7th Class Mathematics Number System and its Operations Number System

Number System

Category : 7th Class

Learning Objectives:

• To understand integers, decimal, rational number and their representation on number line.
• To learn, addition, subtraction, multiplication and division of integers, decimal number and rational numbers.
• To learn how to compare decimal numbers and rational numbers.
• To learn how to convert fraction to decimal and decimal to fraction.

INTEGERS

VARIOUS TYPES OF NUMBERS:

1. Natural numbers: All numbers from 1 to infinite $\infty$ are known as natural numbers. Thus, $1,\text{ }2,\text{ }3....\infty$ are natural numbers.
2. Whole numbers: All natural numbers including zero is known as whole numbers i.e., $0,\text{}1,\text{ }2,\text{ }3....\infty$ are whole numbers 0 + natural numbers = whole numbers

Note: All natural numbers are whole number but zero is the only whole number which is not natural number.

1. Integers: All natural numbers, 0 and negative numbers are called integers. Thus ............. $-\,5,-\,4,-\,3,-1,\,0,\,1,\,2,\,3,$……………..etc, are all integers.

(i) Positive integers: All natural numbers are positive integers such as 1, 2, 3, 4, ............... etc.

(ii) Negative integers: All negative numbers are negative integers such as ...,$-\,4,-3,-2,-1$

(iii) Zero is neither negative nor positive integer.

Note:

1. Both the positive and negative integer are called directed numbers as they indicate direction. These are also known as signed numbers because of the$+$or$-$sign.
2. The sum of any integer and its negative integer is always zero i.e.,$a+\left( -a \right)=0.$

REPRESENTATION OF INTEGERS ON NUMBER LINE:

Every positive integer is greater than the negative integer.

Zero is less than every positive integer but greater than every negative integer.

1. If two positive or two negative integers are added, we add their values without considering their signs and put common sign before the sum.

(i) $36\,\,\,+27$

\begin{matrix}

+ & 36  \\

+ & 27  \\

+ & 63  \\

\end{matrix}

(ii) $-\,36\,\,\,-27$

\begin{matrix}

- & 36  \\

- & 27  \\

- & 63  \\

\end{matrix}

1. To add a positive and a negative integer, we calculate the difference in their numerical values regardless of their signs and put the sign of greater numerical value integer to the value of difference.

(i) $+\,36\,\,\,-27$

\begin{matrix}

+ & 36  \\

- & 27  \\

+ & 9  \\

\end{matrix}

(ii) $-\,\,36\,\,\,+27$

\begin{matrix}

- & 36  \\

+ & 27  \\

- & 9  \\

\end{matrix}

1. Closure property of addition: The sum of two integers is always an integer.

Examples:

(i) $4+3=7,$which is an integer

(ii) $4\,+\left( -\,3 \right)=1,$which is an integer

(iii) $-\,4+3=-1,$which is an integer

(iv) $-\,4+\left( -\,3 \right)=-7,$which is an integer

1. Commutative law of addition: If$x$and$y$ are any two integers, then, $x+y=y+x$

Examples:

(i) $-7+8=1$ and $8+\left( -\,7 \right)=1$

$\therefore$  $-7+8=8+\left( -\,7 \right)$

(ii) $\left( -\,5 \right)+\left( -\,8 \right)=-13$ and $\left( -\,8 \right)+\left( -\,5 \right)=-13$

$\therefore$  $\left( -\,5 \right)+\left( -\,8 \right)=\left( -\,8 \right)+\left( -\,5 \right)$

1. Associative law of addition: If x, y and z are any three integers then$\left( x+y \right)+z=x+\left( y+z \right)$

Example: $\left\{ \left( -\,5 \right)+\left( -\,6 \right) \right\}+7=-11+7=-\,4$

$\left( -\,5 \right)+\left\{ \left( -\,6 \right)+7 \right\}=-\,5+1=-\,\,4$

$\therefore$  $\left\{ \left( -\,5 \right)+\left( -\,6 \right) \right\}+7=-\,5+\left\{ \left( -\,6 \right)+7 \right\}$

1. Existence of additive identity: For any integer x, we have 0 is called the additive identity for integers.

Examples:

(i)   $0+9=9+0=9$

(ii)   $\left( -\,6 \right)+0=0+\left( -\,6 \right)=-\,6$

1. Existence of additive inverse: For any integer $x,$ we have $x+\left( -\,x \right)=\left( -\,x \right)+x=0$

(i) The opposite or additive inverse of$x$is $\left( -\,x \right)$ and $\left( -\,x \right)$ is $x$

(ii) The sum of an integer and its opposite is 0.

Example: $4+\left( -\,4 \right)=0$and $\left( -\,4 \right)+4=0$

So additive inverse of 4 is$(-\,4)$and$(-\,4)$is 4.

SUBTRACTION OF INTEGERS:

For any integers$x$and$y,$we define.

