 The Class 9 Maths Number System MCQs will help you understand the topic well, face the exam confidently, and score the best marks.

What is a number system? A number system is defined as representing numbers consistently using digits or symbols. There are different types of numbers, such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Let us see what each type of number is with some examples.

Natural numbers:  Natural numbers are numbers starting from 1 and ending to infinity. These are positive integer numbers. It includes all whole numbers except 0. These numbers are denoted by the letter N.

Example: 1,2,3,4,5…..

Whole numbers:  Whole numbers are natural numbers, including 0, so they are numbers starting from 0 and ending at infinity. They are all positive numbers. These numbers are denoted by the letter W.

Example: 0,1,2,3,4,5…..

Integers:  Integers are a set of negative numbers, positive numbers including 0. These numbers do not include decimal or fractional numbers. The letter Z denotes these numbers.

Example: 3, 0, -74, etc.

Rational numbers: Any number written in the form of x/y and y≠0 is known as a rational number. The letter Q denotes the rational numbers.

Example: 1/4, 6/8, ⅔, etc.,

There are infinite rational numbers in between any two given rational numbers. To understand this concept, let us see an example.

Example: Find five rational numbers between 2 and 3

We know to find a rational number between x and y, we add x and y and divide it by 2, i.e. x+y/2 so, x+y/2 is the number between x and y

The numbers are 5/2, 4/2, 7/4, 5/4, 1/2.

By looking at this, we realise there are infinite rational numbers between any two given rational numbers.

Irrational numbers: Any number that cannot be represented as a ratio is an irrational number.

Example: √3, √5, etc.

Real numbers: Real numbers consist of those numbers which are rational such as positive integers, negative integers, fractions, and irrational numbers. The letter R denotes these numbers.

Example: 3, 0, 1.8, 6/4, etc.

### Terminating and recurring (non-terminating) decimal numbers

Terminating decimal numbers: Terminating decimal numbers are, as the name says is a decimal number that has an end. If the decimal expression of a fraction x/y ends, then it is known as terminating decimal numbers. These numbers end after a specific number of digits after the decimal point. For example: ⅛ = 0.125, 0.35, 12.64

Recurring decimal numbers or repeating decimal numbers: Decimal numbers in which a number or a set of numbers repeat periodically after the decimal point.

Example: 46.374374374…. , 0.6666….

### Properties of rational numbers:

1. Rational numbers are represented either as a terminating decimal or a non-terminating decimal.
2. All terminating decimal Example: √5are rational numbers.
3. All non-terminating decimal expressions are rational numbers.

### Properties of irrational numbers:

1. The non-terminating decimal numbers are irrational.
2. If x is a positive number, which is not a perfect square, then √x (square root of x ) is an irrational number.

Example: √5

1. If x is a positive integer, which is not a perfect cube, then ∛x (cube root of x) is an irrational number.

Example: ∛5

1. If you add two irrational numbers, the sum need not be an irrational number.

Example: (2+ √5) + (2- √5) = 4

1. If you subtract two irrational numbers, the difference need not be an irrational number.

Example: (7 + √2) – (3 + √2) = 4

1. If you multiply two irrational numbers, the product need not be an irrational number.  Example: √5 x √5 = 5
2. If you divide two irrational numbers, the quotient need not be irrational.

Example:   2√5/√5 =2

1. When you add a rational and an irrational number, the resultant sum is an irrational number.
2. When you subtract a rational and an irrational number, the difference is an irrational number.
3. When you multiply a rational and an irrational number, the product is an irrational number.
4. When you divide a rational and an irrational number, the quotient is an irrational number.

### Operations on real numbers.

Real numbers hold good for commutative law, associative law, distributive law for addition and multiplication, and closure property for rational numbers.

Similarly, real numbers hold good for commutative law, associative law, distributive law for addition and multiplication, and closure property for irrational numbers too.

1.  If you add a rational number and an irrational number, the sum is always an irrational number.
2.  If you subtract a rational number and an irrational number, the difference is always an irrational number.
3.  If you multiply a rational number that is not equal to zero and an irrational number, the product is always irrational.
4. If you divide a rational number that equals zero and an irrational number, the quotient is always an irrational number.
5. If you add, subtract, multiply or divide two irrational numbers, the solution may be rational or irrational.

### Rationalisation

Rationalisation is a process of removing radicals (removing the roots like square root or cube root) in the denominator. This can be done by multiplying the numerator and denominator by an appropriate number. The process of changing an irrational number into a rational number is known as rationalisation.

Example: To rationalise number 1/√x+y, we multiply this by √x+y/√x+y where x and y are integers.

Law of exponents for real numbers

Assume a, b, x and y to be natural numbers. We have the following law of exponents.

i) (ax).ay = a(x+y)

ii) (ax)y = axy

iii) ax/ay = a(x-y)

iv) ax.bx = (ab)x

### Conclusion

Thus, the number system consists of natural numbers, whole numbers, integers, rational numbers, and irrational numbers, as summarised above. This is a reference for the definition of the number system, the types, and their properties.