Multiplying By the Conjugate

Complex Numbers

A Complex Number is a combination of:

a Real Number

Real Numbers are just numbers like:

Nearly any number you can think of is a Real Number

and an Imaginary Number

Imaginary Numbers are special because:

When squared, they give a negative result.

Normally this doesn't happen, because:

• when you square a positive number you get a positive result, and
• when you square a negative number you also get a positive result (because a negative times a negative gives a positive)

But just imagine there is such a number, because we are going to need it!

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

(Read Imaginary Numbers to find out more.)

A Combination

So we have this definition:

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

Examples:

Can a Number be a Combination of Two Numbers?

Can you make up a number from two other numbers? Sure you can! You 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.

Complicated?

Complex does not mean complicated.

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

Adding

To add two complex numbers we add each element separately:

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

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

Multiplying

To multiply complex numbers:

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

Just use "FOIL", which stands for "Firsts, Outers, Inners, Lasts" (see Binomial Multiplication for more details):

	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)

Here is another example:

Example: (1 + i)²

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:

And there you 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

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

So it all makes sense!

Conjugates

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

A conjugate is often written with a bar over it:

Example:

Dividing

The conjugate is used to help division.

The trick is to multiply both top and bottomby theconjugate of the bottom.

Example: Do this Division:

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

Now remember that i² = -1, so:

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

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

DONE!

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

Multiplying By the Conjugate

You can save yourself a little bit of time, though.

In that 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

Remember that when you do division ... it will save you time.

So we could have done it like this:

Example: Do this Division:

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

And then back into a + bi form:

DONE!

Matrices

A Matrix is an array of numbers:

A Matrix
(This one has 2 Rows and 3 Columns)

We talk about one matrix, or several matrices.

There are many things you can do with them ...

Adding

To add two matrices, just add the numbers in the matching positions:

These are the calculations:

The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size.

Example: a matrix with 3 rows and 5 columns can be added to another matrix of 3 rows and 5 columns.

But it could not be added to a matrix with 3 rows and 4 columns (the columns don't match in size)

Negative

The negative of a matrix is also simple:

These are the calculations:

Subtracting

To subtract two matrices, just subtract the numbers in the matching positions:

These are the calculations:

Note: subtracting is actually defined as the addition of a negative matrix: A + (-B)

Date: 2015-12-11; view: 1164