Modulus

You may or may not be aware of what modulus is. If you have come across the modulus (absolute value) of a number before it is likely to only be of real numbers. The notation for the modulus of x is, |x|.

If the number is just a real number then calculating the modulus is incredibly easy and requires no thought. Essentially it is just 'taking the positive value' of the number, for example: |5| is simply 5, and |-3| = 3. It follows that any rational or irrational real number is just the positive value of it. The definition of |x| for any real value of x therefore follows that |x| = √(x²).

Another, potentially easier way to think of the modulus of a number, is to think of the numbers distance from 0 on a number line. -5 is 5 units away from 0 and 5 is also 5 units away from 0. This thought is useful when it comes to finding the modulus of a complex number.

Complex numbers can also be represented (in a way) on a number line, they are represented in a two-dimensional complex plane. A complex number, 5+3i will be 5 units in the x direction and 3 units in the y direction, therefore the general form of a complex number is x+iy.

As I said previously, to find the modulus of a complex number it helps if we think of our previous definition of |x| being the distance from 0 on a number line (or a complex plane, in this case). If we think of it in this manner then simple co-ordinate geometry states that the distance from 0 to x+iy is √[(x-0)²+(y-0)²], or simply the general definition of |x+iy| = √(x²+y²).

|x| = √(x²) when x is a real number, and this able to be proved from our new definition of |x+iy|. |x+iy| = √(x²+y²), and when x+iy is a real number, y has to equal 0, plugging this into the equation we get |x+i0| = √(x²+0²), therefore |x| = √(x²), proved.

If any of this was explained poorly or just went straight over your head please comment and I will do everything I can to make it understandable. Also if you would like more information on imaginary numbers please visit an older post of mine here.