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# 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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#### NTSE Mathematics (Class X)

• Real Numbers
• Polynomials
• Linear Equation in Two Variables
• Trigonometry
• Similar Triangles
• Statistics
• Arithmetic Progressions
• Application of Trigonometry
• Circle
• Co-ordinate Geometry
• Area related to Circle
• Surface Area & Volume
• Constructions
• Probability

#### NTSE Mathematics (Class IX)

• Real Numbers
• Polynomials
• Linear Equation in Two Variables
• Trigonometry
• Similar Triangles
• Statistics