# 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|.