Class 8: Numbers (Lecture Notes)

$\displaystyle \textbf{Natural Numbers:}$

$\displaystyle \text{The counting numbers are called natural numbers.}$

$\displaystyle N=\{1,2,3,4,5,6,7,\ldots\}.$

$\displaystyle \textbf{Whole Numbers:}$

$\displaystyle \text{All natural numbers together with }0\text{ form the set }W\text{ of whole numbers.}$

$\displaystyle W=\{0,1,2,3,4,5,6,7,\ldots\}.$

$\displaystyle \textbf{Integers:}$

$\displaystyle \text{All natural numbers, their negatives, and }0\text{ together form the set }Z\text{ of integers.}$

$\displaystyle Z=\{\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots\}.$

$\displaystyle \textbf{Properties of Addition of Integers}$

$\displaystyle \textbf{Closure Property:}$
$\displaystyle \text{The sum of two integers is always an integer. If }a,b\in Z,\text{ then }a+b\in Z.$

$\displaystyle \textbf{Commutative Law:}$
$\displaystyle \text{If }a\text{ and }b\text{ are integers, then }a+b=b+a.$

$\displaystyle \textbf{Associative Law:}$
$\displaystyle \text{If }a,b\text{ and }c\text{ are integers, then }(a+b)+c=a+(b+c).$

$\displaystyle \textbf{Existence of Additive Identity:}$
$\displaystyle 0\text{ is the additive identity for integers. Therefore, for any integer }a,$
$\displaystyle a+0=0+a=a.$

$\displaystyle \textbf{Existence of Additive Inverse:}$
$\displaystyle \text{For every integer }a,\text{ there exists an integer }-a\text{ such that}$
$\displaystyle a+(-a)=(-a)+a=0.$

$\displaystyle \textbf{Successor and Predecessor:}$
$\displaystyle \text{If }a\text{ is any integer, then }a+1\text{ is called the successor of }a,$
$\displaystyle \text{and }a-1\text{ is called the predecessor of }a.$

$\displaystyle \textbf{Properties of Subtraction of Integers}$

$\displaystyle \textbf{Closure Property:}$
$\displaystyle \text{If }a\text{ and }b\text{ are integers, then }a-b\text{ is also an integer.}$

$\displaystyle \textbf{Subtraction of Zero:}$
$\displaystyle \text{For any integer }a,\text{ we have }a-0=a.$
$\displaystyle \text{However, }0-a=-a,\text{ which is not equal to }a\text{ unless }a=0.$

$\displaystyle \textbf{Order Property:}$
$\displaystyle \text{If }a,b,c\text{ are integers and }a>b,\text{ then }a-c>b-c.$

$\displaystyle \textbf{Properties of Multiplication of Integers}$

$\displaystyle \textbf{Closure Property:}$
$\displaystyle \text{The product of two integers is always an integer. If }a,b\in Z,\text{ then }ab\in Z.$

$\displaystyle \textbf{Commutative Law:}$
$\displaystyle \text{For any two integers }a\text{ and }b,\text{ we have }a\times b=b\times a.$

$\displaystyle \textbf{Associative Law:}$
$\displaystyle \text{For any three integers }a,b\text{ and }c,\text{ we have}$
$\displaystyle (a\times b)\times c=a\times(b\times c).$

$\displaystyle \textbf{Distributive Law of Multiplication over Addition:}$
$\displaystyle \text{For any three integers }a,b\text{ and }c,\text{ we have}$
$\displaystyle a\times(b+c)=a\times b+a\times c.$

$\displaystyle \textbf{Distributive Law of Multiplication over Subtraction:}$
$\displaystyle \text{For any three integers }a,b\text{ and }c,\text{ we have}$
$\displaystyle a\times(b-c)=a\times b-a\times c.$

