1.2 Complex Numbers

A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit. 

• The real number a is called the real part of the complex number a + bi; the real number b is called the imaginary part of a + bi.
• By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part.
• The real part of a complex number z is denoted by Re(z) or ℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). 

For example,

${\displaystyle {\begin{aligned}\operatorname {Re} (-3.5+2i)&=-3.5\\\operatorname {Im} (-3.5+2i)&=2.\end{aligned}}}$

Hence, in terms of its real and imaginary parts, a complex number z is equal to ${\displaystyle \operatorname {Re} (z)+\operatorname {Im} (z)\cdot i}$. This expression is sometimes known as the Cartesian form of z.

An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.

When the number line are real numbers (in the real number system):

You've probably been told  that you couldn't do   yet because of that negative sign.  The reason is that the answer isn't a real number.   There is an answer...  and it's imaginary. 

Here it is:

That's an "i" forimaginary number.

So, what's the i?

For the imaginary number system, we just define it this way:

With this, we can do a lot of cool things we couldn't do before.  One of the coolest things in math is made with imaginary numbers:

FRACTALS!

OK, let's go back to

Here's what really happened with this problem:

So...

We can't pop the  so we just pull the i out

A complex number is made up of a real number and an imaginary number.

Here are a few:

The official form is:

If z = a + bi is a complex number, we define the modulus or magnitude or absolute value of z to be (a²+ b²)^1/2. We denote the modulus by |z|. 

• If we represent a complex number by a point in the complex plane, then the modulus is just the distance from the origin to that point. 

• Any complex number other than 0 also determines an angle  with initial side on the positive real axis and terminal side along the line joining the origin and the point. 

• This angle is known as an argument of the complex number z. 

• The argument is not unique since we may use any coterminal angle. 

• However, we will usually state an argument between 0 and 2. 

• If z is real or pure imaginary then modulus and argument are quite simple to find. 

• The modulus is of z = a real is |z| = |a|, the absolute value of a. 

• The modulus is of z = bi pure imaginary is |z| = |b|, the absolute value of b. 

• An argument can be chosen to be a multiple of /2 since the axes are /2 apart.

EXAMPLE 1 :

Find the modulus for an argument of 5, 8i, -3, and -7i.


Solution:

|5| = 5,  = 0 |8i| = 8,  = /2 |−3| = 3,  =  |−7i| = 7,  = 3/2

Here are two examples that demonstrate finding modulus. These are followed by an exercise on modulus. 

Example 2:

Find the modulus of -3+4i and 4-6i.


Solution:

For z = −3 + 4i: The square of the modulus is the sum of the squares of the real and imaginary parts. We will get |z|2 = (−3)2 + 42 = 25. Hence, |z| = 251/2 = 5. For z = 4 − 6i: |z|2 = 42 + (−6)2 = 52. Hence, |z| = 521/2 (approximately 7.211).