Class 8: Numbers (Lecture Notes)

Natural Numbers: The counting numbers are called natural numbers.

Whole Numbers: All natural numbers together along with $0$ (zero) form the set W of all whole numbers.

Integers: All natural numbers, negative natural numbers and $0$ (zero) together form a set $Z$ of all Integers.

Properties of Addition of Integers

The sum of two integers is always an integer. If $a$ and $b$ are two integers, then: $a + b = b + a$

Associative Law: If $a, b$ , and $c$ are integers, then: $(a + b) + c = a + (b + c)$

Existence of Additive Identity: $0$ is the additive identity for integers. Therefore for any integer $a$ we will have: $a + 0 = 0 + a = a$

Existence of additive inverse: For each integer $a$ , there exists another integer $(-a)$ such that $a + (-a) = 0$

Note: If $a$ be any integer, then $(a+1)$ will be called the successor of $a$ and the predecessor of $a$ is $(a-1)$

Properties of Subtraction

If $a$ and $b$ are two integers, then $(a-b)$ is also an integer.

For any integer $a$ , we have $(a - 0) = a$ . But $(0 - a) \neq a$

If $a, b, c$ are integers and $a > b$ , then $(a - c) > (b - c)$

Properties of Multiplication of Integers

Closure Property: The product of two integer is always an integer

Commutative Law: For any two integers $a$ and $b$ we have: $a \times b = b \times a$

Associative Law: For any three integers $a, b$ , and $c$ we have: $(a \times b) \times c = (a \times (b \times c)$

Distributive Law of Multiplication over Addition: For any there integers $a, b$ , and $c$ we have: $a \times (b + c) = a \times b + a \times c$

Existence of Multiplicative Identity: The integer $1$ is a multiplicative identity for Integers. So, for any integer a we have: $a \times 1 = 1 \times a = a$

Property of Zero: For any integer we have: $a \times 0 = 0 \times a = 0$

Properties of Multiplication of Integers

If $a$ and $b$ are integers, then $\displaystyle ( \frac{a}{b} )$ is not necessarily an integer.

If $a$ is an integer and $a \neq 0$ , then $($ $\frac{a}{a}$ $) = 1$

If $a$ is an integer, then $($ $\frac{a}{1}$ $) = a$

If $a$ is a non-zero integer, then $($ $\frac{0}{a}$ $) = 0$ but $($ $\frac{a}{0}$ $)$ is not defined.

If $a, b, c$ are integers, then $($ $\frac{a}{b}$ $) \div c \neq a \div ($ $\frac{b}{c}$ $)$ , unless $c = 1$

If $a, b, c$ , are integers and $a > b$ then

$(a \div c) > (b \div c)$ , if $c$ is positive

$(a \div c) < (b \div c)$ , if $c$ is negative

Rational Numbers

The numbers that can be expressed in the form $($ $\frac{p}{q}$ $)$ , where $p$ and $q$ are integers and $q \neq 0$ are called rational numbers. Therefore, the set $Q$ of all rational numbers is given by

$\displaystyle Q = \{ \frac{p}{q} : p, q \in Z \ and \ q \neq 0 \}$

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

Note:

Every integer is a rational number, since every integer $a$ can be written as $\frac{a}{1}$
Every fraction is a rational number
The square root of every perfect square is a rational number E.g. $\sqrt{9}, \sqrt[3]{8}$

Every terminating decimal is a rational number E.g., $0.9 =$ $\frac{9}{10}$

Every recurring decimal is a rational number E.g $0.222222 =$ $\frac{2}{9}$

Positive Rational Numbers: A rational number is said to be positive the numerator or denominator are either both positive and both negative. e.g. $\frac{2}{5}, \frac{-2}{-5}$

Negative Rational Numbers: A rational number is said to be negative is any one of the numerator or denominator is negative (they are of opposite sign) e.g. $\frac{2}{5}, \frac{1}{19}$

Equivalent Rational Numbers

If $\frac{p}{q}$ is a rational number and $m$ is a non-zero integer, then we have $\frac{p}{q} = \frac{pm}{qm}$ . We call these numbers $\frac{p}{q}$ and $\frac{pm}{qm}$ as equivalent rational numbers.

Standard form of a Rational Numbers

A rational number $\frac{p}{q}$ is said to be a standard rational number form if $p$ and $q$ are integers having no common divisors other than $1$ and $q$ is positive.

Note: Every rational number can also be represented on a number line.

Addition of Rational Numbers

In order to add rational numbers, we must first convert them into rational numbers with positive denominator.

$\displaystyle \frac{p}{q} + \frac{m}{q} = \frac{p+m}{q}$

When the rational numbers have the same denominator, then these numbers can be added as follows:

$\displaystyle \frac{p}{q} + \frac{m}{n} = \frac{p \times n+m \times q}{q \times n}$

If the denominator is not same (or unequal), we find the LCM of the denominator and express each one of the rational numbers having the LCM as its denominator. You could use the following approach and reduce the number once you add them.