(i) $x-y=x+$(additive inverse of$y$)$=x+(-y)$

(ii) $x-(-y)=x+$ {additive inverse of $(-y)$}$=x+y$

PROPERTIES OF  SUBTRACTION OF INTEGERS:

1. Closure property for subtraction: If x and y are any integer, then$x-y$is always an integer.

Examples:-

(i) $3-5=-\,2,$which is an integer

(ii) $(-\,3)-6=-\,9,$which is an integer

(iii) $3-(-\,6)=9,$which is an integer

(iv) $(-\,3)-(-\,6)=3,$which is an integer

1. Subtraction of an integer is not commutative:

Examples:-

(i) Consider the integers 2 and 4, we have

$(2-4)=2+(-\,4)=-\,2$and $(4-4)=2+(-\,4)=-\,2$

$\therefore \,\,\,(2-4)\ne (4-2)$

(ii) Consider the integers$(-\,5)$and 3 we have

$(-\,5)-3=(-\,5)+(-\,3)=-\,8$ and $3-(-\,5)=8$

$\therefore \,\,\,(-\ 5)-3\ne -\,3-3-(-\,5)$

(iii) Consider the integers$(-\,6)$and $(-\,4),$we have

$(-\,6)-(-\,4)=-\,6+4=-\,2$and$(-\,4)-(-\,6)=-\,4+6=\,2$

$\therefore \,\,(-\,6)-(-\,4)\ne (-\,4)-(-\,6).$

1. Subtraction of integers is not associative:

Consider the integers $4,$ $(-\,5)$ and $(-\,6),$ we have

$\{4-(-\,5)\}-(-\,6)=(4+5)-(-\,6)=9-(-\,6)$

$=9+6=15$

$4-\{(-\,5)-(-\,6)\}=4-\{(-\,5)+6\}=4-1=3$

$\{4-(-\,5)\}-(-\,6)\ne 4-\{(-\,5)-(-\,6)\}$

MULTIPLICATION OF INTEGERS:-

Rule 1: To find the product of two integers with unlike sing, first get their product regardless to their signs, the give minus sign to the product.

Examples:

(i)            $(-\,40)\times 9$

$=-\,360$

(ii)           $20\times (-\,3)$

$=-\,60$

Rule 2: To find the product of two integers with like signs, first find their product regardless to their signs then put plus sign to product.

Examples:

(i)            $3\times 5$

$=+\,\,15$

(ii)           $-\,3\times -\,5$

$=+\,\,15$

PROPERTIES OF MULTIPLICATION OF INTEGERS:

1. Closure properties for multiplication: The product of two integers is always an integer.

Examples:

(i) $3\times 2=6,$ which is an integer

(ii) $(-\,3)\times 2=-\,6,$ which is an integer

(iii) $(-\,3)\times (-\,2)=6,$ which is an integer

(iv) $\,3\times (-\,2)=-\,6,$which is an integer

1. Commutative law of multiplication: For any two integers$x$and$y,$we have.

$(x\times y)=(y\times x)$

Examples:

(i)            $2\times \left( -\,6 \right)=-12$ and $\left( -\,6 \right)\times 2=-12$

$\therefore \,\,\,2\times \left( -\,6 \right)=\left( -\,6 \right)\times 2$

(ii)           $\left( -\,3 \right)\times \left( -\,7 \right)=21$ and $\left( -\,7 \right)\times \left( -\,3 \right)=21$

$\therefore \,\,\,\left( -\,3 \right)\times \left( -\,7 \right)=\left( -\,7 \right)\times \left( -\,3 \right)$

1. Associative law of multiplication: For any integers x, y and z, we have

Examples:

(i)   Consider the integers $3,$ $\left( -\,4 \right)$and $\left( -\,5 \right),$ we have

$\left\{ 3\times \left( -\,4 \right) \right\}\times \left( -\,5 \right)=-\,\,12\times \left( -\,5 \right)=60$ and

$3\text{ }x\text{ }\left\{ \left( -4 \right)\times \left( -5 \right) \right\}=3\times 20=60$

$\therefore$ $\left\{ 3\times \left( -\,4 \right) \right\}\times \left( -\,5 \right)=3\times \left\{ \left( -\,4 \right)\times \left( -\,5 \right) \right\}$

(ii) Consider the integers $\left( -\,5 \right),\left( -\,6 \right)$ and $\left( -\,7 \right),$ we have

$\left\{ \left( -\,5 \right)\times \left( -\,6 \right) \right\}\times \left( -\,7 \right)=30\times \left( -\,7 \right)=-\,210$

$\left( -\,5 \right)\times \left\{ \left( -\,6 \right)\times \left( -\,7 \right) \right\}=\left( -\,5 \right)\times 42=-\,210$

$\therefore$ $\left\{ \left( -\,5 \right)\times \left( -\,6 \right) \right\}\times \left( -\,7 \right)=\left( -\,5 \right)\times \left\{ \left( -\,6 \right)\times \left( -\,7 \right) \right\}$

1. Distributive law of multiplication over addition: For any integer x, y and z, we have:

$x\times \left( y+z \right)=\left( x\times y \right)+\left( x\times z \right)$

Examples: (1) Consider the integers $5,\,\left( -\,6 \right)$ and $\left( -\,7 \right),$we have

$5\times \left\{ \left( -\,6 \right)+(-\,7) \right\}=5\times \left( -\,13 \right)=-\,65$ and $\left\{ 5\times \left( -\,6 \right) \right\}+\left\{ 5\times \left( -\,7 \right) \right\}=-\,30+\left( -\,35 \right)=-\,65$

