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 <\/p>\n
After studying this chapter, you should be able to understand: <\/p>\n
Introduction to Real Numbers <\/p>\n<\/li>\n<\/ul>\n
Rational and Irrational Numbers <\/p>\n<\/li>\n<\/ul>\n
Properties & Operation on Real Numbers <\/p>\n<\/li>\n<\/ul>\n
Decimal Representation & The Number Line <\/p>\n<\/li>\n<\/ul>\n
Introduction to Real Numbers <\/p>\n
Definition <\/p>\n
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). <\/p>\n
The set of real numbers is denoted by the symbol\u202f\u211d. <\/p>\n
When we classify real numbers, we essentially ask: <\/p>\n
"What type of number is this — rational or irrational? Can it be expressed as a fraction?" <\/p>\n
Once we identify the type, we can determine its properties, perform operations, and locate it on the number line. <\/p>\n
Importance of Real Numbers <\/p>\n
Forms the foundation of algebra, calculus, and higher mathematics. <\/p>\n<\/li>\n<\/ul>\n
Enables precise measurement of continuous quantities (length, time, temperature). <\/p>\n<\/li>\n<\/ul>\n
Used in science, engineering, economics, and everyday calculations. <\/p>\n<\/li>\n<\/ul>\n
Bridges the gap between discrete counting numbers and continuous quantities. <\/p>\n<\/li>\n<\/ul>\n
Example <\/p>\n
Group:\u202f{ -3, 0, ½, √2, π, 4.75 }
\nCommon Property:\u202fAll can be placed on a number line.
\nSo, if "√-1" (imaginary number) was given, we could say it does not belong (since it is not a real number).
\n <\/p>\n
Subtopics <\/p>\n
1. Concept of Real Numbers <\/p>\n
Real numbers include every number you normally use in daily life — temperatures, bank balances, measurements, and more. <\/p>\n
Key Points: <\/p>\n
Real numbers can be\u202fpositive, negative, or zero. <\/p>\n<\/li>\n<\/ul>\n
They can be\u202frational\u202f(fractions) or\u202firrational\u202f(non-repeating, non-terminating decimals). <\/p>\n<\/li>\n<\/ul>\n
Every point on the number line corresponds to exactly one real number, and vice versa — this is called the\u202fcompleteness property. <\/p>\n<\/li>\n<\/ul>\n
2. Finding the Group Basis (Property) <\/p>\n
The group basis for real numbers is usually whether a number is\u202frational\u202for\u202firrational, or which subset (natural, whole, integer, rational, irrational) it belongs to. <\/p>\n
Steps to Identify Real Number Subsets: <\/p>\n
Observe\u202fthe number carefully (look for decimal form, square roots, π, etc.). <\/p>\n<\/li>\n<\/ol>\n
Check\u202fif it can be written as p\/q where p, q are integers and q ≠ 0. <\/p>\n<\/li>\n<\/ol>\n
Identify\u202fthe smallest set it belongs to (Natural → Whole → Integer → Rational → Real). <\/p>\n<\/li>\n<\/ol>\n
Apply\u202fproperties like density, closure, commutativity. <\/p>\n<\/li>\n<\/ol>\n
Example 1 – Classifying numbers:
\nNumbers: { -2, 0, 3, ½, √4 } <\/p>\n
√4 = 2, so all are rational. <\/p>\n<\/li>\n<\/ul>\n
Common Property: All are rational numbers. <\/p>\n<\/li>\n<\/ul>\n
Example 2 – Rational vs Irrational:
\nGroup: { √4, \u2153, 0.75, 2 }
\nCommon Property: All are rational (√4 = 2).
\nOdd one out in { √2, √3, √4, √5 } →\u202f√4\u202f(it is rational, others irrational). <\/p>\n
Example 3 – Number line representation:
\nGroup: { -1.5, 0, 2.3, √2 }
\nCommon Property: All can be plotted on a number line. <\/p>\n
Rational Numbers <\/p>\n
Definition <\/p>\n
A rational number is any number that can be expressed in the form\u202fp\/q, where p and q are integers and\u202fq ≠ 0. <\/p>\n
The set of rational numbers is denoted by\u202f\u211a\u202f(from "quotient"). <\/p>\n
Importance of Rational Numbers <\/p>\n
Used in fractions, ratios, proportions, and percentages. <\/p>\n<\/li>\n<\/ul>\n
All terminating and repeating decimals are rational. <\/p>\n<\/li>\n<\/ul>\n
Essential for measurements, cooking, construction, and finance. <\/p>\n<\/li>\n<\/ul>\n
Examples <\/p>\n
Group 1:\u202f{ ½, \u2154, ¾, \u215d } → All are positive fractions. <\/p>\n<\/li>\n<\/ul>\n
Group 2:\u202f{ -3, 0, 5, ½ } → All are rational numbers. <\/p>\n<\/li>\n<\/ul>\n
Subtopics <\/p>\n
1. Integers as Rational Numbers <\/p>\n
Every integer is rational because it can be written with denominator 1.
