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...`

How to identify if a fraction gives terminating decimal or recurring decimal

If the simplified form of a fraction consist the prime factors 2 or 5 only in its denominator, then it is terminating decimal.
OR
If the denominator of the simplified form of any fraction consist at least one factor other than 2 or 5 then the fraction gives a recurring decimal.
Which among the following fractions is/are recurring decimal?
  • `371/128`
  • `403/160`
  • `3721/1024`
  • `8431/3125`
  • `3731/4311`
When observing the prime factors of the denominators in each fraction, it is easy to identify that the denominator of option E is a multiple of 3 and its numerator is not a multiple of 3. Therefore the prime factor 3 of the denominator should be there even after the simplification ( if it is possible).
Hence answer is `3731/4311`
We can check the prime factors of the rest of the denominators.
`128 = 2^7`
`160 = 2^5 * 5`
`1024 = 2^10`
`3125 = 5^5`
Here all of these denominators have the prime factors 2 or 5 only.

 

How to convert from recurring decimal to fraction?

Here we are going to find out the corresponding fraction of given recurring decimal. Group of recurring decimals are classified in to two.

Type I group of recurring decimals

In this type of recurring decimals the repeating digit/a group of digits, starts repeating immediately after the decimal point, ie. There is not any digit in the decimal part as non-repeating. Eg: `0.bar(54), 0.bar(524), 0.bar(1234)` etc

Method of conversion

Find the corresponding fraction of `0.bar(5)`
Let `x = 0.bar(5) -> Equation(1)`
`(1) * 10 -> 10x = 5.bar(5) -> Equation(2)`
`(2)-(1) -> 9x = 5`
`x = 5/9`
Therefore `0.bar(5)=5/9`
Find the corresponding fraction of `0.bar(25)`
Let `x = 0.bar(25) -> Equation(1)`
`(1) * 100 -> 100x = 25.bar(25) -> Equation(2)`
`(2) - (1) -> 99x = 25`
`x = 25/99`
Therefore `0.bar(25) = 25/99`'

Easy approach in converting Type I recurring decimal to fraction

`0.bar(ab) = "repeating group (here ab)"/"as many 9's as the number of digits in repeating group"`

Examples in converting recurring decimal to fraction

`0.bar(13) = 13/99`
`0.bar(251) = 251/999`
`0.bar(1234) = 1234/9999`

Type II group of recurring decimals

In this type the repeating digit/group of digits starts after a digit/ some digits of non-repeating digits. Eg. `0.1bar(6) (0.1666...), 0.03bar(8) (0.03888...), 0.45bar(13) (0.45131313...)`

Method of conversion

Find the corresponding fraction of `0.1bar(6)`
Let `x = 0.1bar(6) -> Equation(1)`
Take the non-repeating digit to the left side (integral portion) of the decimal point.
`(1) * 10 -> 10x = 1.bar(6) -> Equation(2)`
Take on repeating group/digit to the left side of the decimal point.
`(2) * 10 -> 100x = 16.bar(6) -> Equation(3)`
`(3)-(2) -> 90x = 15`
`x = 15/90 = 1/6`
Therefore `0.1bar(6)=1/6`
Find the corresponding fraction of `0.12bar(345)`
Let `x = 0.12bar(345) -> Equation(1)`
Take the non-repeating digit to the left side (integral portion) of the decimal point.
`(1) * 100 -> 100x = 12.bar(345) -> Equation(2)`
Take on repeating group/digit to the left side of the decimal point.
`100000x = 12345.bar(345) -> Equation(3)`
`(3)-(2) -> 99900x = (12345-12) = 12333`
`x = 12333/99900`
Therefore `0.12bar(345)=12333/99900`

Easy approach in converting Type II recurring decimal to fraction

`0.ab bar(cde) = `("entire decimal group" - "non-repeating decimal group")/"as many 9's as the number of repeating digits in the decimal part with as many 0's as the number of non-repeating digits in the decimal part"

Examples in converting Type II recurring decimal to fraction

`0.12bar(345) = (12345 - 12)/99900 = 12333/99900`
`0.8bar(45) = (845-8)/990 = 837/990`

More examples

Find the corresponding fraction of `3.2bar(3)`
`3.2bar(3) = 3 + 0.2bar(3)`
`3 + (23 - 1)/90 = 3 + 21/90 = 3 + 7/30`
`=97/30`
Find `0.3bar(7) + 0.4bar(5) + 0.1bar(6)`
`0.3bar(7) = 34/90`
`0.4bar(5) = 41/90`
`0.1bar(6) = 15/90`
`0.3bar(7) + 0.4bar(5) + 0.1bar(6) = 34/90 + 41/90 + 15/90 = 90/90 = 1`

Fractions

Any number which can be expressed in the form p/q, where p and q are Natural Numbers is called a fraction. Example: `1/2, 3/2, 5` etc. There are three types of fractions.

