Introduction & Types Of Integers

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Unit: Number System

Chapter: Introduction & Types of Integers

Reference: – Introduction to Real Numbers, Classification of Numbers, Rational & Irrational Numbers, Properties of Real Numbers, Operations on Real Numbers, The Number Line, Surds (Radicals), Laws of Exponents for Real Numbers, Density Property, Decimal Representation, Comparison & Ordering, Real-Life Applications

After studying this chapter, you should be able to understand:

• Introduction to Real Numbers
• Rational and Irrational Numbers
• Properties & Operation on Real Numbers
• Decimal Representation & The Number Line

Introduction to Real Numbers

Definition

Real Numbers are the set of all numbers that can be represented on a number line. They include rational numbers (such as integers, fractions, terminating and repeating decimals) and irrational numbers (such as √2, π, e).

The set of real numbers is denoted by the symbol ℝ.

When we classify real numbers, we essentially ask:

“What type of number is this — rational or irrational? Can it be expressed as a fraction?”

Once we identify the type, we can determine its properties, perform operations, and locate it on the number line.

Importance of Real Numbers

• Forms the foundation of algebra, calculus, and higher mathematics.
• Enables precise measurement of continuous quantities (length, time, temperature).
• Used in science, engineering, economics, and everyday calculations.
• Bridges the gap between discrete counting numbers and continuous quantities.

Example

Group: { -3, 0, ½, √2, π, 4.75 }
Common Property: All can be placed on a number line.
So, if “√-1” (imaginary number) was given, we could say it does not belong (since it is not a real number).
 

Subtopics

1. Concept of Real Numbers

Real numbers include every number you normally use in daily life — temperatures, bank balances, measurements, and more.

Key Points:

• Real numbers can be positive, negative, or zero.
• They can be rational (fractions) or irrational (non-repeating, non-terminating decimals).
• Every point on the number line corresponds to exactly one real number, and vice versa — this is called the completeness property.

2. Finding the Group Basis (Property)

The group basis for real numbers is usually whether a number is rational or irrational, or which subset (natural, whole, integer, rational, irrational) it belongs to.

Steps to Identify Real Number Subsets:

1. Observe the number carefully (look for decimal form, square roots, π, etc.).
2. Check if it can be written as p/q where p, q are integers and q ≠ 0.
3. Identify the smallest set it belongs to (Natural → Whole → Integer → Rational → Real).
4. Apply properties like density, closure, commutativity.

Example 1 – Classifying numbers:
Numbers: { -2, 0, 3, ½, √4 }

• √4 = 2, so all are rational.
• Common Property: All are rational numbers.

Example 2 – Rational vs Irrational:
Group: { √4, ⅓, 0.75, 2 }
Common Property: All are rational (√4 = 2).
Odd one out in { √2, √3, √4, √5 } → √4 (it is rational, others irrational).

Example 3 – Number line representation:
Group: { -1.5, 0, 2.3, √2 }
Common Property: All can be plotted on a number line.

Rational Numbers

Definition

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

The set of rational numbers is denoted by ℚ (from “quotient”).

Importance of Rational Numbers

• Used in fractions, ratios, proportions, and percentages.
• All terminating and repeating decimals are rational.
• Essential for measurements, cooking, construction, and finance.

Examples

• Group 1: { ½, ⅔, ¾, ⅝ } → All are positive fractions.
• Group 2: { -3, 0, 5, ½ } → All are rational numbers.

Subtopics

1. Integers as Rational Numbers

Every integer is rational because it can be written with denominator 1.
Examples: 5 = 5/1, -3 = -3/1, 0 = 0/1.

Quick Tip:
Natural numbers (1,2,3…), Whole numbers (0,1,2…), and Integers (…-2,-1,0,1,2…) are all subsets of rational numbers.

2. Fractions (Proper and Improper)

• Proper fraction: Numerator < denominator (e.g., ⅔, ⅞)
• Improper fraction: Numerator ≥ denominator (e.g., 5/3, 7/4)
• Mixed number: Whole number + proper fraction (e.g., 2½)

3. Terminating and Repeating Decimals

• Terminating decimals: Decimal ends after finite digits.
  Example: 0.75 = ¾, 0.125 = ⅛
  Reason: Denominator has only prime factors 2 and/or 5.
• Repeating (recurring) decimals: Decimal repeats a pattern infinitely.
  Example: 0.333… = ⅓, 0.142857142857… = 1/7

Special Note:
All terminating and repeating decimals are rational numbers.

Irrational Numbers

Definition

An irrational number is a real number that cannot be expressed as p/q, where p and q are integers and q ≠ 0. Its decimal expansion is non-terminating and non-repeating.

The set of irrational numbers has no standard symbol but is often written as ℚ‘ or ℝ ℚ.

Importance of Irrational Numbers

• Essential for geometry (diagonals, circles, spirals).
• Appear in physics (π in waves, e in growth/decay).
• Show that the number line has “gaps” that fractions cannot fill.

Examples

• Group 1: { √2, √3, √5, √7 } → All are square roots of non-perfect squares.
• Group 2: { π, e, φ (golden ratio = (1+√5)/2) } → Famous mathematical constants.

Subtopics

1. Square Roots of Non-Perfect Squares

Numbers like √2, √3, √5, √6, √7, √8, √10, etc., are irrational.

Quick Check:
If a positive integer is not a perfect square (1,4,9,16,25…), its square root is irrational.

2. Cube Roots and Higher Roots

Similarly, ∛2, ∛3, ∛5, etc. (where the radicand is not a perfect cube) are irrational.