\nExamples: 5 = 5\/1, -3 = -3\/1, 0 = 0\/1. <\/p>\n
Quick Tip:
\nNatural numbers (1,2,3…), Whole numbers (0,1,2…), and Integers (…-2,-1,0,1,2…) are all subsets of rational numbers. <\/p>\n
2. Fractions (Proper and Improper) <\/p>\n
Proper fraction:\u202fNumerator < denominator (e.g., \u2154, \u215e) <\/p>\n<\/li>\n<\/ul>\n
Improper fraction:\u202fNumerator ≥ denominator (e.g., 5\/3, 7\/4) <\/p>\n<\/li>\n<\/ul>\n
Mixed number:\u202fWhole number + proper fraction (e.g., 2½) <\/p>\n<\/li>\n<\/ul>\n
3. Terminating and Repeating Decimals <\/p>\n
Terminating decimals:\u202fDecimal ends after finite digits.
\n\tExample: 0.75 = ¾, 0.125 = \u215b
\n\tReason:\u202fDenominator has only prime factors 2 and\/or 5. <\/p>\n<\/li>\n<\/ul>\n
Repeating (recurring) decimals:\u202fDecimal repeats a pattern infinitely.
\n\tExample: 0.333… = \u2153, 0.142857142857… = 1\/7 <\/p>\n<\/li>\n<\/ul>\n
Special Note:
\nAll terminating and repeating decimals are rational numbers. <\/p>\n
Irrational Numbers <\/p>\n
Definition <\/p>\n
An irrational number is a real number that\u202fcannot\u202fbe expressed as p\/q, where p and q are integers and q ≠ 0. Its decimal expansion is\u202fnon-terminating and non-repeating. <\/p>\n
The set of irrational numbers has no standard symbol but is often written as\u202f\u211a'\u202for\u202f\u211d \u211a. <\/p>\n
Importance of Irrational Numbers <\/p>\n
Essential for geometry (diagonals, circles, spirals). <\/p>\n<\/li>\n<\/ul>\n
Appear in physics (π in waves, e in growth\/decay). <\/p>\n<\/li>\n<\/ul>\n
Show that the number line has "gaps" that fractions cannot fill. <\/p>\n<\/li>\n<\/ul>\n
Examples <\/p>\n
Group 1:\u202f{ √2, √3, √5, √7 } → All are square roots of non-perfect squares. <\/p>\n<\/li>\n<\/ul>\n
Group 2:\u202f{ π, e, φ (golden ratio = (1+√5)\/2) } → Famous mathematical constants. <\/p>\n<\/li>\n<\/ul>\n
Subtopics <\/p>\n
1. Square Roots of Non-Perfect Squares <\/p>\n
Numbers like √2, √3, √5, √6, √7, √8, √10, etc., are irrational. <\/p>\n
Quick Check:
\nIf a positive integer is\u202fnot\u202fa perfect square (1,4,9,16,25…), its square root is irrational. <\/p>\n
2. Cube Roots and Higher Roots <\/p>\n
Similarly, \u221b2, \u221b3, \u221b5, etc. (where the radicand is not a perfect cube) are irrational. <\/p>\n
3. Famous Irrational Constants <\/p>\n
π (pi)\u202f≈ 3.1415926535… (ratio of circumference to diameter of a circle) <\/p>\n<\/li>\n<\/ul>\n
e (Euler's number)\u202f≈ 2.7182818284… (base of natural logarithms) <\/p>\n<\/li>\n<\/ul>\n
φ (golden ratio)\u202f≈ 1.6180339887… <\/p>\n<\/li>\n<\/ul>\n
4. Sums and Products Involving Irrationals <\/p>\n
√2 + √3 is irrational. <\/p>\n<\/li>\n<\/ul>\n
√2 × √3 = √6 is irrational. <\/p>\n<\/li>\n<\/ul>\n
But √2 × √2 = 2 (rational) — irrational × irrational can be rational. <\/p>\n<\/li>\n<\/ul>\n
Special Note:
\nπ and e are\u202ftranscendental numbers\u202f(a stronger type of irrational — not roots of any polynomial with integer coefficients). √2 is algebraic irrational. <\/p>\n
Decimal Representation of Real Numbers <\/p>\n
Definition <\/p>\n
Every real number has a unique decimal representation (except that terminating decimals can also be written as repeating 9's). <\/p>\n
| \n Type of Number <\/p>\n<\/td>\n | \n Decimal Form <\/p>\n<\/td>\n | \n Example <\/p>\n<\/td>\n<\/tr>\n | |||||||||||||||||||||||||||||||||||
| \n Rational (terminating) <\/p>\n<\/td>\n | \n Ends after finite digits <\/p>\n<\/td>\n | \n 0.75, 2.5, 3.0 <\/p>\n<\/td>\n<\/tr>\n | |||||||||||||||||||||||||||||||||||
| \n Rational (repeating) <\/p>\n<\/td>\n | \n Infinite repeating pattern <\/p>\n<\/td>\n | \n overline{3}, overline{6} <\/p>\n<\/td>\n<\/tr>\n | |||||||||||||||||||||||||||||||||||
| \n Irrational <\/p>\n<\/td>\n | \n Infinite, no repeating pattern <\/p>\n<\/td>\n | \n 1.41421356… (√2) <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n Quick Rule: <\/p>\n
|