Properties of Addition of Rational Numbers

Closure Property: The sum of two rational number is also a rational number. That is, if $\frac{p}{q}$ and $\frac{m}{n}$ are rational numbers than $\frac{p}{q} + \frac{m}{n}$ is also a rational number.

Commutative Law: If $\frac{p}{q}$ and $\frac{m}{n}$ are rational numbers than $\frac{p}{q} + \frac{m}{n}= \frac{m}{n} + \frac{p}{q}$

Associative Law: If $\displaystyle \frac{p}{q}, \frac{m}{n}, \frac{x}{y}$ are rational numbers, then

$\displaystyle (\frac{p}{q}+ \frac{m}{n}) + \frac{x}{y} = \frac{p}{q}+ (\frac{m}{n} + \frac{x}{y})$

Existence of Additive Identity: The rational number 0 is the additive identity for any rational number. Therefore

$\displaystyle ( \frac{p}{q} + 0) = ( 0 + \frac{p}{q}) = \frac{p}{q}$

Subtraction of Rational Numbers

(i) If $\frac{p}{q}$ and $\frac{m}{q}$ are two rational numbers with the same denominator, then

$\displaystyle \frac{p}{q} - \frac{m}{q} = \frac{p-m}{q}$

(ii) If the denominator is not same (or unequal), we find the LCM of the denominator and express each one of the rational numbers having the LCM as its denominator. You could use the following approach and reduce the number once you subtract them. i.e.

$\displaystyle \frac{p}{q} - \frac{m}{n} = \frac{p \times n - m \times q}{q \times n}$

Properties of Multiplication

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

Commutative Law: If $\displaystyle \frac{p}{q}$ and $\displaystyle \frac{m}{n}$ are two rational numbers, then

$\displaystyle \frac{p}{q} \times \frac{m}{n} = \frac{m}{n} \times \frac{p}{q}$

Associative Law: If $\displaystyle \frac{p}{q} , \frac{m}{n}$ and $\displaystyle \frac{x}{y}$ are rational numbers then

$\displaystyle \Big( \frac{p}{q} \times \frac{m}{n} \Big) \times \frac{x}{y} = \frac{p}{q} \times \Big( \frac{m}{n} \times \frac{x}{y} \Big)$

Existence of Multiplicative Identity: The rational number $1$ is a multiplicative identity for rational numbers. For any rational number $\displaystyle \frac{a}{b}$ we have

$\displaystyle \frac{a}{b} \times 1 = 1 \times \frac{a}{b} = \frac{a}{b}$

Existence of Multiplicative Inverse: For every non zero rational number, there exists a multiplicative inverse. For a rational number $\displaystyle \frac{a}{b}$ the multiplicative inverse would be $\displaystyle \frac{b}{a}$ such that $\displaystyle \frac{a}{b} \times \frac{b}{a} = 1$

Distributive Law of Multiplication over Addition: For any three $\displaystyle \frac{p}{q}, \frac{m}{n}$ and $\displaystyle \frac{x}{y}$ we have

$\displaystyle \frac{p}{q} \times \Big( \frac{m}{n} + \frac{x}{y} \Big) = \Big( \frac{p}{q} \times \frac{m}{n} \Big) + \Big( \frac{p}{q} \times \frac{x}{y} \Big)$

Multiplicative Property of Zero: For any rational number $\frac{a}{b}$ that we have

$\displaystyle \frac{a}{b} \times 0 = 0 \times \frac{a}{b} = 0$

Division of Rational Numbers

If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers such that $\frac{c}{d}$ $\neq 0$ . Then, we define $\frac{a}{b}$ $\div$ $\frac{c}{d}$ $=$ $\frac{a}{b}$ $\times$ $\frac{d}{c}$ . When $\frac{a}{b}$ is divided by $\frac{c}{d}$ , then $\frac{a}{b}$ is called the dividend, $\frac{c}{d}$ is called the divisor and the result of the division is called the quotient.

Properties of Division of Rational Numbers

Closure Property: The division of two rational numbers is also a rational number. Therefore if

Properties of Division of Rational Numbers

Closure Property: The division of two rational numbers is also a rational number. Therefore if $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers, then $\frac{a}{b}$ $\div$ $\frac{c}{d}$ is also a rational number given $\frac{c}{d}$ $\neq 0$ .

For any rational number $\frac{a}{b}$ , we have $\frac{a}{b}$ $\div 1 =$ $\frac{a}{b}$

For any non-zero rational number $\frac{a}{b}$ , we have $\frac{a}{b}$ $\div$ $\frac{a}{b}$ $= 1$

How to find a Rational Number between Two Given Rational Numbers?

Let there be two rational numbers $\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ . Then the rational number in between the two would be $\displaystyle \frac{1}{2} \Big( \frac{a}{b}+ \frac{c}{d} \Big)$ .

