Mathematics Class 11 - Number Systems Notes

Number Systems

Welcome to the chapter on Number Systems for Class 11. In this chapter, you will learn about different types of numbers, their properties, and how they are used in mathematics. By the end of this chapter, you will understand the structure of number systems and be able to solve problems involving real and complex numbers.

Key Concepts

• Natural Numbers (ℕ): Counting numbers starting from 1 (1, 2, 3, ...).
• Whole Numbers (𝑥): Natural numbers including 0 (0, 1, 2, 3, ...).
• Integers (ℤ): Positive and negative whole numbers including 0 (..., -2, -1, 0, 1, 2, ...).
• 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.
• Complex Numbers (ℂ): Numbers of the form a + bi, where a and b are real numbers and i = √-1.

Representation of Numbers on the Number Line

Real numbers can be represented on a number line. Rational and irrational numbers both have unique positions on the number line.

Properties of Real Numbers

• Closure: The sum, difference, and product of any two real numbers is a real number.
• Commutativity: a + b = b + a and a × b = b × a for all real numbers a and b.
• Associativity: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
• Distributivity: a × (b + c) = a × b + a × c.
• Identity: 0 is the additive identity, 1 is the multiplicative identity.
• Inverse: For every real number a, there exists -a such that a + (-a) = 0; for a ≠ 0, there exists 1/a such that a × (1/a) = 1.

Irrational Numbers

• Cannot be expressed as a ratio of two integers.
• Their decimal expansion is non-terminating and non-repeating.
• Examples: √2, √3, π, e.

Complex Numbers

• A complex number is written as z = a + bi, where a is the real part and b is the imaginary part.
• i is the imaginary unit, defined as i² = -1.
• Complex numbers can be represented on the Argand plane.

Practice Questions

1. Classify the following numbers as rational or irrational: 3/4, √5, 0.333..., π.
2. Represent -2, 0, and √2 on a number line.
3. Write the additive and multiplicative inverse of 7.
4. Express 5 - 3i in the form a + bi and identify its real and imaginary parts.
5. Is the sum of a rational and an irrational number always irrational? Explain with an example.

Challenge Yourself

• Prove that √2 is an irrational number.
• If z₁ = 2 + 3i and z₂ = 1 - 4i, find z₁ + z₂ and z₁ × z₂.

Did You Know?

• The set of real numbers is uncountable, while the set of rational numbers is countable.
• Complex numbers are used in engineering, physics, and signal processing.

Glossary

• Natural Numbers: Counting numbers starting from 1.
• Rational Numbers: Numbers that can be written as a fraction.
• Irrational Numbers: Numbers that cannot be written as a fraction.
• Complex Numbers: Numbers of the form a + bi.
• Argand Plane: A plane to represent complex numbers graphically.

Answers to Practice Questions

1. 3/4: Rational, √5: Irrational, 0.333...: Rational, π: Irrational
2. -2 and 0 are marked on the number line; √2 is between 1 and 2.
3. Additive inverse: -7, Multiplicative inverse: 1/7
4. a = 5, b = -3; Real part: 5, Imaginary part: -3
5. Yes. Example: 2 (rational) + √2 (irrational) = 2 + √2 (irrational)

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Understanding number systems is the foundation for higher mathematics. Practice and explore more to master this topic!