Real Numbers and Laws of Exponents jee notes

Real Numbers (Conceptual)

• Real numbers include all rational and irrational numbers.

• They can be represented on a number line.

• Examples: 5, -3, 0.75, √2, π

Important Points About Real Numbers

1. Closure Property – Adding, subtracting, or multiplying two real numbers always gives a real number.

2. Commutative Property – The order of addition or multiplication does not affect the result.

3. Associative Property – Changing the grouping of numbers does not affect the result.

4. Distributive Property – Multiplication can be distributed over addition or subtraction.

5. Identity –

   • Addition identity = 0 → adding 0 does not change the number

   • Multiplication identity = 1 → multiplying by 1 does not change the number

6. Inverse –

   • Additive inverse → number that adds to zero

   • Multiplicative inverse → number that multiplies to one

2. Laws of Exponents (Conceptual)

Exponents are shortcuts to repeated multiplication.

Important Concepts

1. Product of Powers → When multiplying numbers with the same base, you add their exponents.

2. Quotient of Powers → When dividing numbers with the same base, you subtract their exponents.

3. Power of a Power → When an exponent is raised to another exponent, you multiply the exponents conceptually.

4. Zero Exponent → Any non-zero number raised to zero is 1.

5. Negative Exponent → A negative exponent represents reciprocal (flipping the number).

Key Points for Exams

• Real numbers = rational + irrational; can be shown on a number line

• Properties of real numbers: closure, commutative, associative, distributive, identity, inverse

• Exponents: shortcut for repeated multiplication

• Laws of exponents:

  • Product → add exponents

  • Quotient → subtract exponents

  • Power of power → multiply exponents

  • Zero exponent → 1

  • Negative exponent → reciprocal