$\therefore$ $5\times \{(-\,6)+(-\,7)\}=\{5\times (-\,6)\}+\{5\times (-7)\}$

(2) Consider the integers (-5), (-6) and (-7), we have $\left( -\,5 \right)\times \left\{ \left( -\,6 \right)+\left( -\,7 \right) \right\}=\left( -\,5 \right)\times \left( -13 \right)=65$ and

$\left\{ \left( -\,5 \right)\times \left( -\,6 \right) \right\}+\left\{ \left( -\,5 \right)\times \left( -\,7 \right) \right\}=\left( -\,30 \right)+\left( -\,35 \right)=65$

$\therefore \,\,\left( -\,5 \right)\times \left\{ \left( -\,6 \right)+\left( -\,7 \right) \right\}=\left\{ \left( -\,5 \right)\times \left( -\,6 \right) \right\}+\left\{ \left( -\,5 \right)\times \left( -\,7 \right) \right\}$

1. Existence of multiplicative identity: For every integer $x$ we have: $(x\times 1)=(1\times x)=x,$ 1 is known as the multiplicative identity for integers.

Examples:

(i) $13\times 1=13$

(ii) $(-12)\times 1=-12$

1. Existence of multiplicative inverse: Multiplicative inverse of a non-zero integer$x$is

$\frac{1}{x}$as $x\left( \frac{1}{x} \right)=\left( \frac{1}{x} \right)\cdot x=1$

Examples:

(i)   Multiplicative inverse of $5=\frac{1}{5}$

(ii) Multiplicative inverse of $-\,5=-\,\frac{1}{5}$

1. Property of zero: For every integer, $x$ we have:

$(x\times 0)=(0\times x)=5$

Examples:

(i) $5\times 0=0\times 5\text{ }=0$

(ii) $(-\,5)\times 0=0\times (-\,5)=0$

IMPORTANT RESULTS:

1. $(-\,{{x}_{1}})\times (-\,{{x}_{2}})\times (-\,{{x}_{3}})\times ..........\times (-{{x}_{n}})$ $=-\,({{x}_{1}}\times {{x}_{2}}\times {{x}_{3}}\times .........\times {{x}_{n}})$ when $n$ is odd.
2. $(-\,{{x}_{1}})\times (-\,{{x}_{2}})\times (-\,{{x}_{3}})\times ..........\times (-{{x}_{n}})$$=({{x}_{1}}\times {{x}_{2}}\times {{x}_{3}}\times .........\times {{x}_{n}})$ when $n$ is even.
3. $(-\,x)\times (-\,x)\times (-\,x)\times ..........\times n\,$times$=-\,{{x}^{n}}$ when $n$ is odd.
4. $(-\,x)\times (-\,x)\times (-\,x)\times .....\times n$ times $={{x}^{n}}$ when $n$ is even.
5. $(-1)\times (-1)\times (-1)........\,n$ times $=-\,1$ when $n$ is odd.
6. $(-1)\times (-1)\times (-1)........\,n$ times $=1$ when $n$ is even.

DIVISION OF INTEGERS:

Rule 1: To find the division of two integers with unlike sign, first get their quotient regardless to their signs, then give minus sign to the product.

Example:

(i)            $-\,74\div 2$

$=\frac{-\,74}{2}$

$=-\,37$

(ii)           $96\div (-\,3)$

$=\frac{96}{-\,3}$

$=-\,32$

Rule 2: To find the division of two integers with like signs, first find their product regardless to their signs then put plus sign to quotient.

Example:

(i)            $48\div 6$

$=\frac{48}{6}$

$=8$

(ii)           $-\,155\div (-\,5)$

$=\frac{-\,155}{-\,5}$

$=\,\,31$

PROPERTIES OF DIVISION OF INTEGERS:

1. If $x$ and $y$ are integers then $(x\div y)$ is not necessarily an integer

Example:

(i) $12$ and $5$ are both integer but $(12\div 5)$ is not an integer.

(ii) $(-12)$ and $5$are both integer. But $[(-12)\div 5]$ is not an integer.

1. If $x$ is an integer and $x\ne 0,$ then $x\div x=1$

Examples:

(i) $10\div 10=1$                            (ii) $(-\,5)\div (-\,5)=1$

1. If $x$ is an integer, then $(x\div 1)=x$

Examples:

(i) $5\div 1=5$                 (ii) $(-\,5)\div 1=(-\,5)$

1. If $x$ is an integer and $x\ne 0,$ then $(0\div x)=0$ but $(x\div 0)$ is not meaningful.

Examples:

(i) $0\div 9=0$

(ii) $0\div (-\,9)=0$

(iii) $6\div 0=$ Meaning less

1. If $x,y$ and $z$ are integers, then $(x\div y)\div z$ $\ne x\div (y\div z)$ unless $z=1.$

Thus division on integers is not associative.

