**What is a real number?**

We call a group of rational numbers and irrational numbers a real number or in other words we can say that a square whose square is always a positive number is called a real number. We understand this with the example given below-

Exp- 2/3, 4.5, -9, -5.6, 5√3, 75√3 etc. are examples of real numbers. But √-3 is not an example of real numbers!

There are 2 types of real numbers.

**(1) Rational number: –** The number which can be expressed as p / q. Are called rational numbers. But the numbers are zero. They are not rational numbers.

Exp-1, 3, 4, 3/4, 6/7, 2/3, 4.5, -9, -5.6 etc.

**(2) Irrational number: –** The number which cannot be expressed as p / q, is called the irrational number.

Exp-5√3, 75√3, √-3 etc.

**Equivalent rational number: –** To find the rational number equivalent to a given positive positive number, multiply by the same number in the numerator and denominator.

Exp- 2/5 = 2/5 * 2/2 = 4/10 (where / = is discount | * = multiplication)

**(1) Natural Number: –** The number of count is called natural number.

Exp- N = 1,2,3,4,5,6,7,8,9, ………………….)

**(2) Whole Number: –** If we include 0 in the group of natural numbers, then it is called a whole number, it is indicated as W in the symbol.

Exp-W = 0,1,2,3,4,5,6,7,8,9 ………………)

**(3) Even Number: –** Natural numbers which are 2 completely divisible are called even numbers. Even numbers can be written as 2n where n is a natural number.

Exp-2,4,6,8,10,12,14, ………………………)

**(4) Odd Number: –** Those natural numbers which are not completely divisible by 2, make them odd numbers. It can be written by 2n-1. Where natural n is the number.

Exp-2,3,5,7,9,11,13,15 …………………….)

Note-2 is the smallest odd number.

**(5) Prime or root number: –** acquitted and natural numbers from 1 which are not divisible by 1 or any other number except themselves are called prime or root numbers.

Exp-2,3,5,7,11,13,17, ………………………….)

**(6) Composite Number: –** Those natural numbers which are either 1 or omitted and divided by the number of the other are called divisible or compound numbers.

Exp-4,6,8,9,10,12,14, …………………….)

**(7) Co-Prime Number: –** There are two numbers or more than two numbers whose HCF is one i.e. the maximum common divisor of those numbers is called a co-prime number.

Exp- (5,7) (9,7) …………………………….

**(8) Integer Number: –** Positive natural numbers are called negative natural numbers and groups of zeros.

Exp-1,2,3,4, -5, -7, -8, -9 …………………)

Types of Integers

Integers can be divided into two types. There are two types of integers: positive integers, negative integers

**(a) Positive Integers: –** Such integer numbers, which are positive, are called positive integers.

Exp- 1, 2, 3, 4, 5 ,. . . . . . . . Etc. are all positive integers.

**(b). Negative Integers: –** Integers that have preceded the negative sign are called negative integers.

Exp- Hence, –1, –2, –3, –4, –5 ,. . . . . . . . ) Etc. are negative integers.

Note- Zero is an integer, but zero is neither positive nor minus

**(9) Fraction: –** The number obtained by dividing by an integer m i is called the fraction m / n where m is called the numerator and n is denominator n.

Exp-2/3, 3/5, 7/6, 6/7, 8/7, 9/8. . . . . . . . .)

**(10) Algebraic rule in emphasis and multiplication of rational numbers**

**(a) Closurelaw: –** Even if m and n are two rational numbers, it will be rational.

Exp- m + n = 2 + 3 = 5,

m * n = mn = 2 * 3 = 6 are rational numbers.

**(b) Commutative law: –** If a and b are two rational numbers.

a + b = b + a (the ordinal law of addition)

a.b = b.a (ordinal law of multiplication)

**(c) Asssociative law: –** If a, b and c are three rational numbers.

(a + b) + c = a + (b + c) = a + b + c (joint associative rule)

(ab) c = a (bc) = abc (multiplication associative rule)

**(d) Existeencennce of identity element: –** If 0 and 1 are two rational numbers.

m + 0 = 0 + m = m (The sum-sum element is 0.)

a * 1 = 1 * a = a (Multiplying elements are 1.)

**(e) Inverse element: –** The inverse element of a is -a.

Exp-1 is -1, 2 is -2. etc.

**(f) Cancellation law: –**

If m + n = m + p then n = p

If m + c = n + c then m = n

If m # 0 and mn = mp then n = p (where # = no)

If m # 0 and bm = cm then b = c (where # = no)

**(g) Distributive law: –** If a, b and c are three rational numbers.

a (b + c) = ab + ac

(b + c) a = ab + ac

**(h) Finding Rational Numbers Between Two Numbers: –** If a and b are two rational numbers. (a <b) then a + b / 2 is the rational number between them.

Exp- a = 2 is the number between b = 3.

a + b / 2 = 2 + 3/2 = 5 \ 2.

**Important formulas and powerful formulas –** Important Formulas of Surds and Indices

**1. 5 consecutive natural numbers N = 1, 2, 3, 4, 5. . . . . . . . . . . . . . .**

**2. 5 consecutive whole numbers W = 0, 1, 2, 3, 4.. . . . . . . . . . . . . .**

**3. 5 consecutive even numbers 2, 4, 6, 8, 10. . . . . . . . . . . . . . .**

**4. 5 consecutive odd numbers 1, 2, 3, 5, 7. . . . . . . . . . . . . . .**

**5. 5 consecutive composite or compound numbers 4, 6, 8, 9, 10 ,. . . . . . . . . . . . . . .**

**6. 5 consecutive prime or conservative numbers 2, 3, 5, 7, 11. . . . . . . . . . . . . . . . .**

**7. 5 consecutive consecutive composite numbers (9, 19) (5, 9), (1, 2) (2, 3), (4, 5). . . . … . . .**

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