What is the Absolute Value of a Number?

The absolute value of a number means how far that number is from zero on the number line. Questions also uses terms like “magnitude” or “modulus” to represent the absolute…
by Talha Omer, M.Eng., Cornell Grad
in Math Concepts

The absolute value of a number means how far that number is from zero on the number line.

Questions also uses terms like “magnitude” or “modulus” to represent the absolute value of a number.

 

 

“6” is 6 away from zero,
and “?6” is also 6 away from zero.

So the absolute value of 6 is 6,
and the absolute value of ?6 is also 6

This is the same as saying….

 

The magnitude of 6 is 6 The modulus of 6 is 6
The magnitude of ?6 is also 6 The modulus of ?6 is also 6

 

More Examples:

  • The absolute value of ?11 is 11
  • The absolute value of 5 is 5
  • The absolute value of 0 is 0
  • The absolute value of ?142 is 142

In effect, all negative numbers become positive and all non-negative numbers (? 0) remain unaltered.

No Negatives!

You can also imagine the absolute value of a number as the distance of that number from zero on the number line.

Since distance cannot be negative, this means that absolute value is always ? 0, i.e. non-negative.

 

The absolute value of -67 – 33 + 100 is?

Solution:

First, simplify.

-67 – 33 + 100 =

-100 + 100 =

0

The absolute value of 0 is 0.

 

Let’s try another one.

What is the magnitude of ?(69 ? 67)2 ? 100?

  1. ?104
  2. ?96
  3. 0
  4. 96
  5. 104

Solution:

First, we will simplify the expression: ?(69 ? 67)2 ? 100

Using PEMDAS,

?(69 ? 67)2 ? 100

?(2)2 ? 100

?4 ? 100

?104

The magnitude of ?104 is 104.

 

Absolute Value Symbol

To show that we want the absolute value of something, we put “|” marks either side, like |-5| = 5 or |5| = 5.

Revisiting the earlier examples, we can rewrite them using the absolute value symbol as follows:

  • The absolute value of ?11 is 11 ? |-11| = 11
  • The absolute value of 5 is 5 ? |5| = 5
  • The absolute value of 0 is 0 ? |0| = 0
  • The absolute value of ?142 is 142 ? |-142| = 142

Sometimes absolute value is also written as “abs()”, so abs(?1) = 1 is the same as |?1| = 1. However, the questions almost always uses the “|” marks to represent the absolute value operator.

 

|?3×6| = 

Solution:

This question wants you to find the magnitude of the product of ?3 and 6. First, we take the product of ?3 and 6, which is ?18. Then we find the absolute value of ?18, which is 18.

= |?3×6| 

= |?18| 

= 18

 

Is the answer of |8 ? 3| the same as |3 ? 8|?
  • Yes
  • No

Solution:

It doesn’t matter which way around we do a subtraction, the absolute value will always be the same:

|8?3| = |5| = 5
|3?8| = |?5| = 5

 

?|2?5| =
  • 3
  • ?3

Solution:

?|2?5|  = ?|?3|  = ?3

Notice that there is a negative on the outside of absolute value. Since it is on the outside the absolute value, it will not be affected.

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