Mathematics Class 9 - Number Systems Notes

Number Systems

In this chapter, you will learn about different types of numbers, their properties, and how to use them in calculations. By the end of this chapter, you will understand the number system, its types, and how numbers are represented and used in mathematics.

Key Concepts

• Natural Numbers (N): Counting numbers starting from 1 (1, 2, 3, ...).
• Whole Numbers (W): Natural numbers including 0 (0, 1, 2, 3, ...).
• Integers (Z): All positive and negative whole numbers, including 0 (..., -3, -2, -1, 0, 1, 2, 3, ...).
• Rational Numbers (Q): Numbers that can be written as p/q, where p and q are integers and q ≠ 0.
• Irrational Numbers: Numbers that cannot be written as p/q (e.g., √2, π).
• Real Numbers (R): All rational and irrational numbers.

Representation of Numbers on the Number Line

All real numbers can be shown on a number line. Integers are marked at equal distances, and fractions/decimals are placed between them.

Properties of Rational and Irrational Numbers

• Rational numbers can be expressed as terminating or repeating decimals.
• Irrational numbers have non-terminating, non-repeating decimal expansions.
• Sum, difference, and product of two rational numbers is always rational.
• Sum or product of a rational and an irrational number is always irrational (except when the rational number is zero).

Operations on Real Numbers

• Addition, subtraction, multiplication, and division can be performed on real numbers.
• The result of these operations is also a real number (except division by zero).

Important Points

• Every natural number is a whole number, but not every whole number is a natural number.
• Every integer is a rational number, but not every rational number is an integer.
• Every rational and irrational number is a real number.

Practice Questions

1. Classify the following numbers as natural, whole, integer, rational, or irrational: 0, -5, 3/4, √3, 7.
2. Write the decimal expansion of 1/3. Is it rational or irrational?
3. Is π a rational or irrational number? Why?
4. Place -2, 0, 1.5, and √2 on a number line.
5. What is the sum of a rational and an irrational number?

Challenge Yourself

• Find two irrational numbers whose sum is rational.
• Give an example of a non-terminating, repeating decimal and a non-terminating, non-repeating decimal.

Did You Know?

• The decimal expansion of π never ends and never repeats!
• The set of real numbers includes all numbers you use in daily life, except imaginary numbers.

Glossary

• Natural Numbers: Counting numbers starting from 1.
• Whole Numbers: Natural numbers plus zero.
• Integer: Positive and negative whole numbers, including zero.
• Rational Number: Number that can be written as p/q.
• Irrational Number: Number that cannot be written as p/q.
• Real Number: All rational and irrational numbers.

Answers to Practice Questions

1. 0: whole, integer, rational, real; -5: integer, rational, real; 3/4: rational, real; √3: irrational, real; 7: natural, whole, integer, rational, real.
2. 1/3 = 0.333... (repeating); it is rational.
3. π is irrational because its decimal expansion is non-terminating and non-repeating.
4. Mark -2, 0, 1.5, and √2 (between 1 and 2) on the number line.
5. The sum is always irrational (unless the rational number is zero).

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Understanding number systems is the foundation for all higher mathematics!