Proper Fractions

If the numerator of a fraction is lesser than the denominator, then it is a proper fraction. Hence the value of a proper fraction should lie in between 0 and 1. Eg. `1/3, 2/5, 6/13` etc

Improper fractions

If the numerator of a fraction is greater than or equal to its denominator then it is an improper fraction. Hence the value of any improper fraction is greater than or equal to 1. Eg. `3/2, 10/7, 5` etc

Mixed Fractions

Basically a mixed fraction is another expression of a corresponding improper fraction. Example: `5/4` is an improper fraction. I can be expressed in the following manner too.
`5/4 = (4+1)/4 = 1 + 1/4 = 1 1/4`, Here 1 is the natural number part and `1/4` is the proper fractional part.

Comparison of fractions

Comparison of different types of fractions is a basic requirement in Data Interpretation. Some direct questions from the comparison concept also can be expected in your exam. Hence you must be familiar with different methods for the comparison of fractions.

Fraction comparison using cross multiplication

Let's look at example to see how to compare two fractions quickly using cross multiplication.

Which is greater, `3/7` or `7/16` ?

For the comparison of any two fractions, it is quite easy to apply the cross multiplication method. Here, multiply the numerator of the first fraction with the denominator of the second fraction. This product is representing the first fraction. ie. `3 xx 16 = 48`

Similarly multiply the numerator of the second fraction with the denominator of the first fraction and this product is representing the second fraction. `ie. 7 xx 7 = 49`

Here `49 > 48`, therefore `7/16 > 3/7`.
Compare `101/200` and `51/100` ?
`101 xx 100 = 10100 -> "Represents 1st Fraction"`
`51 xx 200 = 10200 -> "Represents 2nd Fraction"`
`10200 > 10100, :. 51/100 > 101/200`.

Fraction comparison using denominator equalization

In a group of fractions, if the denominators of all the factions are equal then the largest fraction consist the largest numerator.

Let's look at example to see how to compare two fractions quickly using denominator equalization.

Compare the fractions, `3/5`, `7/10` and `21/25` ?
Here we can equate the denominators of all fractions to the LCM of denominators.
LCM of denominators = LCM (5, 10, 25) = 50
`3/5 = (3xx10)/(5xx10) = 30/50`
`7/10 = (7xx5)/(10xx5) = 35/50`
`21/25 = (21xx2)/(25xx2) = 42/50`
`:. 21/25 > 7/10 > 3/5`

Fraction comparison using numerator equalization

In a group of fractions if the numerators are equal then the largest fraction consist of the least denominator.

Let's look at example to see how to compare two fractions quickly using numerator equalization.

Compare the fractions `3/5`, `7/10` and `21/60`?
Here we can equate the numerators of all fractions to the LCM of numerators.
LCM of denominators = LCM (3, 7, 21) = 21
`3/5 = (3xx7)/(5xx7) = 21/35`
`7/10 = (7xx3)/(10xx3) = 21/30`
`21/60 ("no need to change")`
`:. 7/10 > 3/5 > 21/60`

Fraction comparison using special property of proper fractions

In a set of proper fractions if the differences between the numerator and denominator in each fraction are equal, then the greatest fraction has the greatest numerator and denominator.

Let's look at an example to see how to compare two fractions quickly using special property of proper fractions.

Compare `3/7`, `11/15`, `23/27`, `25/29` and `107/111`.?
By observation, the differences between numerator and denominator in each fraction are equal to 4. Hence `107/111 > 25/29 > 23/27 > 11/15 > 3/7`.

Fraction comparison using special property of improper fractions

In a set of improper fractions if there is a common difference between the numerator and denominator in each fraction, then the largest fraction has the least numerator and denominator.

Let's look at an example to see how to compare two fractions quickly using special property of improper fractions.

Compare `19/17`, `5/3`, `103/101`, `1015/1013` and `7/5`.?
Here the differences between the numerator and denominator in each fraction are equal. Therefore, `5/3 > 7/5 > 19/17 > 103/101 > 1015/1013`.