3. Famous Irrational Constants

• π (pi) ≈ 3.1415926535… (ratio of circumference to diameter of a circle)
• e (Euler’s number) ≈ 2.7182818284… (base of natural logarithms)
• φ (golden ratio) ≈ 1.6180339887…

4. Sums and Products Involving Irrationals

• √2 + √3 is irrational.
• √2 × √3 = √6 is irrational.
• But √2 × √2 = 2 (rational) — irrational × irrational can be rational.

Special Note:
π and e are transcendental numbers (a stronger type of irrational — not roots of any polynomial with integer coefficients). √2 is algebraic irrational.

Decimal Representation of Real Numbers

Definition

Every real number has a unique decimal representation (except that terminating decimals can also be written as repeating 9’s).

Quick Rule:

• If decimal terminates or repeats → Rational
• If decimal never terminates and never repeats → Irrational

Properties of Real Numbers

Real numbers follow several important properties under addition and multiplication.

Key Point:
Real numbers are closed under addition, subtraction, multiplication, and division (except division by zero).

Operations on Real Numbers

1. Addition and Subtraction
Combine like terms. For irrationals, only combine if the irrational part is identical.

Example: 3√2 + 5√2 = 8√2
But 3√2 + 4√3 cannot be simplified further.

2. Multiplication
Multiply coefficients and multiply radicands separately.

Example: (2√3)(5√6) = 10√18 = 10 × 3√2 = 30√2

3. Division
Rationalize the denominator when needed.

Example: 1/√2 = √2/2

4. Rationalisation
Process of removing a radical from the denominator using conjugate.

Example: 1/(√3+√2) = (√3−√2)/((√3+√2)(√3−√2)) = (√3−√2)/(3−2) = √3−√2

The Number Line

Every real number corresponds to exactly one point on the number line.

Density Property:
Between any two distinct real numbers, there exists infinitely many rational numbers and infinitely many irrational numbers.

Surds (Radicals)

Definition

A surd is an irrational root of a rational number.
Example: √2, ∛5, √(10) are surds.
√4 = 2 is not a surd (it’s rational).

Types of Surds:

• Pure surd: √a where a has no factor that is a perfect power. Example: √3
• Mixed surd: k√a where k is rational. Example: 2√3

Operations with Surds:

• Addition: 3√5 + 2√5 = 5√5
• Multiplication: √a × √b = √(ab)
• Division: √a / √b = √(a/b)

Simplification: √72 = √(36×2) = 6√2

Laws of Exponents for Real Numbers

For a, b > 0 (real numbers) and rational exponents p, q:

Comparison & Ordering of Real Numbers

To compare two real numbers:

1. If both rational → convert to decimals or common denominator.
2. If one irrational → approximate decimal value.
3. Use square comparison: For a,b > 0, if a² > b² then a > b.

Example: Compare √7 and 2.8
√7 ≈ 2.64575 < 2.8

Example: Compare √5 + √3 and √6 + √2
Square both sides → (√5+√3)² = 5+3+2√15 = 8+2√15 ≈ 8+7.746=15.746
(√6+√2)² = 6+2+2√12 = 8+2√12 ≈ 8+6.928=14.928
Since 15.746 > 14.928, √5+√3 > √6+√2.

Real-Life Applications of Real Numbers

• Measurements: Length, weight, volume, temperature (all continuous quantities)
• Time: Hours, minutes, seconds (and fractional seconds)
• Finance: Interest rates, stock prices, currency exchange
• Science: Velocity, acceleration, force, energy
• Engineering: Dimensions, tolerances, stress/strain calculations
• Medicine: Dosages, vital signs, lab results
• Everyday life: Speed, distance, fuel efficiency, cooking measurements

Example Problem Set – Odd One Out (Classification Style)

Examine the six numbers below. Exactly one does NOT belong with the rest. Identify it and give three independent reasons (A) rational/irrational classification, (B) decimal expansion property, (C) algebraic / surd simplification property.

Items:

1. √16
2. √2
3. 0.333…
4. 22/7
5. 1.41421356…
6. (√3)²

Solution:

(A) Rational / Irrational Classification

• √16 = 4 → Rational
• √2 → Irrational
• 0.333… = ⅓ → Rational
• 22/7 → Rational
• 1.41421356… (non-terminating, no pattern) → Irrational
• (√3)² = 3 → Rational

So √2 and 1.41421356… are irrational; others rational. This alone doesn’t single out one.

(B) Decimal Expansion Property

• √16 = 4.000… (terminating)
• √2 = 1.41421356… (non-terminating, non-repeating)
• 0.333… = 0.{3} (repeating)
• 22/7 = 3.142857142857… (repeating pattern of length 6)
• 1.41421356… (non-terminating, non-repeating)
• (√3)² = 3.000… (terminating)

Here, two numbers (√2 and 1.41421356…) share non-terminating non-repeating property. Still not unique.

(C) Surd Simplification / Exact form

• √16 = 4 (exact integer)
• √2 = surd (cannot simplify)
• 0.333… = ⅓ (exact fraction)
• 22/7 = exact fraction
• 1.41421356… = approximation of √2, not exact representation
• (√3)² = 3 (exact integer)

Conclusion:
The number 1.41421356… is the odd one out because it is presented as a decimal approximation of √2 rather than in its exact surd form √2. All others are given in exact form (integer, fraction, repeating decimal with bar notation, or surd symbol). This is a semantic/representational oddity.

Thus the odd item is Item 5 (1.41421356…).