Quantitative Aptitude > Real Numbers

The set of real numbers consists of all rational numbers and all irrational numbers . The real numbers include all integers, fractinons, 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. On the number line, all numbers to the left of 0 are negative and all numbers to the right of 0 are positive. Only the number 0 is neither negative nor positive.

A real number is less than a real number if is to the left of on the number line, which is written as . A real number is greated than a real number if is to the right of on the number line, which is written as . For e.g.

To say that a real number is between 2 and 3 on the number line means that and , which can also be written as the double inequality . The set of all real numbers that are between 2 and 3 is called an interval, and the double inequality is often used to represent that interval. Note that the endpoints of the interval, 2 and 3 are not included in the interval. The following inequalities represent four types of intervals, depending on whether or not the endpoints are included.

The distance between a number and 0 on the number line is called the absolute value of , written as . Therefore, 3 = 3 and 3 = 3 because each of the numbers 3 and 3 is at a distance of 3 from 0. Note that if is positive, then  = and if is negative, then  = ; and lastly, 0 = 0.It follows that the absolute value of any nonzero number is positive. Here are a few examples.

There are several general properties of real numbers that are used frequently. If a, b and c are real numbers, then

and .

and


and

If , then either or both.

Division by 0 is not defined; for example, and are undefined.

If both and are positive, then both and are positive.

If both and are negative, then is negative and is positive.

If is positive and is negative, then is negative.

. This is known as triangle inequality.


If , then If , then .