General approaches for fraction comparison

More often it is not possible to find any particular relations between the numerators and denominators in the given set of fraction. Then we have to approach the comparison in a general way of comparison.

Approach I: Decimal Value form

In this method we are converting the given set of fractions in its corresponding decimal values.

Compare the fractions `5/11`, `1/3`, `4/9`, `3/7`, `2/5` and `5/12`.?
Convert the fractions into its decimal values.
`5/11 = 0.bar(45)`
`1/3 = 0.bar(3)`
`4/9 = 0.bar(4)`
`3/7 = 0.bar(428571)`
`2/5 = 0.4`
`5/12 = 0.41bar(6)`
`:. 5/11 > 3/7 > 5/12 > 4/9 > 2/5 > 1/3`
Which among the given fractions is greatest. `1047/523`, `501/251`, `869/436`, `2041/1021`?
In the given set of fractions it is not an easy task to find out the corresponding decimal value of each fraction. Hence we are applying another one method for finding the largest among them. From the given fractions, it is easy to approximate them in the following manner;
`1047/523 > 2`
`501/ 251 < 2`
`869/ 436 < 2`
`2041/1021 < 2`
Therefore the largest fraction is `1047/523`.
Approach II: Percentage comparison

In this method, we are finding the approximate percentage value represented by each of the fractions and then comparing them.

Arrange the following fractions in ascending order. `9/24`, `5/17`, `92/366`, `70/176`?
`9/24`: 9 is approximately 34% of 24
`5/17`: 5 is approximately 30% of 17
`92/366`: 92 is approximately 25% of 366
`70/176`: 70 is approximately 40% of 176
`:. 92/366 < 5/17 < 9/24 < 70/176`.

Integers and its properties

The set of integers is denoted by the letter 'Z'. Z = { .....-4, -3, -2, -1, 0, 1, 2, 3, 4.....}.

Subsets of Z (set of integers)

  • Odd numbers: {......-5, -3, -1, 1, 3, 5......}
  • Even numbers: {......-4, -2, 0, 2, 4, 6, 8......}
  • Negative integers: {......-4, -3, -2, -1}
  • Non-negative integers (Whole Numbers): {0, 1, 2, 3, 4......}
  • Positive integers (Natural Numbers): {1, 2, 3, 4, 5......}
  • Prime Numbers: {2, 3, 5, 7, 11, 13, 17......}
  • Composite numbers: {4, 6, 8, 9, 10, 12, 14, 15......}

 

Properties of odd and even numbers

Property I: Addition

  • Sum of odd number of odd numbers is odd. Eg: 1 + 3 + 5 = 9.
  • Sum of even number of odd numbers is even. Eg: 3+ 11 + 7 + 5 = 26.
  • Sum of any number of even numbers is even.
  • In a certain summation of odd and even numbers, then the sum is:
    • Odd: when there is odd number of odd numbers in the selected group of integers
    • Even: when there is even number of odd numbers in the selected group of integers

 

Property II: Subtraction

What are the properties considered in the addition of integers, the same properties are applicable in the subtraction in integers.

  • If `a + b` is odd for any two integers a and b, then; `a - b`, `-a + b`, `-a - b` are all odd.
  • If `a + b` is even, then `a - b`, `-a + b`, `-a - b` are all even.

Property III: Multiplication

If a group of integers consist at least one even number then the product of the given integers should be even, otherwise the product become odd (ie. all integers are odd).

Property IV: Division

Consider a rational number `x/y`, where x and y are two integers and `y != 0`.

Case I: Numerator is even and denominator is odd:

In a division in the form `x/y`, if the numerator is an even multiple of the odd denominator y, then the result of `x/y` is always an even number. Example: `36/9 = 4` (it is even).

Case II: Both the numerator and denominator are even:

In `x/y`, where x and y are integers and x is divisible by y, then it is not possible to make a conclusion of the nature of the value of `x/y`. Example: `10/5 = 2` (it is even)

Case III: Dividing an odd number by an even number:

In `x/y` , if x is odd and y is even, then x can't be any multiple of y. Hence the value of `x/y` in this situation can't be an integer.

