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Complex numbers are represented in the form 'a + ib', where 'a' is the real part and 'b' is the imaginary part. Adding and subtracting complex numbers involves adding or subtracting the real parts separately from the imaginary parts. Multiplying complex numbers follows standard multiplication rules, while dividing complex numbers requires multiplying both the numerator and denominator by the complex conjugate of the denominator to make it real.

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Complex numbers: real and imaginary parts

Every complex number has the following shape: a+ib.

The first part of that complex number is real: the real constant a. The second part is

imaginary: the real constant b multiplied by i.

The constant a is referred to as "the real part", not too controversially, and the constant b

is referred to as "the imaginary part".

NOTICE: the imaginary part is just the constant b, not ib.

Adding and subtracting complex numbers

Add

If you know how to add and subtract real numbers then you can add and subtract complex

numbers. All you have to remember is to keep the real parts and the imaginary parts

separate. Here's what I mean.

Suppose we want to add the two complex numbers 2+3i and 4-i. What do you think the

answer will be? Adding 2+3i and 4-i

First look at the real parts of each of the complex numbers.

The first complex number, 2+3i, has real part 2.

The second complex number, 4-i, has real part 4.

Add the real parts together, to get the real part of the result.

So the result has real part 6.

Next, look at the imaginary parts of each complex number.

The first complex number, 2+3i, has imaginary part 3. (NOT 3i)

The second complex number, 4-i, has imaginary part -1. (NOT -i)

Add the imaginary parts together, to get the imaginary part of the result.

So the result has imaginary part 2.

Thus our result is: (2+3i)+(4-i)=6+2i.

Every complex number has the following shape: a+ib.

The first part of that complex number is real: the real constant a. The second part is

imaginary: the real constant b multiplied by i.

The constant a is referred to as "the real part", not too controversially, and the constant b

is referred to as "the imaginary part".

NOTICE: the imaginary part is just the constant b, not ib.

Adding and subtracting complex numbers

Add

If you know how to add and subtract real numbers then you can add and subtract complex

numbers. All you have to remember is to keep the real parts and the imaginary parts

separate. Here's what I mean.

Suppose we want to add the two complex numbers 2+3i and 4-i. What do you think the

answer will be? Adding 2+3i and 4-i

First look at the real parts of each of the complex numbers.

The first complex number, 2+3i, has real part 2.

The second complex number, 4-i, has real part 4.

Add the real parts together, to get the real part of the result.

So the result has real part 6.

Next, look at the imaginary parts of each complex number.

The first complex number, 2+3i, has imaginary part 3. (NOT 3i)

The second complex number, 4-i, has imaginary part -1. (NOT -i)

Add the imaginary parts together, to get the imaginary part of the result.

So the result has imaginary part 2.

Thus our result is: (2+3i)+(4-i)=6+2i.

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Subtraction

Similarly, we can subtract two complex numbers by dealing with their real parts and

imaginary parts separately. For example, what's 2+3i minus 4-i? Write down what you

think it should be

2+3i minus 4-i

By subtracting the real part of the second complex number from the real part of the first,

we get the real part of the result to be 2-4=-2.

By subtracting the imaginary part of the second complex number from the imaginary part

of the first, we get the imaginary part of the result to be 3-(-1)=4.

So the full result is: (2+3i)-(4-i)=-2+4i.

Multiplying complex numbers

Suppose we have the two complex numbers: 2+3i and 1+4i and we want to multiply them

together. If I write them like this: (2+3i)(1+4i)

(2+3i)(1+4i)

We multiply them by multiplying out the brackets as follows.

For each of the parts in the first brackets, we multiply by each of the parts in the second

brackets. This gives: (2+3i)(1+4i)=2x1+2x4i+3ix1+3ix4i.

Now we can simplify that: (2+3i)(1+4i)=2+8i+3i-12. (NOTE: the "-12" at the end is

negative because we have i2, which is -1.)

Simplifying still more, by gathering together all the real parts and all the imaginary parts,

we obtain the final result: (2+3i)(1+4i)=-10+11i.

Division of complex numbers

We have looked at adding, subtracting and multiplying complex numbers, so naturally we

now turn to dividing complex numbers. This is slightly less straightforward than the other

operations.

