Absolute Value of Real Numbers

Absolute value of real numbers `a` is called the number itself, if `a>=0` , and opposite number `-a`, if `a<0`. The absolute value of number is denoted by `|a|`.

Definition. Absolute value of a is `|a|={(a if a>=0),(-a if a<0):}` .

For example, `|pi-3|=pi-3`, since `pi-3>0 (pi=3.14...)` ; `|-3.7|=-(-3.7)=3.7`, since `-3.7<0`.

Geometrically `|a|` means distance on the coordinate line of point `a` from point `O` .

Let′s consider the properties of absolute values:

  1. `|a|>=0`
  2. `|a|=|-a|`
  3. `|ab|=|a|*|b|`
  4. `|a/b|=|a/b|, b!=0`
  5. `|a|^2=a^2`

Now let's see what ineaqualities hold for absolute values.

Inequality 1. |a|<b (where, of course, b>0) is equivalent to double inequality -b<a<b.

Proof.

If |a|<b then simultaneously a<b and -a<b, i.e. a>-b. So, -b<a<b.

If -b<a<b then a<b and a>-b, i.e. -a<b. But |a| is either a or -a, so |a|<b.

Similarly can be proved

Inequality 2. `|a|<=b` (where, of course, b>0) is equivalent to double inequality `-b<=a<=b`.

Inequality 3 (Triangle Inequality). `|a+b|<=|a|+|b|`.

Proof.

It is evident that `-|a|<=a<=|a|` and `-|b|<=b<=|b|`. Adding these inequalities we have that `-(|a|+|b|)<=a+b<=|a|+|b|`. This by Inequality 2 is equivalent to `|a+b|<=|a|+|b|`.

This property can be proved with the help of mathematical induction for sum of n numbers: `|a_1+a_2...+a_n|<=|a_1|+|a_2|+...+|a_n|`.

Inequality 4. `|a-b|<=|a|+|b|`.

Proof.

Follows from property 3 if we set -b instead of b: `|a-b|=|a+(-b)|<=|a|+|-b|=|a|+|b|`.

Inequality 5. `|a+b|>=|a|-|b|`.

Proof.

Since `a=(a+b)-b` then we can apply Inequality 3 to this: `|a|=|(a+b)-b|<=|a+b|+|-b|=|a+b|+|b|` or `|a+b|>=|a|-|b|`.

Similarly can be proved

Inequality 6. `|a-b|>=|a|-|b|`.

Proof.

Since `a=(a-b)+b` then we can apply Property 3 to this: `|a|=|(a-b)+b|<=|a-b|+|b|` or `|a-b|>=|a|-|b|`.

Inequality 7. `||a|-|b||<=|a-b|`.

Proof.

From property 6 we have that `|a|-|b|<=|a-b|` . If we interchange a and b we will obtain that `|b|-|a|<=|b-a|` or `|a-|b|>=|a-b|` . Now, by Inequality 2 `||a|-|b||<=|a-b|`.