Absolute Value Formulas You Should Know - BYJUS

# Absolute Value Formulas

The absolute value of a number is a magnitude of that number which explains the distance of the number from 0 on a number line. Absolute value of any number is expressed using the symbol | | and pronounced as “modulus”. Here we will focus on finding absolute values.   The absolute value of a number is defined as the distance of the number from the origin on a number line. Since the absolute value is a distance and as we know that the distance is not negative hence the absolute value of any number is always a positive number.     In the above figure the distance of - 3 from the origin(0) is 3 units which is the same as the distance of 3 from the origin, therefore the absolute value of - 3 is equal to the absolute value of 3.   |- 3| = 3...Read MoreRead Less

### Formula to find the absolute value of a number

Consider a number a, which is plotted on the number line below. Absolute value of a, |a| = a – 0 = a

### Rapid Recall

• The absolute value of a written as |a| or abs(a)
• Absolute value of a is pronounced as “mod a” or “modulus of a”
• The absolute value of any number is the magnitude of the given number, excluding the sign.
• Absolute value of zero is zero.

### Solved Examples

Example 1: Find the value of |8 – 4|.

Solution : |8 – 4| = |4| = 4

Example 2: Compare 3 and |-4|.

Solution: Graph 3 and |-4| on the number line. |-4| = 4

Here 3 is to the left of 4

Hence, 3 < 4

So, 3 < |-4|.

Example 3: Compare -2  and |2|.

Solution: Graph -2 and |2| on the number line. 2 = 2

Here – 2 is to the left of 2

Hence – 2 < 2

So,– 2< |2|.

Example 4: Order the values from least to greatest – 2, 4, |- 3|, 0 and onwards |2|.

Solution: Graph – 2, 4, |- 3|, 0 and |2| on the number line. – 3 = 3

2 = 2

Numbers that lie to the left are lesser than the numbers that lie to the right.

Hence the arrangement is – 2 < 0 < |2| < |- 3| < 4.

Example 5: Order the values from greatest to least -|4|,- 1,|5|,- |- 2| and 3.

Solution: Graph – |4|,- 1,|5|,- |- 2| and 3 on the number line. – |4|= – 4

|-2| = – 2

|5| = 5

Numbers that lie to the left are lesser than the numbers that lie to the right.

Hence the arrangement is – |4|< -|- 2| < – 1 < 3 < |5|.

Example 6: Jacob was asked by his teacher, “Which number has an absolute value of 4 that lies to the left of the origin?” Can you help him solve the problem?

Solution:  Negative numbers lie to the left of origin.

Since |- 4| = 4

So, – 4 is a number that lies to the left of the origin and whose absolute value is 4.

Example 7: A person inside the submarine observes marine animals with the help of a device.  The respective positions of animals from the submarine were found and plotted in the table below. Animals to the right of the submarine will have a positive position and those to the left will have a negative position.

1. Which animal is farthest from the submarine?
2. Which animal is closest to the submarine? Solution : First let’s graph each elevation. The position of the submarine is 0, it is given that the positions of the animals to the right are taken as positive and to the left as negative. Hence to find their distances from the submarine, we need to find the absolute values

|- 80| = 80

|- 20| = 20

50 = 50

20 = 20

So the distances from the submarine when arranged from the least to the greatest: 20 < 30 < 50 < 80

1.  The maximum distance of an animal from a submarine is 80 feet. So, the Shrimp is farthest from the submarine.

2.  The least distance of an animal from a submarine is 20 feet. So, the sea lion is closest to the submarine.

As we know that |- x| = x, so the absolute value of any negative number is a positive number.

The Absolute Value of a number is defined as the distance of number from origin. Distance is a non-negative quantity, hence the Absolute value of any number is always a positive number.

The distance of 0 from origin(0) is 0 so the Absolute Value of 0 is 0.

For example – 3 and 3 are two different numbers but |- 3| = 3 and |3| = 3. So, yes two different numbers have the same Absolute Value.

The absolute value of fractions is the distance of the fraction from the origin. Let’s understand with an example,

$$\left|\frac{-7}{8}\right|=0-(\frac{-7}{8})=\frac{7}{8}$$

Hence, $$\left|\frac{-7}{8}\right|=\frac{7}{8}$$