$\displaystyle \textbf{Existence of Multiplicative Identity:}$
$\displaystyle 1\text{ is the multiplicative identity for integers. Therefore, for any integer }a,$
$\displaystyle a\times1=1\times a=a.$

$\displaystyle \textbf{Property of Zero:}$
$\displaystyle \text{For any integer }a,\text{ we have }a\times0=0\times a=0.$

$\displaystyle \textbf{Properties of Division of Integers}$

$\displaystyle \textbf{Integers are Not Closed under Division:}$
$\displaystyle \text{If }a\text{ and }b\text{ are integers and }b\neq0,\text{ then }\frac{a}{b}\text{ need not be an integer.}$

$\displaystyle \textbf{Division of a Non-Zero Integer by Itself:}$
$\displaystyle \text{If }a\text{ is a non-zero integer, then }\frac{a}{a}=1.$

$\displaystyle \textbf{Division by }1\textbf{:}$
$\displaystyle \text{If }a\text{ is an integer, then }\frac{a}{1}=a.$

$\displaystyle \textbf{Division of Zero and Division by Zero:}$
$\displaystyle \text{If }a\text{ is a non-zero integer, then }\frac{0}{a}=0,\text{ but }\frac{a}{0}\text{ is not defined.}$

$\displaystyle \textbf{Division is Not Associative:}$
$\displaystyle \text{For non-zero integers }b\text{ and }c,\text{ in general,}$
$\displaystyle (a\div b)\div c\neq a\div(b\div c).$

$\displaystyle \textbf{Order Property:}$
$\displaystyle \text{If }a>b\text{ and }c>0,\text{ then }\frac{a}{c}>\frac{b}{c}.$
$\displaystyle \text{If }a>b\text{ and }c<0,\text{ then }\frac{a}{c}<\frac{b}{c}.$

$\displaystyle \\$

$\displaystyle \textbf{Rational Numbers}$

$\displaystyle \text{Numbers that can be expressed in the form }\frac{p}{q},\text{ where }p\text{ and }q\text{ are integers}$
$\displaystyle \text{and }q\neq0,\text{ are called rational numbers.}$

$\displaystyle \text{Therefore, the set }Q\text{ of all rational numbers is given by}$

$\displaystyle Q=\left\{\frac{p}{q}:p,q\in Z\text{ and }q\neq0\right\}.$

$\displaystyle \text{Examples: }\frac{2}{5},\ \frac{1}{19},\ -\frac{1}{5},\ \frac{-117}{-11},\ \frac{0}{5},\ \frac{23}{15}.$

$\displaystyle \textbf{Note:}$

$\displaystyle \text{Every integer is a rational number, since every integer }a\text{ can be written as }\frac{a}{1}.$

$\displaystyle \text{Every fraction of the form }\frac{p}{q},\text{ where }p,q\in Z\text{ and }q\neq0,\text{ is a rational number.}$

$\displaystyle \text{Every root of a perfect power that gives an integer value is a rational number.}$
$\displaystyle \text{For example, }\sqrt{9}=3\text{ and }\sqrt[3]{8}=2.$

$\displaystyle \text{Every terminating decimal is a rational number.}$
$\displaystyle \text{For example, }0.9=\frac{9}{10}.$

$\displaystyle \text{Every recurring decimal is a rational number.}$
$\displaystyle \text{For example, }0.222\ldots=\frac{2}{9}.$

$\displaystyle \textbf{Positive Rational Numbers}$

$\displaystyle \text{A non-zero rational number is positive if its numerator and denominator have the same sign.}$

$\displaystyle \text{For example, }\frac{2}{5}\text{ and }\frac{-2}{-5}\text{ are positive rational numbers.}$

$\displaystyle \textbf{Negative Rational Numbers}$

$\displaystyle \text{A rational number is negative if its numerator and denominator have opposite signs.}$

$\displaystyle \text{For example, }\frac{-2}{5}\text{ and }\frac{2}{-5}\text{ are negative rational numbers.}$

