## CBSE Class 9 Maths Chapter 1 Number Systems Revision Notes

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## Real numbers

Real numbers are all numbers that can be represented on a number line and include all rational numbers like integers, fractions, decimals, and also all irrational numbers. The real numbers include all the rational numbers, such as the integer − 5 and the fraction 4/3, and all the irrational numbers, such as √2 (1.41421356…, the square root of 2, an irrational algebraic number).

**Types of real numbers**

- 0: whole number, an integer but not a natural number.
- Every natural number N has a successor N + 1.
- Every integer has a successor Z + 1 and predecessor Z – 1.
- Every integer is a rational number but vice-versa may not be true.
- 2/1 is an integer
- 3/2 = 1.5 and 1/3 are not integers

## Irrational Numbers

An irrational number is a real number that cannot be expressed as a ratio of integers, i.e. as a fraction. Therefore irrational numbers when written as decimal numbers, do not terminate nor do they repeat.

- Irrational numbers cannot be written in the form p/q, where p and q are integers and q ≠ 0
- There are infinitely many irrational numbers. For example √2, √3, √5 are all irrational.
- The ratio of the length of the circumference of a circle to the length of its diameter is always constant. It is an irrational number and denoted by π.

## Operations

The basic arithmetic operations are the addition, subtraction, multiplication, and division for Real Numbers. The basic arithmetic properties are closure, commutative, associative, and distributive properties.

**Laws:**

1. Commutative law of addition: a + b = b + a

2. Commutative law of multiplication: a × b = b × a

3. Associative law of addition: a + (b + c) = (a + b) + c

4. Associative law of multiplication: a × (b × c) = (a × b) × c

5. Distributive law of multiplication over addition :a × (b + c) = (a × b) + (a × c) or (a + b) × c = (a × c) + (b × c)

CBSE Class 9 Maths Chapter 1 Number Systems Revision Notes PDF