Number System Tutorial I: Integers, Fractions, Prime & Composite

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Numbers is one of the main sources of questions in all of the B-School entrance exams. Especially CAT exam consist a lots of questions related to numerical properties. In the old format of CAT, a lion's share of the questions asked from the area of numbers and geometry. Basic number properties and its advanced application is one of the main testing points in CAT, MAT, XAT etc.

If you are really a serious aspirant for any of the of the B-school entrance exams, you can't avoid the topic 'Numbers'. It is very important to understand the different number groups and its properties. In most of the questions required an awareness plus logical approach in both average level and high level. Finally, Numbers is the most challenging and very interesting area in these exams.

Objectives

  • Classification of real numbers
  • Terminating and Recurring decimals
  • Conversion from recurring decimal to fraction
  • Comparison of fractions
  • Integers (odd and even) and its properties
  • Prime and composite numbers
  • Summation results

 

Classification of Real Numbers

It is possible to design a family tree of numbers in the following manner. 

Rational Numbers

Any number which can be expressed in the form p/q, where p and q are integers (both + and -) and `q != 0`, is called a rational number. Eg: `1/2, (-3)/4, 5, 0, -7` etc.

Irrational Numbers

Obviously this is the number which can't be expressed in the form `p/q`, where `p` and `q` are integers (both + and -) and `q != 0`. Eg: `sqrt(2), pi, e` etc. For a clear understanding about irrational numbers, it is better to consider the classification of Decimal Numbers.

Terminating and Non-terminating Decimals

If the decimal part of a number consisting a finite number of digits, then it is terminating decimal, otherwise it is a non-terminating decimal.
Eg: `0.2, 1.25, 123.12345` etc are the examples for terminating decimals.
`0.333..., 0.1542782157`... are the examples for non-terminating decimal.

Recurring Decimal

In a non-terminating decimal if the decimal part is an infinite repetition of a number or a group of numbers, and then it is called a recurring decimal.
Eg: 0.33333... (only 3 repeating), this can be expressed as `0.bar(3)`
0.545454... (pair 54 repeating), this can be expressed as `0.bar(54)`
0.12366666... (only 6 repeating), this can be expressed as `0.123bar(6)`

Definition for irrational numbers
If a decimal is non-terminating and non-recurring, then it is an irrational number. Eg: `sqrt(2) = 1.414213562..., sqrt(3)=1.7320508...`
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