$\displaystyle \textbf{Equivalent Rational Numbers}$

$\displaystyle \text{If }\frac{p}{q}\text{ is a rational number and }m\text{ is a non-zero integer, then}$

$\displaystyle \frac{p}{q}=\frac{pm}{qm}.$

$\displaystyle \text{Thus, }\frac{p}{q}\text{ and }\frac{pm}{qm}\text{ are called equivalent rational numbers.}$

$\displaystyle \text{For example, }\frac{2}{3}=\frac{2\times2}{3\times2}=\frac{4}{6}.$

$\displaystyle \textbf{Standard Form of a Rational Number}$

$\displaystyle \text{A rational number }\frac{p}{q}\text{ is said to be in standard form if }p\text{ and }q\text{ have no}$
$\displaystyle \text{common factor other than }1\text{ and }q>0.$

$\displaystyle \text{For example, }\frac{6}{8}=\frac{3}{4},\text{ and }\frac{3}{4}\text{ is in standard form.}$

$\displaystyle \text{Note: Every rational number can be represented on a number line.}$

$\displaystyle \textbf{Addition of Rational Numbers}$

$\displaystyle \text{Before adding rational numbers, it is convenient to write them with positive denominators.}$

$\displaystyle \textbf{Rational Numbers with the Same Denominator}$

$\displaystyle \text{If two rational numbers have the same denominator, add their numerators and retain}$
$\displaystyle \text{the common denominator.}$

$\displaystyle \frac{p}{q}+\frac{m}{q}=\frac{p+m}{q},\qquad q\neq0.$

$\displaystyle \text{For example, }\frac{2}{7}+\frac{3}{7}=\frac{2+3}{7}=\frac{5}{7}.$

$\displaystyle \textbf{Rational Numbers with Different Denominators}$

$\displaystyle \text{If the denominators are different, find the LCM of the denominators and express both}$
$\displaystyle \text{rational numbers with the LCM as their common denominator. Then add the numerators.}$

$\displaystyle \frac{p}{q}+\frac{m}{n}=\frac{pn+mq}{qn},\qquad q\neq0,\ n\neq0.$

$\displaystyle \text{For example, }\frac{2}{3}+\frac{1}{4}=\frac{8}{12}+\frac{3}{12}=\frac{11}{12}.$

$\displaystyle \text{The resulting rational number should be reduced to its standard form, if required.}$

$\displaystyle \\$

$\displaystyle \textbf{Properties of Addition of Rational Numbers}$

$\displaystyle \textbf{Closure Property:}$
$\displaystyle \text{The sum of two rational numbers is also a rational number.}$
$\displaystyle \text{If }\frac{p}{q}\text{ and }\frac{m}{n}\text{ are rational numbers, then }\frac{p}{q}+\frac{m}{n}\text{ is also rational.}$

$\displaystyle \textbf{Commutative Law:}$
$\displaystyle \text{If }\frac{p}{q}\text{ and }\frac{m}{n}\text{ are rational numbers, then}$
$\displaystyle \frac{p}{q}+\frac{m}{n}=\frac{m}{n}+\frac{p}{q}.$

$\displaystyle \textbf{Associative Law:}$
$\displaystyle \text{If }\frac{p}{q},\frac{m}{n}\text{ and }\frac{x}{y}\text{ are rational numbers, then}$
$\displaystyle \left(\frac{p}{q}+\frac{m}{n}\right)+\frac{x}{y}=\frac{p}{q}+\left(\frac{m}{n}+\frac{x}{y}\right).$

$\displaystyle \textbf{Existence of Additive Identity:}$
$\displaystyle \text{The rational number }0\text{ is the additive identity for rational numbers. Therefore,}$
$\displaystyle \frac{p}{q}+0=0+\frac{p}{q}=\frac{p}{q}.$