If you want to find a large number of rational numbers between two given rational numbers, then first make the denominator equal by taking LCM. After that you can just insert values between the two numerators.

Irrational Numbers

A number that can neither be expressed as a terminating nor as a repeating decimals, is called an irrational number. These are non-terminating and non-repeating numbers. Examples

Non-terminating and non-repeating numbers such as
- $82485863738485738 \ldots$ or $1.2456345789560 \ldots$
Square root of positive integers that are not perfect squares
- $\sqrt{2}, \sqrt{5}, \sqrt{7}, \ldots$
Cube root of numbers that are not perfect cubes
- $\displaystyle \sqrt[3]{2}, \sqrt[3]{5}, \sqrt[3]{7}, \ldots$
Number $\pi$ is irrational. It has a value that is non-terminating and non-repeating. We only approximate it to .

Properties of Irrational Numbers

Sum of two irrational numbers need not be an irrational number

Take $(5+ \sqrt{2})$ and $(5- \sqrt{2})$ as two irrational numbers. The sum is $10$ which is a rational number.

Difference of two irrational numbers need not be an irrational number

Take $(5+ \sqrt{2})$ and $(7+ \sqrt{2})$ as two irrational numbers. The difference is $12$ which is a rational number

Product of two irrational numbers need not be an irrational number

Take $(5+ \sqrt{2})$ and $(5- \sqrt{2})$ as two irrational numbers. The product is $23$ which is a rational number.

Quotient of two irrational numbers need not be an irrational number

Take $(9\sqrt{2})$ and $(3\sqrt{2})$ as two irrational numbers. The quotient is $3$ which is a rational number.

Commutative Law:

$\displaystyle \text{ If } \frac{p}{q} \text{ and } \frac{m}{n} \text{ are irrational numbers, then } \frac{p}{q} \times \frac{m}{n}= \frac{m}{n} \times \frac{p}{q}$
$\displaystyle \text{ If } \frac{p}{q} \text{ and } \frac{m}{n} \text{ are rational numbers, then } \frac{p}{q} + \frac{m}{n} = \frac{m}{n} + \frac{p}{q}$

Associative Law

$\displaystyle \text{ If } \frac{p}{q} \ , \ \frac{m}{n} \text{ and } \frac{x}{y} \text{ are irrational numbers, then}$

$\displaystyle \Big( \frac{p}{q} \times \frac{m}{n} \Big) \times \frac{x}{y} = \frac{p}{q} \times \Big( \frac{m}{n} \times \frac{x}{y} \Big)$

$\displaystyle \text{ If } \frac{p}{q} \ , \ \frac{m}{n} \text{ and } \frac{x}{y} \text{ are irrational numbers, then}$

$\displaystyle \Big( \frac{p}{q} + \frac{m}{n} \Big) + \frac{x}{y} = \frac{p}{q} + \Big( \frac{m}{n} + \frac{x}{y} \Big)$

Distributive Law

$\displaystyle \text{ If } \frac{p}{q} \ , \ \frac{m}{n} \text{ and } \frac{x}{y} \text{ are irrational numbers, then}$

$\frac{p}{q}$ $\times \Big($ $\frac{m}{n}$ $+$ $\frac{x}{y}$ $\Big) = \Big($ $\frac{p}{q}$ $\times$ $\frac{m}{n}$ $\Big) + \Big($ $\frac{p}{q}$ $\times$ $\frac{x}{y}$ $\Big)$

The sum or difference of a rational and an irrational numbers is always irrational.

Example $2 + \sqrt{5}$ is irrational
Example $2 - \sqrt{5}$ is irrational

The product or quotient of a rational and an irrational numbers is always irrational

Example $2 \times \sqrt{5}$ is irrational
Example $2 \div \sqrt{5}$ is irrational

Real Numbers

All rational and irrational numbers forms the set of all real numbers.

Real number Line

The real number line represents all real numbers.
On a real number line, each point corresponds to a point and conversely, every point on the real number line corresponds to a real number.
Between any two real numbers, there exist infinite real numbers.

Properties of Real Numbers

Closure property

The sum of two real numbers is always a real number.
The product of two real numbers is always a real number.

Commutative law

$a +b=b+a$
$a \times b=b \times a$

Associative law

$(a+b )+c=a+( b+c)$
$(a \times b) \times c=a \times ( b \times c)$

Distributive Laws of multiplication over addition

$a \times (b+c)=a \times b+a \times c$
$\displaystyle \text{ If } a \neq 0 \text{ , then } \frac{1}{a} \text{ is called the reciprocal of } a \text{ and } a \times \frac{1}{a} =1$

Comparison of two irrational numbers

If a and b are two irrational numbers such that $a < b$ , then $\sqrt{a} < \sqrt{b}$

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Class 8: Numbers (Lecture Notes)

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