Examples: Let $x=-\,6,y=3,\,z=-\,2$ then,

$(x\div y)\div z=\{(-\,6)\div 3\}\div (-\,2)=(-\,2)\div (-\,2)=1$

$x\div \{(y)\div (z)\}=(-\,6)\div \{3\div (-\,2)\}=(-\,6)\div (-1.5)=4$

$\therefore$  $(x\div y)\div z\ne x\div \{(y)\div (z)\}$

If $x=-\,6,y=3$and$z=1$then.

$(x\div y)\div z=\{(-\,6)\div 3\}\div 1=(-\,2)\div 1=-\,2$

$x\div \{(y)\div z\}=(-\,6)\div \{3+1\}=(-\,6)\div 3=-\,2$

So, $(x\div y)\div z=x\div \{y\div z\}$

1. If $x,y,z$are non-zero integers and$x>y$then

(i) $(x\div z)>(y\div z)$ if $z$ is positive

(ii) $(x\div z)<(y\div z)$ if $z$ is negative

Example:-

(i) If$x=27,y=18$and$z=9$

$(x\div z)>(y\div z)$

$(27\div 9)>(18\div 9)$

$3>2$

(ii) If $x=27,y=18$and $z=-\,9$

$(x\div z)<(y\div z)$

$\{27\div (-\,9)\}<\{18\div (-\,9)\}$

$-\,3<-\,2$

Use of Brackets:

We know that the priority order of different mathematical operations is

(1) Division                          (2) Multiplication

Sometimes in complex expressions, we require a set of operations to be performed prior to the others.

Here we make use of brackets. Brackets which are commonly used are: -

 Brackets Name ( ) Parentheses or common brackets { } Braces or curly brackets [ ] Square or Box brackets — Vinculum
• Proper fraction: A proper fraction is a fraction that represents a part of a whole number. Here denominator is greater than the numerator.
• Improper fraction: An improper fraction is a combination of a whole and a proper fraction. Here numerator is greater than the denominator.
• Mixed fraction: An improper fraction can be written as a mixed fraction, e.g., $\frac{7}{4}=1\frac{3}{4}$.
• Like fractions: Fractions having the same denominators are called like fractions.
• Unlike fractions: Fractions with different denominators are called unlike fractions.
• Addition and subtraction of fractions: For addition or subtraction, we first find out the LCM of the denominators, then we convert each fraction into an equivalent fraction whose denominator is equal to the LCM. Finally, we add or subtract the like fractions obtained above.

For examples, $\frac{3}{5}+\frac{7}{2}$

LCM of $5,\,2=10$

Like fraction $\Rightarrow \frac{3}{5}=\frac{6}{10}$and$\frac{7}{2}=\frac{35}{10}$

$\frac{6}{10}+\frac{35}{10}=\frac{41}{10}$

MULTIPLICATION AND SUBTRACTION OF FRACTIONS

• MULTIPLICATION OFAFRACTION BYAWHOLE NUMBER: To multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same.

To multiply a mixed fraction by a whole number, first convert it into an improper fraction and then multiply.

FRACTIONAS AN OPERATOR ‘OF’

Observe this figure. The two squares are exactly same.

Each shaded portion represents $\frac{1}{2}$of 1.

So, both the shaded portions together will represent $\frac{1}{2}$ of 2.

Combine the 2 shaded$\frac{1}{2}$parts. It represents 1.

So, we say $\frac{1}{2}$ of 2 is 1. We can also get it as$\frac{1}{2}\times 2=1.$

Thus, $\frac{1}{2}$ of $2=\frac{1}{2}\times 2=1$

$\text{In}\,\text{general}\,\text{product}\,\text{of}\,\text{fractions}$

$\,\text{=}\,\,\frac{\text{Product}\,\text{of}\,\text{Numerators}}{\text{Product}\,\text{of}\,\text{Denominators}}$

• Value of products of 2 fractions: When two proper fractions are multiplied, the product is less than each of the fractions.

The value of the product of two improper fractions is more than each of the two fractions.

DIVISION OF FRACTIONS

• Division of whole number by a fraction: For dividing a whole number by a fraction. First obtain the reciprocal of the fraction and then multiply it with the whole number.

For example, $6\div \frac{2}{3}\Rightarrow 6\times \frac{3}{2}=9$

• Reciprocal of a fraction: When a fraction is inverted its reciprocal is obtained.

The non-zero numbers whose product with each other is 1 are called reciprocals of each other.

• Division of a fraction by another fraction: For dividing a fraction by another fraction reverse one of the fractions and multiply it with the other.

H.C.F. AND L.C.M.

• The highest common factor (HCF) of two or more numbers is the greatest number which divides each number exactly.
• The least common multiple (LCM) of two or more numbers is the smallest number which is exactly divisible by each number separately.

The H.C.F. of given numbers is not greater than any of the given numbers.

The L.C.M. of given numbers is not less than any of the given numbers.

H.C.F. of two numbers always divides their L.C.M.

The H.C.F. of two co-prime numbers is 1.

The L.C.M. of two co-prime numbers is product of the numbers.

The product of the H.C.F. and the L.C.M. of the two given numbers is equal to the product of those numbers, i.e.,

H.C.F.$\times$L.C.M. = Product of the given numbers.

