Absolute Value

An introduction to the concept of Absolute value

In one word, Absolute value is “distance”.

Absolute value of a number is the distance of the number from zero.

Now, guess smart, what can the term Absolute value of a number signify?

 A number without a sign!

Yes, you guessed it right! Why so?

Because it’s distance and distance is just a number, neither positive nor negative (i.e., a number without a + or — sign).

Whether it is +3 or -3, the absolute value of both the numbers is just 3.

And again, since distance is never expressed in negative terms, therefore the absolute value of a number is never negative
(You do not say your school is - 3 miles away from your home, do you?)

(Not even positive for that matter, do you feel distance is positive, or just a number?)

Denotation of Absolute value of a number:

The vertical bars |   | are used to denote the absolute value of a number.

For example,

| 5 | = 5 and | — 5 | = 5,

i.e. the absolute value of both the numbers 5 (i.e. +5) and —5 is 5.

Note:

— | 5 | = — 5, because the negative sign is outside of the bars.

But, | —5 |≠ —5, because the negative sign is inside the bars.

Representing Absolute Value of a number on a Number Line

On the Number line above, the two numbers 5 and -5 are both a distance of 5 from zero.

This distance from zero is what Absolute value of a number denotes.

Absolute value of a number is, therefore, the geometrical concept of distance; distance of a number from zero.

Example 1:

Evaluate the following:

1. | 4 | 2. | - 4 |   3.  | -2 | + | -5 |     4. | -9 | - | 10 |     5. | 2 | × | -3 |

6. | 5 |/| -6 | 7. | 0 |

Answers:

1.  | 4 | is the distance of 4 from zero, therefore | 4 |    = 4

2.  | -4 | is distance of -4 from zero, therefore | - 4 |     = 4

3.  | -2 | + | -5 |     = 2 + 5 = 7

4.  | -9 | - | 10 |     = 9 - 10 = -1

5. | 2 | × | -3 | = 2 × 3 = 6

6. | 5 |/| -6 | = 5/6

7. | 0 | = 0

Important Note:

1. Why is | 0 | = 0?

Recall what absolute value of a number signifies?

It is distance of the number from 0.

As a consequence, what should be the distance of Zero from Zero?

Of course, 0!

2. Very important note:

As a convention, √x stands for the non-negative root of the number x.

By the same vein, √x² denotes the non-negative square root of x²

Now, since | x | is non-negative (≥0), it becomes, therefore, possible to write:

√x² = | x |

Example 2:

Is the question below true or false?

| — 6 | = — 6

Answer:  False

Example 3:

Evaluate || — 6 ||

Solution:

Move out from inside: | — 6 | = 6

 || — 6 || = | 6 | = 6

Example 4:

Is | 2 — x | = | x — 2 |?

Answer: Yes

First of all, we can write | 2 — x | as |— (x — 2) | {since, 2 —x = —(x—2)}

Now, as an example, since | — 6 | = | 6 | = 6, therefore,

| 2 — x | = | (x — 2) |

Finally, | 2 — x | = | (x — 2) | = x — 2

The Algebraic definition of Absolute value (A.V.) of any real number x

| x | = x , if x ≥ 0, (i.e., x is non-negative number) and

| x | = — x, if x < 0 (i.e. if x is negative number)

At the outset, let there not be the question or the doubt as to how | x | can ever be negative.

For, though the above definition does indeed create such a doubt, but it is only apparently, not actually as the following explanation will clarify:

1. What is | x |, if x is 3?

Since x is 3, and as 3 is a positive number (i.e. 3 > 0), therefore, the first part of the definition of A.V is used to write | 3 |

| 3 | = 3

2. What is | x |, if x is —3?

This time, since x is —3, and as —3 < 0, i.e. — 3 is a negative number, therefore, the second part of the definition of A.V. is used to write | —3 |

| —3 | = — (—3) = 3

Therefore, | 3 | = | —3 | = 3

Note:

| —3 | ≠ —3, since the absolute value of a number can never be a negative number.

Why?  You know it, because absolute value of a number, from its definition, is the distance of a number from zero, and distance is never expressed as a negative number.

Note:

If | x | = a, then the equation has two solutions, i.e., there can exist two values of x which satisfy the absolute value equation.

And they are:

x = a or x = —a

For example, the absolute value equation | x | = 5, has two solutions:

One x = 5 and the other x = —5

Important:

x is called the argument of the absolute value | x |

Again, note the following

1. If | x |= | a |, then x = ± a

For example, if | x |= | 3 |, then x = ± 3, i.e. x = 3 or x = —3

Problems:

1. Solve for x: | x — 2| = 5

Solution:

The argument x — 2 can be either 5 or —5, so

x — 2 = 5 or x — 2 = —5, i.e.

x = 2 + 5 or x = 2 — 5

So, x = 7 or — 3

Important:

What is the geometrical meaning of | x— 2 | = 5?

|x — 2 | = 5 represents the distance of a number x from 2 is 5