$\displaystyle \textbf{Existence of Additive Inverse:}$
$\displaystyle \text{For every rational number }\frac{p}{q},\text{ there exists an additive inverse }-\frac{p}{q}\text{ such that}$
$\displaystyle \frac{p}{q}+\left(-\frac{p}{q}\right)=0.$

$\displaystyle \textbf{Subtraction of Rational Numbers}$

$\displaystyle \textbf{Rational Numbers with the Same Denominator:}$
$\displaystyle \text{If }\frac{p}{q}\text{ and }\frac{m}{q}\text{ have the same denominator, then}$
$\displaystyle \frac{p}{q}-\frac{m}{q}=\frac{p-m}{q}.$

$\displaystyle \textbf{Rational Numbers with Different Denominators:}$
$\displaystyle \text{If the denominators are different, find the LCM of the denominators and express both}$
$\displaystyle \text{rational numbers with the LCM as their common denominator. Then subtract the numerators.}$

$\displaystyle \frac{p}{q}-\frac{m}{n}=\frac{pn-mq}{qn},\qquad q\neq0,\ n\neq0.$

$\displaystyle \text{The resulting rational number should be reduced to its standard form, if required.}$

$\displaystyle \textbf{Properties of Multiplication of Rational Numbers}$

$\displaystyle \textbf{Closure Property:}$
$\displaystyle \text{The product of two rational numbers is also a rational number. Therefore, if}$
$\displaystyle \frac{a}{b}\text{ and }\frac{c}{d}\text{ are rational numbers, then }\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}\text{ is rational.}$

$\displaystyle \textbf{Associative Law:}$
$\displaystyle \text{If }\frac{p}{q},\frac{m}{n}\text{ and }\frac{x}{y}\text{ are rational numbers, then}$
$\displaystyle \left(\frac{p}{q}\times\frac{m}{n}\right)\times\frac{x}{y}$
$\displaystyle =\frac{p}{q}\times\left(\frac{m}{n}\times\frac{x}{y}\right).$

$\displaystyle \textbf{Existence of Multiplicative Identity:}$
$\displaystyle \text{The rational number }1\text{ is the multiplicative identity for rational numbers. Therefore,}$
$\displaystyle \frac{a}{b}\times1=1\times\frac{a}{b}=\frac{a}{b}.$

$\displaystyle \textbf{Existence of Multiplicative Inverse:}$
$\displaystyle \text{Every non-zero rational number has a multiplicative inverse.}$
$\displaystyle \text{The multiplicative inverse of }\frac{a}{b}\text{ is }\frac{b}{a},\text{ where }a\neq0.$
$\displaystyle \frac{a}{b}\times\frac{b}{a}=1.$

$\displaystyle \textbf{Distributive Law of Multiplication over Addition:}$
$\displaystyle \text{For any three rational numbers }\frac{p}{q},\frac{m}{n}\text{ and }\frac{x}{y},\text{ we have}$
$\displaystyle \frac{p}{q}\times\left(\frac{m}{n}+\frac{x}{y}\right)$
$\displaystyle =\left(\frac{p}{q}\times\frac{m}{n}\right)+\left(\frac{p}{q}\times\frac{x}{y}\right).$

$\displaystyle \textbf{Distributive Law of Multiplication over Subtraction:}$
$\displaystyle \frac{p}{q}\times\left(\frac{m}{n}-\frac{x}{y}\right)$
$\displaystyle =\left(\frac{p}{q}\times\frac{m}{n}\right)-\left(\frac{p}{q}\times\frac{x}{y}\right).$

$\displaystyle \textbf{Multiplicative Property of Zero:}$
$\displaystyle \text{For any rational number }\frac{a}{b},\text{ we have}$
$\displaystyle \frac{a}{b}\times0=0\times\frac{a}{b}=0.$

$\displaystyle \textbf{Division of Rational Numbers}$

$\displaystyle \text{If }\frac{a}{b}\text{ and }\frac{c}{d}\text{ are rational numbers and }\frac{c}{d}\neq0,\text{ then}$

