In mathematics, absolute value is a value that describes the given number's distance on the number line from 0. The absolute value of a number is always positive even if the number is negative. It is also known as the numerical value or the magnitude of the given number.
In this article, we are going to learn about the meaning of absolute value, the symbol of absolute value, and the properties of absolute value.
|1.||Meaning of Absolute Value|
|2.||Absolute Value Symbol|
|3.||Absolute Value Functions|
|6.||FAQs on Absolute Values|
Meaning of Absolute Value
The absolute value of a number or integer is the distance of the number from zero, in a number line irrespective of the direction. It is the distance from 0 and is expressed as a positive integer. The symbol used to represent an absolute value are vertical bars, i.e., |x| where x is an integer. The distance on the number line from the origin is always a non-negative quantity. For example: the absolute value of 4 is written as |4| and the absolute of -4 is also written as |4|.
The following figure represents the absolute value of the integer 4 from the origin 0 on a number line. Irrespective of the direction, the absolute value of the integer is always positive.
Absolute Value Symbol
To represent the absolute value of a number x (or a variable), we write a vertical bar on either side of the number, i.e., |x|, where x is an integer and is pronounced as Mod x or Modulus of x. The word modulus is a Latin word that means measure. Since the absolute value is the distance of a number from the origin it can show both negative and positive numbers. If the integer is positive then the absolute value will be a positive number. If the number is negative, even then the absolute value of this number will be a positive number. For example, the absolute value of 7 is written as |7| and the absolute value of -7 is written as |-7|. This means that the absolute value of any number can never be negative because the distance is never represented in a negative form. Hence, |7| = 7 and |-7| = 7
The following figure represents how the absolute value is the same for the positive and negative numbers.
Absolute Value Functions
If x is a real number, the absolute value will satisfy the following functions:
| x | = x, if x ≥ 0
| x | = – x, if x < 0
For example: |3| = 3 and |-3| = -(-3) = 3
The graph of the absolute value function is shown below. The v-shaped line represents the distance between both the x-axis and y-axis.
The absolute value function is used to measure the distance between two numbers. Thus, the distance between x and 0 is |x − 0| = |x|, and the distance between x and y are |x − y|.
For example, the distance from −3 to −5 is |−3 − (−5)| = |−3 + 5| = |2| = 2, and the distance from −4 to 7 is |−4 − 7| = |−11| = 11.
Listed below are a few topics related to absolute value.
Example 1: Megan wants to find the values of the following. Can you help her using the definition of absolute value?
b) - |- 3|
c) |2(-3) +4 |
We know that the absolute value of a number is always positive.
a) |-13/15| = 13/15
b) - | - 3| = -3
c) |2(-3) +4 | = |-6 +4| = |-2| = 2
Example 2: Mia is instructed by her teacher to solve the following absolute value equation using the definition of the absolute function. |x - 2| = 4 Can we help her?
The given equation is:
|x - 2| = 4
Using the definition of the absolute value function, we remove the absolute value sign on one side of the equation. We then get \(\pm\) sign on the other side.
x - 2 = \(\pm\)4
This results in two equations, which we solve separately.
x - 2 = 4 which gives x = 6
x - 2 = -4 which gives x = -2
FAQs on Absolute Value
What is an Absolute Value?
The absolute value of a number or integer is the distance of the given number from zero, in a number line, irrespective of the direction. It is always expressed as a positive integer.The symbol used to represent an absolute value are vertical bars, i.e., |x| where x is an integer. For example: the absolute value of 6 is written as |6|, and the absolute value of -6 is also written as |6|.
Why is the Absolute Value Always Positive?
Absolute value means the distance of the number from the origin 0. The number represented on a number line can be negative but the absolute value is always positive since the distance is never negative. For example: the absolute value of -9 is 9 which means that the distance from the number -9 to the origin 0 is 9 units on a number line.
What is that One Basic Rule of an Absolute Value?
The basic rule of an absolute value says that the absolute value of any number is always positive. If the number is a positive integer, the absolute value of the number is positive, i.e., |15| = 15. If the number is a negative integer, even then, the absolute value is positive, i.e., |-20| = 20.
What is the Absolute Value of 10?
The absolute value of 10 is 10 itself. It is written as |10| = 10.
What is the Absolute Value of |-11|?
The absolute value of |-11| is 11 because absolute value is never negative. If the given number is negative, its absolute value is expressed as the number without the negative sign.