# GRE Arithmetic – Real Numbers

• Last Updated : 08 Feb, 2021

The set of Real Numbers consists of all Rational Numbers and all irrational numbers. The Real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the Real Number Line. Every Real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. All numbers to the left of 0 on the number line are negative. Similarly, all numbers to the right of 0 on the number line are positive.

A Real number m is less than a real number n if m is to the left of n on the number line which is written as m < n. example:

```Example 1: 1/2 > 0
Example 2: 3 < -1
Example 3: 2 < 0 < 2```

Absolute Value: The distance between the number x and 0 on the number line is called the absolute value of x, written as |x|. Therefore, |3|=3 and |-3|=3 because each of the numbers 3 and -3 is a distance of 3 from 0. Note that if x is positive, then |x|=x, if x is negative, then |x|=-x, and lastly, |0|=0. It follows that the absolute value of any non-zero number is positive, Here are some examples:-

```Example 1: |5| = 5
Example 2: -|-23| = 23
Example 3: -|-10.2| = 10.```

Properties of Real Numbers

Here are twelve general properties of Real numbers that are used frequently. In each property r, s, and t are real numbers.

Property 1: Commutative Property

Addition: r + s = s + r

Multiplication: r∗s = s∗r

Example:

```8 + 2 = 2 + 8 = 10
(-3)(17) = (17)(-3) = -51```

Property 2: Associative Property

Addition: (r+s) + t = r + (s+t)

Multiplication: (r∗s)∗t = r∗(s∗t)

Example:

```( 7 + 3 ) + 8 = 7 + ( 3 + 8 ) = 18
( 7∗2 )∗4 = 7∗( 2∗4 ) = 56```

Property 3: Distributive Property

r( s+t ) = rs + rt

Example:

`5( 3 + 16 ) = 5(3) + 5(16) = 15 + 80 = 95`

Property 4: Identity Property

Multiplication: r∗1 = r

Property 5: Zero Property of Multiplication

If r∗s = 0, then either r=0 or s=0 or both.

`If -2∗s = 0, then s = 0`

Property 6: Division by 0 is undefined

```5/0 = undefined
0/0 = undefined```

Property 7: If both r and s are positive, then both r+s and r∗s are positive.

Property 8: If both r and s are negative, then r+s are negative and r∗s is positive.

Property 9: If r is positive and s is negative then r∗s is negative.

Property 10: |r+s| ≤ |r| + |s|. This is known as Triangle inequality

```If r = 5 and s = -2 then
| 5 + (-2) | = | 5 - 2 | = |3| = 3 and
|5| + |-2| = 5 + 2 = 7. Therefore,
| 5 + (-2) | <= |5| + |-2|```

Property 11: |r||s| = |rs|

`|5||-2| = | (5)(-2) | = |-10| = 10 `

Property 12: If r>1, then r2 > r. If 0<s<1, then s2<s

```52 = 25 > 5, but
(1/5)2 = 1/25 < 1/5 ```
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