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.

Nicely defined and well explained modulus,maths requires practice more than other subjects and in my opinion you should provided some examples of modulus so one can understand it easily.

ReplyDeleteI am delighted that the examples I provided helped you to understand modulus. Hopefully you are able to see the beauty of mathematics.

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