#### Mathematics (NTSE/Olympiad)

# Real Numbers

__Number System__

**Natural Numbers :**

The simplest numbers are 1, 2, 3, 4....... the numbers being used in counting. These are called natural numbers.

**Whole numbers :**

The natural numbers along with the zero form the set of whole numbers i.e. numbers 0, 1, 2, 3, 4 are whole numbers. W = {0, 1, 2, 3, 4....}

**Integers :**

The natural numbers, their negatives and zero make up the integers. Z = {....–4, –3, –2, –1, 0, 1, 2, 3, 4,....}. The set of integers contains positive numbers, negative numbers and zero.

**Rational Number :**

(i) A rational number is a number which can be put in the form p/q , where p and q are both integers and q ≠ 0.

(ii) A rational number is either a terminating or non-terminating and recurring (repeating) decimal.

(iii) A rational number may be positive, negative or zero.

**Complex numbers :**

Complex numbers are imaginary numbers of the form a + ib, where a and b are real numbers and i = √–1, which is an imaginary number.

**Factors :**

A number is a factor of another, if the former exactly divides the latter without leaving a remainder (remainder is zero) 3 and 5 are factors of 12 and 25 respectively.

**Multiples :**

A multiple is a number which is exactly divisible by another, 36 is a multiple of 2, 3, 4, 9 and 12.

**Even Numbers :**

Integers which are multiples of 2 are even number (i.e.) 2,4, 6, 8............... are even numbers.

**Odd numbers :**

Integers which are not multiples of 2 are odd numbers.

**Prime and composite Numbers :**

All natural number which cannot be divided by any number other than 1 and itself is called a prime number. By convention, 1 is not a prime number. 2, 3, 5, 7, 11, 13, 17 ............. are prime numbers. Numbers which are not prime are called composite numbers.

**The Absolute Value (or modulus) of a real Number :**

If a is a real number, modulus a is written as |a| ; |a| is always positive or zero.It means positive value of ‘a’ whether a is positive or negative |3| = 3 and |0| = 0, Hence |a| = a ; if a = 0 or a > 0 (i.e.) a ≥ 0

|–3| = 3 = – (–3) . Hence |a| = – a when a < 0

Hence, |a| = a, if a > 0 ; |a| = – a, if a < 0

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