Example: The HCF of two numbers is 9 and their LCM is 270. If the sum of the numbers is 99, their difference is equal to

Solution:

Let the numbers be $x$ and $99-x.$

So $x\times (99-x)=9\times 270$

On solving the above equation, we get; $x=54,$ or$45$

So difference will be$54-45=9.$

FACTOR THEOREM

If $\left( x+a \right)$ is a factor of polynomial$P(x),$then remainder$=0$

$\Rightarrow P(-\,a)=0$

Example: Show that $(x-3)$ is a factor of the polynomial

$P(x)={{x}^{3}}-3{{x}^{2}}+4x-12$

lf $(x-3)$ is a factor of polynomial

$P(x)={{x}^{3}}-3{{x}^{2}}+4x-12,$ then remainder

$P(3)=0$

$P(3)=0$

$P(3)={{3}^{3}}-3\times {{3}^{2}}+4\times 3-12$

$=27-27+12-12=0$

As remainder$P(3)=0$

$\therefore$ $(x-3)$ is a factor of polynomial$P\,(x)$.

PROPERTIES OF DECIMALS

1. Numbers in form of 1.18, 22.36, and 853.4448 are known as decimal numbers.
2. A decimal number have two parts. A whole number part and decimal number part. These two parts are separated by a dot. This dot is known as decimal point. The whole number part is to the left of the decimal point and the decimal part to the right of decimal point.

Thus, in 63.825

1. The number of digits contained in the decimal part shows its decimal places.

Example: 6.438 have three digits in decimal part which is known as three decimal places.

1. Decimals having the same number of decimal places are called like decimals.

Example: 6.238, 38.937, 48.385 are like decimals.

1. Decimals having different number of decimal places are called unlike decimals.

Example: 6.238, 32.73, 65.3984 are unlike decimals.

Remember: On adding zeros after the last digit of the decimal part of any decimal number does not change the value of the decimal.

Example: 6.28 can be 6.280 or 6.2800 but the value of all three are same.

We use this property to convert the unlike decimal into like decimal.

COMPARINGDEOMALS:

Suppose we have to compare two given decimals. We follow the following steps:

1. Convert the given decimals into like decimals.
2. First compare the whole number part. The decimal with greater whole number part is greater.
3. If the whole-number parts are equal, compare the tenths digits. The decimal with bigger digit in tenths place is greater.
4. If the tenths digits are also equal, compare the hundredths digits, and so on.

Example: Compare the following digits.

5.74, 6.03, 0.8, 0.648 and 8.2.

Solution: Converting the given numbers into like decimals we get them as:

5.740, 6.030, 0.800, 0.648 and 8.200

Clearly 0.648 < 0.800 < 5.740 < 6.030 < 8.200

CONVERSION OFDEOMAL INTO FRACTION:

1. Write the given decimal number without decimal point as the numerator of the fraction.
2. In the denominator, write 1 followed by as many zeros as there are decimal places in the given decimal.
3. Reduce the fraction into simplest form.

Example: Convert 2.85 into fraction.

Solution: $\frac{285}{100}=\frac{57}{20}$

CONVERSION OFA FRACTION INTO DECIMAL:

1. Divide the numerator by the denominator till non-zero remainder is obtained.
2. Put a decimal point in the divided as well as in the quotient:
3. Put a zero on the right of the decimal point in the dividend as well as on the right of the remain
4. Divide again just as we do in whole numbers.
5. Repeat steps 3 & 4, till the remainder is zero.

Example: Convert $\frac{27}{4}$ and $2\frac{3}{8}$ into decimal fraction.

Solution: $\frac{27}{4}$

$4\,\underset{\times \,\times }{\mathop{\underline{\underset{\,\,\,\,-\,20}{\mathop{\underset{\,\,\,\,\,\,\,\,\,\,\,20}{\mathop{\overset{6.75}{\mathop{\underline{\underset{-28}{\mathop{\underset{30}{\mathop{\overline{\underline{\underset{-\,24}{\mathop{)\,\,\,27.00}}\,\,\,\,\,}}}}\,}}\,}}}\,}}\,}}\,}}}\,$

$\therefore$  $\frac{27}{4}=6.75$

and        $2\frac{3}{8}=\frac{19}{8}$

$8\,\underset{\times \,\times }{\mathop{\underline{\underset{-\,40}{\mathop{\underset{40}{\mathop{\underline{\underset{\,\,\,\,-\,56}{\mathop{\underset{\,\,\,\,\,\,\,\,\,\,\,60}{\mathop{\overset{2.375}{\mathop{\underline{\underset{-24}{\mathop{\underset{30}{\mathop{\overline{\underline{\underset{-\,16}{\mathop{)\,\,19.000}}\,\,\,\,\,}}}}\,}}\,}}}\,}}\,}}\,}}}\,}}\,}}}\,$

$\therefore$  $2\frac{3}{8}=2.375$

Method:

(i)            Write the given numbers one below the other with decimal points in vertical line.

(ii)           Equal the digits of numbers by adding zeros at right extremes of decimal parts as required.

(iii)          Then add or subtract the numbers and put decimal in result directly under the other decimal points.