$\displaystyle \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{ad}{bc}.$

$\displaystyle \text{Thus, to divide by a non-zero rational number, multiply by its reciprocal.}$

$\displaystyle \text{In }\frac{a}{b}\div\frac{c}{d},\text{ the rational number }\frac{a}{b}\text{ is called the dividend,}$
$\displaystyle \frac{c}{d}\text{ is called the divisor, and the result is called the quotient.}$

$\displaystyle \textbf{Properties of Division of Rational Numbers}$

$\displaystyle \textbf{Closure Property:}$
$\displaystyle \text{The quotient of two rational numbers is a rational number, provided the divisor is non-zero.}$
$\displaystyle \text{If }\frac{a}{b},\frac{c}{d}\in Q\text{ and }\frac{c}{d}\neq0,\text{ then }\frac{a}{b}\div\frac{c}{d}\in Q.$

$\displaystyle \textbf{Division by }1\textbf{:}$
$\displaystyle \text{For any rational number }\frac{a}{b},\text{ we have }\frac{a}{b}\div1=\frac{a}{b}.$

$\displaystyle \textbf{Division of a Non-Zero Rational Number by Itself:}$
$\displaystyle \text{For any non-zero rational number }\frac{a}{b},\text{ we have }\frac{a}{b}\div\frac{a}{b}=1.$

$\displaystyle \textbf{Division of Zero:}$
$\displaystyle \text{For any non-zero rational number }\frac{a}{b},\text{ we have }0\div\frac{a}{b}=0.$

$\displaystyle \textbf{Division by Zero:}$
$\displaystyle \text{Division of any rational number by }0\text{ is not defined.}$

$\displaystyle \textbf{How to Find a Rational Number between Two Given Rational Numbers?}$

$\displaystyle \text{Let }\frac{a}{b}\text{ and }\frac{c}{d}\text{ be two distinct rational numbers. A rational number between}$
$\displaystyle \text{them is their arithmetic mean, given by}$

$\displaystyle \frac{1}{2}\left(\frac{a}{b}+\frac{c}{d}\right).$

$\displaystyle \text{In fact, there are infinitely many rational numbers between any two distinct rational numbers.}$

$\displaystyle \text{To find several rational numbers between two given rational numbers, first express them with}$
$\displaystyle \text{a suitable common denominator. If necessary, multiply both numerator and denominator}$
$\displaystyle \text{by the same positive integer to create enough integers between the two numerators.}$

$\displaystyle \text{For example, to find rational numbers between }\frac{1}{3}\text{ and }\frac{2}{3},\text{ write}$

$\displaystyle \frac{1}{3}=\frac{5}{15}\qquad\text{and}\qquad\frac{2}{3}=\frac{10}{15}.$

$\displaystyle \therefore \frac{6}{15},\frac{7}{15},\frac{8}{15}\text{ and }\frac{9}{15}\text{ are rational numbers between them.}$

$\displaystyle \\$

$\displaystyle \textbf{Irrational Numbers}$

$\displaystyle \text{A number whose decimal expansion is non-terminating and non-repeating is called an irrational number.}$

$\displaystyle \text{An irrational number cannot be expressed in the form }\frac{p}{q},\text{ where }p,q\in Z\text{ and }q\neq0.$

$\displaystyle \textbf{Examples of Irrational Numbers}$

$\displaystyle \text{Non-terminating and non-repeating decimals, such as }1.2456345789560\ldots$

$\displaystyle \text{Square roots of positive integers that are not perfect squares, such as}$
$\displaystyle \sqrt{2},\ \sqrt{5},\ \sqrt{7},\ldots$

$\displaystyle \text{Cube roots of integers that are not perfect cubes, such as}$
$\displaystyle \sqrt[3]{2},\ \sqrt[3]{5},\ \sqrt[3]{7},\ldots$

