Number System

Integers, Rational Numbers, Exponents and Powers

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Overview

The Number System forms the foundational backbone of upper-primary mathematics and is a consistently tested area in PSTET Paper II. This topic builds upon whole numbers learned in primary classes and extends understanding to negative numbers, fractions expressed as ratios, and the compact representation of repeated multiplication through exponents.

For PSTET, you must demonstrate both content mastery and pedagogical awareness. Questions typically test conceptual clarity on integers and rational numbers, computational fluency with exponents, and the ability to identify common student misconceptions. Expect 3–5 direct questions from this topic, often integrated with problem-solving scenarios.

Mastering this topic requires understanding the hierarchy: Natural Numbers ⊂ Whole Numbers ⊂ Integers ⊂ Rational Numbers. Each expansion of the number system was historically motivated by the need to solve equations that couldn't be solved in the previous system.

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Key Concepts

• **Integers** include all whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Zero is neither positive nor negative.

• **Rational numbers** are numbers expressible as p/q where p and q are integers and q ≠ 0. Every integer is rational (e.g., 5 = 5/1), but not every rational is an integer.

• **Additive inverse**: For any integer a, there exists -a such that a + (-a) = 0. This concept explains why subtraction of integers is rewritten as addition of the inverse.

• **Multiplicative inverse (reciprocal)**: For any non-zero rational p/q, its reciprocal is q/p such that (p/q) × (q/p) = 1.

• **Exponents** represent repeated multiplication: aⁿ means 'a' multiplied by itself 'n' times. Here, 'a' is the base and 'n' is the exponent (or power).

• **Laws of exponents** provide shortcuts for operations involving powers with the same base or same exponent.

• **Negative exponents** represent reciprocals: a⁻ⁿ = 1/aⁿ. This extends the exponent system to include division operations.

• **Standard form (scientific notation)** expresses very large or very small numbers as a × 10ⁿ where 1 ≤ a < 10.

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Formulas / Key Facts

**Integer Operations:**

• a + (-b) = a - b
• (-a) + (-b) = -(a + b)
• (-a) × (-b) = +ab
• (-a) × (+b) = -ab
• Division follows the same sign rules as multiplication

**Rational Number Operations:**

• Addition/Subtraction: a/b ± c/d = (ad ± bc)/bd
• Multiplication: (a/b) × (c/d) = ac/bd
• Division: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc

**Laws of Exponents (a, b ≠ 0; m, n are integers):**

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• aᵐ × bᵐ = (ab)ᵐ
• aᵐ ÷ bᵐ = (a/b)ᵐ
• a⁰ = 1 (for a ≠ 0)
• a⁻ⁿ = 1/aⁿ

**Key Facts:**

• Between any two rational numbers, infinite rational numbers exist (density property)
• Rational numbers can be represented on a number line
• Every terminating or repeating decimal is a rational number

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Worked Examples

**Example 1: Integer Operation** Evaluate: (-15) + 8 - (-12) + (-5)

Step 1: Rewrite subtraction as addition of inverse = (-15) + 8 + 12 + (-5)

Step 2: Group positive and negative numbers Positives: 8 + 12 = 20 Negatives: (-15) + (-5) = -20

Step 3: Combine = 20 + (-20) = 0

**Example 2: Rational Number Division** Simplify: (-3/7) ÷ (9/14)

Step 1: Convert division to multiplication by reciprocal = (-3/7) × (14/9)

Step 2: Multiply numerators and denominators = (-3 × 14)/(7 × 9) = -42/63

Step 3: Simplify by finding HCF (HCF of 42 and 63 is 21) = -42/63 = -2/3

**Example 3: Exponents** Simplify: (2³ × 2⁵) ÷ 2⁴ × 2⁻²

Step 1: Apply multiplication law in numerator 2³ × 2⁵ = 2³⁺⁵ = 2⁸

Step 2: Apply division law 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴

Step 3: Multiply with 2⁻² 2⁴ × 2⁻² = 2⁴⁺⁽⁻²⁾ = 2² = 4

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Common Mistakes

• **Wrong thinking**: (-5) - (-3) = -8 because "two negatives make more negative."

**Correct fix**: Subtracting a negative means adding the positive. (-5) - (-3) = (-5) + 3 = -2.

• **Wrong thinking**: 0/5 = undefined, same as 5/0.

**Correct fix**: 0/5 = 0 (zero divided by any non-zero number is zero). Only 5/0 is undefined (division by zero).

• **Wrong thinking**: a² × a³ = a⁶ (multiplying exponents).

**Correct fix**: When bases are same and we multiply, we add exponents: a² × a³ = a⁵.

• **Wrong thinking**: (2 + 3)² = 2² + 3² = 4 + 9 = 13.

**Correct fix**: First compute inside brackets: (2 + 3)² = 5² = 25. Exponents don't distribute over addition.

• **Wrong thinking**: 2⁻³ = -8 (negative exponent means negative answer).

**Correct fix**: Negative exponent means reciprocal: 2⁻³ = 1/2³ = 1/8 (positive).

• **Wrong thinking**: -3/4 and 3/-4 are different numbers.

**Correct fix**: Both represent the same rational number. Standard form places the negative sign with the numerator: -3/4.

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Quick Reference

• **Integer signs**: Same signs multiply/divide to positive; different signs give negative.

• **Rational number**: p/q where q ≠ 0; includes all integers and fractions.

• **Exponent multiplication**: Same base → add powers; same power → multiply bases.

• **Zero exponent**: Any non-zero number raised to power 0 equals 1.

• **Negative exponent**: a⁻ⁿ = 1/aⁿ (flip to denominator, make exponent positive).

• **Standard form**: Shift decimal to get one non-zero digit before it; count shifts as power of 10.