Class 9 Number Systems JEE NOTES

What is a Number System?

A number system is a way to represent numbers. Different number systems are used in daily life, computers, and mathematics.

2. Types of Numbers

1. Natural Numbers (N)

   • Counting numbers starting from 1.

   • Example: 1, 2, 3, 4…

2. Whole Numbers (W)

   • Natural numbers including 0.

   • Example: 0, 1, 2, 3…

3. Integers (Z)

   • Whole numbers including negative numbers.

   • Example: -3, -2, -1, 0, 1, 2, 3…

4. Rational Numbers (Q)

   • Numbers that can be expressed as fraction of two integers.

   • Example: 1/2, -3/4, 5/1

5. Irrational Numbers

   • Numbers that cannot be written as fractions.

   • They have non-repeating, non-terminating decimal forms.

   • Example: √2, π

6. Real Numbers (R)

   • All rational and irrational numbers together.

   • Represents any number on the number line.

7. Imaginary Numbers (for reference)

   • Numbers involving √-1, used in advanced mathematics.

3. Important Concepts

1. Even and Odd Numbers

   • Even → divisible by 2.

   • Odd → not divisible by 2.

2. Prime Numbers

   • Numbers greater than 1 with only two factors: 1 and itself.

3. Composite Numbers

   • Numbers having more than two factors.

4. Factors and Multiples

   • Factor → number that divides another number completely.

   • Multiple → result of multiplying a number with any other whole number.

4. Representation of Numbers

1. Decimal System (Base 10) – Numbers using digits 0–9.

2. Binary System (Base 2) – Numbers using 0 and 1 (used in computers).

3. Octal System (Base 8) – Numbers using 0–7.

4. Hexadecimal System (Base 16) – Numbers using 0–9 and A–F.

5. Key Notes for Exams

• Natural → counting numbers.

• Whole → natural + 0.

• Integers → whole numbers + negatives.

• Rational → fractions; Irrational → non-repeating decimals.

• Real numbers → rational + irrational.

• Prime → only two factors; Composite → more than two factors.

• Number systems: Decimal, Binary, Octal, Hexadecimal.