$\displaystyle \pi\text{ is an irrational number. Its decimal expansion is non-terminating and non-repeating.}$
$\displaystyle \text{Its approximate value is }\pi\approx3.14159.$

$\displaystyle \textbf{Properties of Irrational Numbers}$

$\displaystyle \textbf{Sum of Two Irrational Numbers:}$

$\displaystyle \text{The sum of two irrational numbers need not be irrational.}$
$\displaystyle \text{For example, }(5+\sqrt{2})+(5-\sqrt{2})=10,\text{ which is rational.}$

$\displaystyle \textbf{Difference of Two Irrational Numbers:}$

$\displaystyle \text{The difference of two irrational numbers need not be irrational.}$
$\displaystyle \text{For example, }(7+\sqrt{2})-(5+\sqrt{2})=2,\text{ which is rational.}$

$\displaystyle \textbf{Product of Two Irrational Numbers:}$

$\displaystyle \text{The product of two irrational numbers need not be irrational.}$
$\displaystyle \text{For example, }(5+\sqrt{2})(5-\sqrt{2})=25-2=23,\text{ which is rational.}$

$\displaystyle \textbf{Quotient of Two Irrational Numbers:}$

$\displaystyle \text{The quotient of two irrational numbers need not be irrational.}$
$\displaystyle \text{For example, }\frac{9\sqrt{2}}{3\sqrt{2}}=3,\text{ which is rational.}$

$\displaystyle \textbf{Commutative Laws:}$

$\displaystyle \text{For any real numbers }a\text{ and }b,$
$\displaystyle a+b=b+a.$
$\displaystyle a\times b=b\times a.$

$\displaystyle \textbf{Associative Laws:}$

$\displaystyle \text{For any real numbers }a,b\text{ and }c,$
$\displaystyle (a+b)+c=a+(b+c).$
$\displaystyle (a\times b)\times c=a\times(b\times c).$

$\displaystyle \textbf{Distributive Law:}$

$\displaystyle \text{For any real numbers }a,b\text{ and }c,$
$\displaystyle a\times(b+c)=a\times b+a\times c.$

$\displaystyle \textbf{Operations involving Rational and Irrational Numbers}$

$\displaystyle \text{The sum or difference of a rational number and an irrational number is always irrational.}$

$\displaystyle \text{For example, }2+\sqrt{5}\text{ is irrational.}$
$\displaystyle \text{Also, }2-\sqrt{5}\text{ is irrational.}$

$\displaystyle \text{The product of a non-zero rational number and an irrational number is always irrational.}$

$\displaystyle \text{For example, }2\times\sqrt{5}=2\sqrt{5}\text{ is irrational.}$

$\displaystyle \text{The quotient of a non-zero rational number by an irrational number is irrational,}$
$\displaystyle \text{provided the quotient is defined.}$

$\displaystyle \text{For example, }2\div\sqrt{5}=\frac{2}{\sqrt{5}}\text{ is irrational.}$

$\displaystyle \textbf{Real Numbers}$

$\displaystyle \text{The collection of all rational and irrational numbers forms the set of real numbers.}$

$\displaystyle \text{The set of real numbers is denoted by }R.$

$\displaystyle \textbf{Real Number Line}$

$\displaystyle \text{The real number line represents all real numbers.}$

$\displaystyle \text{Each point on the real number line corresponds to exactly one real number, and conversely,}$
$\displaystyle \text{every real number corresponds to exactly one point on the real number line.}$

$\displaystyle \text{Between any two distinct real numbers, there are infinitely many real numbers.}$

$\displaystyle \textbf{Properties of Real Numbers}$

$\displaystyle \textbf{Closure Property:}$

$\displaystyle \text{The sum, difference and product of two real numbers are always real numbers.}$
$\displaystyle \text{The quotient of two real numbers is also a real number, provided the divisor is non-zero.}$