Example:

(i) Add: 0.872, 3.46, 4.309, and 3.17

Solution:

\underline{\begin{align} & \underline{\begin{align} & \,\,\,\,0.872 \\ & \,\,\,\,3.460 \\ & \,\,\,\,4.309 \\ & +\,3.170 \\ \end{align}} \\ & \,\,11.811 \\ \end{align}}

(ii) Subtract: 17.182 from 360.05

\underline{\begin{align} & \underline{\begin{align} & \,\,\,\,360.050 \\ & -\,017.182 \\ \end{align}} \\ & 342.868 \\ \end{align}}

MULTIPLICATION OF DECIMAL NUMBERS:

(1)          To multiply a decimal number by 10 or any power of 10 move the decimal point as many places to the right as there are zeros in the multiplier or we can say that multiplying a decimal number by ${{10}^{n}}$ moves the decimal point$n$places to the right.

Examples:

(2)          To multiply a decimal number by a whole number or a decimal number multiply the numbers first as there were no decimal point at all and fix the position by the rule that there are as many decimal places in the product as they are in multiplier and multiplicand put together.

Example:

(1) $\underset{\text{1}\,\,\text{place}}{\mathop{0.6}}\,\times \underset{\text{2}\,\,\text{places}}{\mathop{0.02}}\,=\underset{\text{3}\,\,\text{places}}{\mathop{0.012}}\,$

(2) $\underset{0\,\,\text{place}}{\mathop{7\,}}\,\,\,\,\times \underset{3\,\,\text{places}}{\mathop{\,0.003}}\,\,\,\,\,=\underset{\text{3}\,\,\text{places}}{\mathop{\,0.021\,}}\,$

DIVISION OF DECIMAL NUMBERS

(1)          Dividing a whole number or decimal number by $10$ or any higher power of $10\,({{10}^{n}})$ move the decimal point that number of places $(n)$ to the left.

Examples: (1) $53\div 10=5.3$

(2) $53\div 100=0.53$

(3) $53\div 1000=0.053$

(2)          To divide a decimal number by a whole number proceed as with whole numbers, but place the decimal point in the quotient directly above or below the decimal point in the dividend.

Example:  (1) Divide 15.064 by 28

$\therefore$ $15.064\div 28=0.538$

(2) Divide 24.2 by 55000

$\because \,\,\,24.50\div 55=0.44$ 

$\because \,\,\,24.20\div 55000=0.0004$

(3)          To divide a decimal number by a decimal number move the decimal point of divisor to the right until it becomes a whole number (i.e., multiply it by 10 or a power of 10). Then move the decimal point of the dividend the same number of places to the right, adding zero if necessary.

Example: Divide 49.08 by 0.012

Solution: $49.08\times 1000=49080$

$0.012\times 1000=12$ 

RATIONAL NUMBER

A number in form of $\frac{a}{b}$ where $a$ and $b$ are integers and $b\ne 0$ is known as rational number.

Examples: $\frac{1}{2},\frac{-\,6}{7},\frac{3}{-\,2},\frac{-\,5}{-\,8}$is a rational number.

Remember:

1. Zero is a rational number, since we can write it as$\frac{0}{1}.$
2. Every natural number is rational number but a rational number need not to be natural number as $\frac{1}{1},\frac{2}{1},\frac{3}{1},\frac{4}{1}$ etc. are natural numbers and rational numbers but $\frac{2}{3},\frac{6}{8},\frac{1}{3}$ etc. are rational number which can't be natural numbers.
3. Every integer is a rational number but a rational number need not to be an integer.

Example: $1=\frac{1}{1},2=\frac{2}{1},3=\frac{3}{1}\,\,.......\,\,\frac{n}{1}$are integers but rational number like $\frac{5}{7},\frac{-\,8}{9},\frac{11}{-13}$ are not integers.

1. Every fraction is a rational number but a rational number need not to be a fraction.

Let $\frac{a}{b}$ is a fraction where $a$ and $b$ are natural numbers. Since every natural number is $a$ integer so $a$ and $b$ are integers so fraction $\frac{a}{b}$ where $b\ne 0$ is a rational number.

Example:

A number like $\frac{5}{-\,6}$ is a rational number is not a fraction because its denominator $-\,6$ is not a natural number.

Positive rational number: A rational number is said to be positive rational number if both the numerator and denominator are positive or negative.

Examples: $\frac{5}{7},\frac{-18}{-\,27},\frac{-16}{-13},\frac{8}{9}$

Negative rational numbers: A rational number is said to be negative if its numerator and denominator are such that one of them is positive and another is negative.

Example: $\frac{-\,3}{4},\frac{8}{-\,6},\frac{-\,28}{11}$

Remember:

1. Every negative integer is a negative rational number.

Example: $-1,-\,2,-\,3.......$ may be written as: $-\frac{1}{1},\frac{-\,2}{1},\frac{-\,3}{1}........$ are all negative rational numbers.

1. The rational number 0 is neither positive nor negative.

TWO IMPORTANT PROPERTDES OF RATIONAL NUMBERS:

Property 1: Equivalent rational numbers $\div$ If $\frac{a}{b}$ is a rational number and $n$ is a non-zero integer then on multiplying the numerator and denominator of rational number by $n$ we will get its equivalent rational number as $\frac{a}{b}=\frac{a\times n}{b\times n}$

Examples: $-\frac{3}{4}=\frac{(-\,3)\times 2}{4\times 2}=\frac{(-\,3)\times 3}{4\times 3}=\frac{(-\,3)\times 4}{4\times 4}=.....$

All these rational numbers are equal to one another and are called equivalent rational numbers.

