Quantitative Aptitude > Real Numbers

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  • 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 .


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