Welcome to a mini-series of lessons on the topic Modulus.
Modulus will be one of the most important tools in your high-school math toolbox. It will find its way in almost every domain – algebra, trigonometry, calculus and coordinate geometry. So, learn to use it well and keep it handy.
Definition of Modulus
The modulus of a number x is represented as |x|. Here’s how we define it:
\( |x| = \begin{cases}
x& \text{ if } x\ge 0 \\
-x& \text{ if } x < 0
\end{cases} \)
Let’s evaluate |5|, |–6| and |0|.
(i) |5|
Here, x = 5 and x > 0. Therefore, |x| = x = 5.
(ii) |–6|
Here, x = –6 and x < 0. Therefore, |x| = –x = –(–6) or 6.
(iii) |0|
Here, x = 0. Therefore, |x| = 0.
|x| is also sometimes represented by abs(x), denoting the absolute value of x.
On the number line, |x| denotes the distance of the x from 0.
Therefore, |5| is the distance of 5 from 0, which equals 5.
And, |–6| is the distance of –6 from 0, which is 6.
Finally, |0| is the distance of 0 from 0, which is 0.
I’ll end this lesson with a few simple properties of modulus.
Properties of Modulus
1. |x| is always non-negative, i.e. |x| ≥ 0, for all values of x. This is because |x| denotes distance, something that is always non-negative.
This also follows from the definition. When x ≥ 0, |x| = x, which makes |x| ≥ 0. When x < 0, |x| = –x. Hence, |x| > 0 in this case too.
2. The only value of x for which |x| = 0 is 0. Mathematically, |x| = 0 iff x = 0.
3. |x| = |–x| for all values of x. This follows from its definition again.
See you in the next lesson where I’ll cover a few examples relating to modulus.