Case IV: Divide an odd number by another odd number:

In the division `x/y`, both x and y are odd numbers and x is divisible by y, then the value of `x/y` is always odd Example: `15/3 = 5` (it is odd)

Some interesting properties of odd numbers

  • Cube of all natural numbers greater than '1' can be expressed as the sum of a certain number of consecutive odd numbers.
    `n^3 = [(n - 1). n + 1] + [(n - 1). n + 3] + [(n - 1). n + 5] + ... + [(n-1). n + (2n - 1)]`
    Examples:
    • `23 = 3 + 5`
    • `33 = 7 + 9 + 11`
    • `43 = 13 + 15 + 17 + 19`
  • All odd natural numbers except '1' can be one of the three values of a Pythagorean triplet.
    • 32 + 42 = 52
    • 52 + 122 = 132
    • 72 + 242 = 132
    • 92 + 402 = 412
    • 112 + 602 = 612
    • 132 - 122 = 52
    • 152 + 82 = 172
    • 172 - 152 = 82

Prime and Composite numbers

 

Prime number

A natural number which doesn't have any factor other than 1 and itself is a prime number. Example: 2, 3, 5, 7, 11, 13, 17 etc.

Properties of prime numbers

  • 2 is the least prime number
  • 2 is the one and only one even prime.
  • All primes greater than 3 can be expressed in either of the form `6k - 1` or `6k + 1`, where `k` is any natural number, but all the numbers which are in the above mentioned forms are not necessarily prime always. Eg when k = 4, `6k + 1 = 25`, it is not a prime.
  • Goldbach's Conjecture: Every even natural number greater than 2 can be expressed as the sum of two primes ( not necessarily distinct). Eg: `4 = 2 + 2`, `6 = 3 + 3`, `8 = 3 + 5`, `10 = 3 + 7` and so on.
  • Relatively prime / co-prime numbers: If two natural numbers doesn't have a common factor other than 1 are called relatively prime/co-prime numbers. Eg. 8 and 9 are relatively prime. 4 and 5 are co-prime.
  • 1 is relatively prime to all the other natural numbers.
  • Twin Primes: If any two consecutive odd numbers are prime, ten they are called twin primes. Eg. (3,5), (5,7), (11,13), (17,19), etc.
  • Prime triplet: If three consecutive odd numbers are prime, then they are called prime triplets. There is only one prime triplet and that is (3,5,7).
  • Fermatt's theorem: If 'a' and 'b' are any two prime numbers, then `a^b - a` is always divisible by b. Eg: `2^3 - 2 = 6` is divisible by 3.
    `3^5 - 3 = 240` is divisible by 5
  • Wilson's theorem: For any prime 'p', `(p - 1)! + 1` is divisible by p. Eg: Let p = 5, `(p - 1)! + 1 = 4! + 1 = 25` is divisible by 5.
    Let p = 11, `(p - 1)! + 1 = 10! + 1 = 3628800 + 1 = 3628801` is divisible by 11.
  • Progression of prime numbers: Some set of prime numbers are forming an Arithmetic Progression. Eg: 11,17,23,29
    199,409,619,829, 1039, 1249, 1459, 1669, 1879, 2089.
  • How to identify a prime number? Consider 173. For checking 173 a prime or not, consider the square of any prime number which is immediately less tan 173. Here `13^2 = 169` is immediately less than 173.
    Check the divisibility of all the prime numbers up to 13 with 173. None of the prime numbers 2, 3, 5,7, 11 and 13 is not a factor of 173. Therefore 173 is a prime number.
    Example: Is 391 a prime number?
    Square of a prime number, which is immediately less than 391 is `19^2 = 361`. While checking the divisibility of all primes up to 19, we will get 17 is a factor of 391 (Ie. `391 = 17 xx 23`). Therefore 391 is not a prime number.
  • In the set of natural numbers from 1 to 50 there are exactly 15 prime numbers and from 51 to 100, there are 10 prime numbers.
  • For any prime `p > 3`, `p^2 - 1` is a multiple of 24.

Distribution of prime numbers in the first thousand natural numbers

Range of natural numbersNumber of prime numbers
1 to 100 25
101 to 200 21
201 to 300 16
301 to 400 16
401 to 500 17
501 to 600 14
601 to 700 16
701 to 800 14
801 to 900 15
901 to 1000 14

Composite number

If a natural number has at least one factor other than 1 and itself is called a composite number. Eg: 4, 6, 8, 9, 10, 12, 14, 15 etc.

Properties of composite numbers

  • 4 is the least composite number.
  • Wilson's theorem: For any composite number `c > 4`, `(c - 1)!` is divisible by c.
    Eg: Let c = 6
    `(c - 1)! = (6 - 1)! = 5! = 24` is divisible by 6.
    Let c = 12
    `(c - 1)! = 11! = 39916800` is divisible by 12.

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