)i  pattern.

This rule is certainly faster, but if you forget it, just remember the FOIL method.

Let us try i²

Just for fun, let's use the method to calculate i²

Example: i²

i can also be written with a real and imaginary part as 0 + i

i2 = (0 + i)2		= (0 + i)(0 + i)
		= (0×0 − 1×1) + (0×1 + 1×0)i
		= −1 + 0i
		= −1

And that agrees nicely with the definition that i² = −1

So it all works wonderfully!

Conjugates

We will need to know about conjugates in a minute!

A conjugate is where we change the sign in the middle like this:

A conjugate is often written with a bar over it:

Example:

5 − 3i   =   5 + 3i

Dividing

The conjugate is used to help complex division.

The trick is to multiply both top and bottom by the conjugate of the bottom.

Example: Do this Division:

	2 + 3i
	4 − 5i

Multiply top and bottom by the conjugate of 4 − 5i :

	2 + 3i	×	4 + 5i	=	8 + 10i + 12i + 15i2
	4 − 5i		4 + 5i		16 + 20i − 20i − 25i2

Now remember that i² = −1, so:

	=	8 + 10i + 12i − 15
		16 + 20i − 20i + 25

Add Like Terms (and notice how on the bottom 20i − 20i cancels out!):

	=	−7 + 22i
		41

We should then put the answer back into a + bi form:

	=	−7	+	22	i
		41		41

DONE!

Yes, there is a bit of calculation to do. But it can be done.

Multiplying By the Conjugate

There is a faster way though.

In the previous example, what happened on the bottom was interesting:

(4 − 5i)(4 + 5i) = 16 + 20i − 20i − 25i²

The middle terms cancel out!
And since i² = −1 we ended up with this:

(4 − 5i)(4 + 5i) = 4² + 5²

Which is really quite a simple result

In fact we can write a general rule like this:

(a + bi)(a − bi) = a² + b²

So that can save us time when do division, like this:

Example: Let's try this again

	2 + 3i
	4 − 5i

Multiply top and bottom by the conjugate of 4 − 5i :

	2 + 3i	×	4 + 5i	=	8 + 10i + 12i + 15i2
	4 − 5i		4 + 5i		16 + 25
				=	−7 + 22i 41

And then back into a + bi form:

	=	−7	+	22	i
		41		41

DONE!

HOTS QUESTIONS !

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