Property 2: Reducing to simpler form

If $\frac{a}{b}$ is a rational number and $n$ is a common divisor of $a$ and $b$ then $\frac{a}{b}=\frac{a\div n}{b\div n}$

Examples: $-\frac{48}{60}=\frac{-48\div 2}{60\div 2}=\frac{-48\div 3}{60\div 3}=\frac{-48\div 4}{60\div 4}$

$=\frac{-48\div 6}{60\div 6}=\frac{-48\div 12}{60\div 12}$

$\Rightarrow -\frac{48}{60}=\frac{-24}{30}=\frac{-16}{20}=\frac{-12}{15}=\frac{-8}{10}=\frac{-4}{5}$

The rational number $\frac{-\,4}{5}$ is in lowest number.

STANDARD FORM:

A rational number is said to be in its standard form when its denominator is positive and it is in its lowest term.

Rational number can be in its standard form by following steps-

1. Make the denominator of rational number positive.
2. Divide both the numerator and denominator by their HCF.

Example: Convert the following in their standard form.

(i)            $2\frac{4}{9}$

$=\frac{22}{9}$

(ii)           $-\frac{8}{2}$

$-\frac{8\div 2}{2\div 2}=-\,4$

(iii)          $\frac{4}{-11}$

$=\frac{4}{-11}\times \frac{-1}{-1}=\frac{-\,4}{11}$

(iv)         $\frac{9}{15}$

HCF of 9 and 15 is 3

so $\frac{9\div 3}{15\div 3}=\frac{3}{5}$

Note: If the denominator of rational number is negative then multiply both the numerator and denominator by $-1$ to make denominator positive.

RATIONAL NUMBER ON NUMBER LINE:

To express rational numbers on appropriate number lines, divide the unit length into the number of equal parts as the denominator of the rational number and then mark the numbers on the line.

Example: $P=\frac{7}{8},\,\,Q=\frac{-\,3}{8},\,\,R=\frac{1}{8},\,\,S=\frac{1}{-\,8}$

EQUALITY OF RATIONAL NUMBERS:

Two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ are equal $\frac{a}{b}=\frac{c}{d}$ if $a\times d=b\times c.$

Example: $\frac{-24}{27}=\frac{8}{-9}$

$-\,24\times -\,9=27\times 8$

$216=216$

COMPARISON OF TWO RATIONAL NUMBERS:

Positive rational numbers are always greater than negative rational numbers. But if we have to compare two positive or two negative rational numbers we compare them by two methods.

First Method:

(a)          Express each rational number with its positive denominator.

(b)          Find LCM of positive denominators.

(c)           Express each given rational numbers with LCM as the common denominator.

(d)          The number having greater numerator is greater.

Example: Compare:

(i)            $\frac{9}{15}$ and $\frac{11}{6}$

LCM of 15 and $6=3\times 5\times 2=30$

$\frac{9\times 2}{15\times 2}=\frac{18}{30},\frac{11\times 5}{6\times 5}=\frac{55}{30}$

$\frac{55}{30}>\frac{18}{30},$So, $\frac{11}{6}>\frac{9}{15}$

(ii)           $\frac{3}{-14}$and$-\frac{5}{21}$

$\frac{3}{-14}$and$-\frac{5}{21}$

LCM of 14 and $21=2\times 7\times 3=42$

$-\frac{3\times 3}{14\times 3}=\frac{-9}{42},\frac{-5\times 2}{21\times 2}=\frac{-10}{42}$

$\frac{-9}{42}>\frac{-10}{42}$

So, $\frac{-3}{14}>\frac{-5}{12}$

Second Method:

To compare two rational numbers $\frac{a}{b}$ and $\frac{c}{d},$ we compare the products $a\times d$ and $b\times c$ define their inequality accordingly.

If $a\times d>b\times c$                            If $a\times d<b\times c$

then $\frac{a}{b}>\frac{c}{d}$                 then $\frac{a}{b}<\frac{c}{d}$

Example: Compare $\frac{-5}{9}$ and $\frac{11}{-16}$

First $-\frac{5}{9}$ and $-\frac{11}{16}$

Now, $-\,5\times 16$ and $9\times -\,11$

So, $-\,80>-\,99$

So, $-\,\frac{5}{9}>-\,\frac{11}{16}$

ORDER PROPERTIES OF RATIONAL NUMBERS:

Property 1: For each rational number $a,$ exactly one of the following is true.

(i) $a>0$                            (ii) $a=0$

(iii) $a<0$

Property 2: For any two rational numbers $a$ and $b$ exactly, one of the following is true.

(i) $a>b$                            (ii) $a=0$

(iii) $a<b$

Property 3: lf a, b and c are any three rational numbers such that a > b and b > c, then a > c

ABSOLUTE VALUE OF RATIONAL NUMBERS:

The absolute value of an integer is an integer similarly the absolute value of a rational number is a rational number regardless to their signs.