Similarly, we can subtract two complex numbers by dealing with their real parts and

imaginary parts separately. For example, what's 2+3i minus 4-i? Write down what you

think it should be

2+3i minus 4-i

By subtracting the real part of the second complex number from the real part of the first,

we get the real part of the result to be 2-4=-2.

By subtracting the imaginary part of the second complex number from the imaginary part

of the first, we get the imaginary part of the result to be 3-(-1)=4.

So the full result is: (2+3i)-(4-i)=-2+4i.

Multiplying complex numbers

Suppose we have the two complex numbers: 2+3i and 1+4i and we want to multiply them

together. If I write them like this: (2+3i)(1+4i)

(2+3i)(1+4i)

We multiply them by multiplying out the brackets as follows.

For each of the parts in the first brackets, we multiply by each of the parts in the second

brackets. This gives: (2+3i)(1+4i)=2x1+2x4i+3ix1+3ix4i.

Now we can simplify that: (2+3i)(1+4i)=2+8i+3i-12. (NOTE: the "-12" at the end is

negative because we have i2, which is -1.)

Simplifying still more, by gathering together all the real parts and all the imaginary parts,

we obtain the final result: (2+3i)(1+4i)=-10+11i.

Division of complex numbers

We have looked at adding, subtracting and multiplying complex numbers, so naturally we

now turn to dividing complex numbers. This is slightly less straightforward than the other

operations.

Suppose we have two complex numbers: 2+6i and 4+i, and we want to know their ratio:

(2+6i)/(4+i). It is not obvious how we can divide by the complex number 4+i.

It would be easy to do the division if the number on the bottom were real, for instance

(2+6i)/2 is simply 1+3i: we divide the real part by 2 and we divide the imaginary part by

2. But what if the number on the bottom is not real?

The solution to this question lies with complex conjugates. Recall that when you multiply

a complex number by its complex conjugate the result is REAL. So ....... if we multiply

the bottom number of our division by its complex conjugate, we'll end up with something

real on the bottom, and then we can do the division.

Of course, if we multiply the bottom of the ratio by anything then we have to multiply the

top by the same thing, otherwise we're changing the value of the whole expression.

So let's try that plan with our example: (2+6i)/(4+i).

First, write down the complex conjugate of 4+i. That's 4-i.

Second multiply both the top and bottom by that number.

The bottom: (4+i)(4-i)=16-4i+4i+1 = 17. (real as expected).

The top: (2+6i)(4-i)=8-2i+24i+6 = 14+22i.

Third, carry out the division, now that the bottom is real.

The ratio is now: (14+22i)/17, so that's just 14/17 + 22i/17.

So that's the answer.

If we make our numbers more general, we can write down a general formula

(a+ib)/(c+id)=(ac+bd)+i(bc-ad))/(c2+d2)

(2+6i)/(4+i). It is not obvious how we can divide by the complex number 4+i.

It would be easy to do the division if the number on the bottom were real, for instance

(2+6i)/2 is simply 1+3i: we divide the real part by 2 and we divide the imaginary part by

2. But what if the number on the bottom is not real?

The solution to this question lies with complex conjugates. Recall that when you multiply

a complex number by its complex conjugate the result is REAL. So ....... if we multiply

the bottom number of our division by its complex conjugate, we'll end up with something

real on the bottom, and then we can do the division.

Of course, if we multiply the bottom of the ratio by anything then we have to multiply the

top by the same thing, otherwise we're changing the value of the whole expression.

So let's try that plan with our example: (2+6i)/(4+i).

First, write down the complex conjugate of 4+i. That's 4-i.

Second multiply both the top and bottom by that number.

The bottom: (4+i)(4-i)=16-4i+4i+1 = 17. (real as expected).

The top: (2+6i)(4-i)=8-2i+24i+6 = 14+22i.

Third, carry out the division, now that the bottom is real.

The ratio is now: (14+22i)/17, so that's just 14/17 + 22i/17.

So that's the answer.

If we make our numbers more general, we can write down a general formula

(a+ib)/(c+id)=(ac+bd)+i(bc-ad))/(c2+d2)

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