Number Systems Notes for Class 9

The number system is the method of using numbers to perform tasks like counting, measuring, and calculating. It is an essential part of mathematics, and it has many practical applications in our daily lives, from simple counting to more complex math problems.

Natural Numbers

Natural numbers are the numbers which we use for counting and the list starts from 1, 2, 3, 4, and so on.

• The set of natural numbers is written as N = {1, 2, 3, 4, …}.
• The smallest natural number is 1.
• There is no largest natural number because the list goes on forever.

Natural numbers are the base for all other types of numbers.

Whole Numbers

Whole numbers are the set of numbers that include all the natural numbers along with 0.

• The set of whole numbers is W = {0, 1, 2, 3, 4, …}.
• Whole numbers start from 0 and go on without an end.

Every natural number is a whole number, but zero is not a natural number.

Integers

Integers are an extension of the whole numbers and include all the whole numbers as well as their negative counterparts.

• The set of integers is Z = {…, -3, -2, -1, 0, 1, 2, 3, …}.
• Positive numbers are to the right of zero, and negative numbers are to the left.

Integers help in real-life situations like temperatures below zero or money borrowed.

Rational Numbers

A rational number is a number that can be expressed in the form p/q where p and q are integers and q ≠ 0.
Examples: ½, -¾, 5/2, and 0.

Rational numbers can be written as decimals too.

• Terminating decimals: End after a few digits, like 0.75.
• Non-terminating but repeating decimals: Repeat in a pattern, like 0.3333…

All integers and whole numbers are also rational numbers because they can be written as fractions (for example, 5 = 5/1).

Irrational Numbers

A number that cannot be written in the form of p/q is called an irrational number.

• Their decimal form never ends and never repeats.
• Example: √2 = 1.414213…

Irrational numbers fill the gaps between rational numbers on the number line.

Square Root Spiral

The square root spiral is a way to show square roots on paper.

• Start with a right-angled triangle with both sides equal to 1.
• The hypotenuse becomes √2.
• Keep adding triangles in the same way to get √3, √4, √5, and so on.

This makes a spiral pattern that helps us see how square roots increase step by step.

Rationalization

When a fraction has a square root in its denominator, we remove it using rationalization.
Example:

1/√3 = √3/3.

We multiply both the top and bottom by √3 to make the denominator rational. This process helps in solving equations easily.