Thus$\left| \frac{5}{7} \right|=\frac{5}{7};\left| \frac{-5}{7} \right|=\frac{\left| -5 \right|}{\left| 7\, \right|}=\frac{5}{7};\left| \frac{65}{-23} \right|=\frac{\left| 65 \right|}{\left| -23 \right|}=\frac{65}{23}$

OPERATIONS ON RATIONAL NUMBERS:

Case 1. When denominators of given rational numbers are equal

Let $\frac{p}{q}$ and $\frac{r}{q}$ are two rational numbers. Then $\frac{p}{q}+\frac{r}{q}$or $\left( \frac{p+r}{q} \right)$

Example: Add- $\frac{7}{-11}$ and $\frac{3}{11}$

$\frac{-7}{11}+\frac{3}{11}=\frac{-7+3}{11}=\frac{-4}{11}$

Case 2: When denominator of given numbers are unequal.

1. Take the LCM of denominators of the given rational numbers.
2. Express each of the given rational numbers with above LCM as the common denominator.
3. Now add the numbers as case I.

Example: Add: $\frac{7}{-\,27}+\frac{11}{18}-\frac{7}{27}+\frac{11}{18}$

LCM of 27 and $18=3\times 3\times 3\times 2=54$

$-\frac{7\times 20}{27\times 2}=-\frac{14}{54}$and$\frac{11\times 3}{18\times 3}=\frac{33}{54}$

$\therefore \,\,\,\frac{-14}{54}+\frac{33}{34}=\frac{-14+33}{54}=\frac{19}{54}$

2              Subtraction of Rational Number: If $\frac{a}{b}$ and  $\frac{c}{d}$ are two rational numbers, then

$\frac{a}{b}-\frac{c}{d}=\frac{a}{b}+\left\{ \text{Additive}\,\text{inverse}\,\text{or}\,\text{negative}\,\text{of}\frac{c}{d} \right\}$ $\frac{a}{b}-\frac{c}{d}=\frac{a}{b}+\left( -\frac{c}{d} \right)$

Example: Subtract (i) $\frac{7}{8}$ from $\frac{5}{12}$

$\frac{5}{12}-\frac{7}{8}=\frac{5}{12}+\frac{-\,7}{8}$

LCM of 12 and $8=2\times 2\times 3\times 2=24$

$=\frac{5\times 2+(-7)\times 3}{24}=\frac{10+(-21)}{24}=\frac{-11}{24}$

(ii)           $\frac{-4}{9}$from $\frac{-7}{18}$

$\frac{-7}{18}-\left( \frac{-\,4}{9} \right)$

$\frac{-7}{18}+\left[ -\left( \frac{-\,4}{9} \right) \right]$

$-\frac{7}{18}+\frac{4}{9}$

LCM of 18 and $9=9\times 2=18$

$\frac{(-7)\times 1+4\times 2}{18}=\frac{-7+8}{18}=\frac{1}{18}$

1. Multiplication of Rational Numbers:

$\text{Product of two rational numbers}$

$=\frac{\text{Product}\,\text{of}\,\text{numerators}}{\text{Product}\,\text{of}\,\text{denominator}}$

Thus if $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers, then $\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$

Example: (i) $\frac{-7}{15}\times \frac{5}{-14}=\frac{-7\times 5}{15\times -14}=\frac{-35}{-210}=\frac{1}{6}$

(ii) $\frac{-7}{9}\times \frac{-4}{5}=\frac{28}{45}$

RECIPROCALOR MULTIPLICATIVE INVERSE OF A RATIONAL NUMBER:

The reciprocal of a rational number $\frac{a}{b}$ is $\frac{b}{a}$ or ${{\left( \frac{a}{b} \right)}^{-1}}=\frac{b}{a}$

Note:    (i) Reciprocal of 0 does not exist.

(ii) Reciprocal of 1 is 1

(iii) Reciprocal of$-1$ is$-1$.

Example: Write reciprocal of:

(i) $\frac{12}{7}$                                            (ii) $\frac{-7}{9}$

(iii) $-5$

Solution:

(i) $\frac{12}{7}$

(ii) $\frac{-7}{9}$

reciprocal$=\frac{7}{12}$                           $\frac{9}{-7}$

(iii) $-5\,{{(-5)}^{-1}}=\frac{1}{-5}$

DIVISION OF RATIONAL NUMBERS:

If $\frac{a}{b}$ and $\frac{c}{d}$ are two rational numbers such that $\frac{c}{d}$ then,

$\frac{a}{b}-\frac{c}{d}=\frac{a}{b}\times \left( \text{reciorocal}\,\text{of}\,\frac{c}{d} \right)=\frac{a}{b}\times \frac{d}{c}$

Example:

(i) $\frac{7}{15}\div \frac{2}{3}=\frac{7}{15}\times \left( \text{reciorocal}\,\text{of}\,\frac{2}{3} \right)=\frac{7}{15}\times \frac{3}{2}=\frac{7}{10}$

(ii) $\frac{16}{21}\div \frac{-4}{3}=\frac{16}{21}\times \left( \text{reciorocal}\,\text{of}\,\frac{-4}{3} \right)=\frac{16}{21}\times \frac{3}{-4}=\frac{-4}{7}$

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