Number Systems forms the base for Quantitative Ability and clearing of concepts is important for CAT and other related exams. A measurement carried out, of any quantity, leads to a meaningful value called the Number. This value may be positive or negative depending on the direction of the measurement and can be represented on the number line.

Following table gives a brief introduction to system of numbers: Natural Numbers (N)

The numbers 1, 2, 3, 4, 5, ... are known as natural numbers. The set of natural numbers is denoted by N. The natural numbers are further divided as even, odd, prime, etc.

Whole Numbers (W)

All natural numbers together with ‘0’ are collectively called whole numbers. The set of whole numbers is denoted by W.

W = {0, 1, 2, 3, ......}

Integers (Z)

The set including all whole numbers and their negatives is called a set of integers. It is denoted by Z.

Z = {– ∞, ... – 3, – 2, – 1, 0, 1, 2, 3, ....... ∞}

They are further classified into Negative integers, Neutral integers and positive integers.

Even Numbers

All numbers divisible by 2 are called even numbers. For example, 2, 4, 6, 8, 10 ... Even numbers can be expressed in the form 2n, where n is an integer. Thus 0, – 2, − 6, etc. are also even numbers.

Odd Numbers

All numbers not divisible by 2 are called odd numbers. For example, 1, 3, 5, 7, 9... Odd numbers can be expressed in the form (2n + 1) where n is any integer. Thus – 1, − 3, − 9, etc. are all odd numbers.

Prime Number

A prime number is a natural number which has only two distinct divisors: 1 and itself.

The number 1 is not a prime number.

There are 25 prime numbers under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Important observation about prime numbers: A prime number greater than 3, when divided by 6 leaves either 1 or 5 as the remainder. Hence, a prime number can be expressed in the form of 6K± 1. But the converse of this observation is not true, that a number leaving a remainder of 1 or 5 when divided by 6 is not necessarily a prime number.

Prime Factorization Theorem: This is the area where prime numbers are used. This theorem states that any integer greater than 1 can be written as a unique product of prime numbers.

Examples:

• 550 = 2 × 52 × 11
• 1200 = 24 × 3 × 52

Thus, prime numbers are the basic building blocks of any positive integer. This factorisation will also help in finding GCD and LCM quickly.

Composite Numbers

A composite number has other factors besides itself and unity. For example, 8, 72, 39, etc. On the basis of this fact that a number with more than two factors is a composite we have only 34 composite from 1 to 50 and 40 composite from 51 to 100.

Perfect Numbers

A number is a perfect number if the sum of its factors, excluding itself and but including 1, is equal to the number itself.

The sum of all the possible factors of the number is equal to twice the number.

Examples: 6 (1 + 2 + 3 = 6), 28 (1 + 2 + 4 + 7 + 14 = 28)

Other examples of perfect numbers are 496, 8128, etc. There are 27 perfect numbers discovered so far.

Co-Prime Numbers

Two numbers (prime or composite) are co-prime to each other, if they do not have any common factor except 1.

Examples: 25 and 9, since they don’t have a common factor other than 1. Another example is 35 and 12, since they both don’t have a common factor among them other than 1.

Fractions

A fraction denotes part or parts of a unit. Several types are:

1. Common Fraction: Fractions whose denominator is not 10 or a multiple of it. For example, 2/3, 17/18

2. Decimal Fraction: Fractions whose denominator is 10 or a multiple of 10.

3. Proper Fraction: In this the numerator is less than denominator. Hence its value < 1.

4. Improper Fraction: In these the numerator is greater denominator. Hence its value > 1.

5. Mixed Fractions: When a improper fraction is written as a whole number and proper fraction it is called mixed fraction.

Rational Numbers

Rational Number is defined as the ratio of two integers i.e. a number that can be represented by a fraction of the form p/q where p and q are integers and q ≠ 0. They also can be defined as the non-terminating recurring decimal numbers.

Irrational Numbers

Any number which can not be represented in the form p/q where p and q are integers and q ≠ 0 is an irrational number. On the basis of non-terminating decimals, irrational numbers are non-terminating non recurring decimals.

Non-Terminating Decimal Numbers

When you divide any number by other number, either you get a terminating number or a non-terminating number. A non-terminating number on the basis of occurrence of digits after decimal can classified as:

1. Pure Recurring Decimals: A decimal in which all the figures after the decimal point repeat, is called a pure recurring decimal.
2. Mixed Recurring Decimals: A decimal in which some figures do not repeat and some of them are repeated is called a mixed recurring decimal.
3. Non-Recurring Decimals: A decimal number in which the figure don’t repeat themselves in any pattern are called non-terminating non-recurring decimals and are termed as irrational numbers.

Points To Remember

1. The number 1 is neither prime nor composite.

2. The number 2 is the only even number which is prime.

3. (xn + yn) is divisible by (x + y), when n is an odd number.

4. (xn – yn) is divisible by (x + y), when n is an even number.

5. (xn – yn) is divisible by (x – y), when n is an odd or an even number.