# 6th Class Mathematics Number System and its Operations Number System and Its Operations

Number System and Its Operations

Category : 6th Class

Number System and Its Operations

Numbers are the symbolic representation of counted objects. There are infinite counting numbers from 1. Some arc-divisible by another whereas some are not divisible. Numbers are differentiated according to their divisibility and factors. A numeral system is a writing system for expressing numbers. The most commonly used system of numerals is Hindu-Arabic numeral system. In this chapter, we will learn about various numeral systems, types of numbers and operation on numbers.

Indian or Hindu-Arabic Number System

This number system was introduced by Indians, and is therefore, called Indian Number System. In this number system 10 is considered as the base.

10 ones = 10, 10 tens = 1 hundred, 10 hundreds = 1 thousand

Hindu - Arabic number system is based on the place value of digits in number

Indian Place Value Chart

 crores Ten Lakes Lakes Ten Thousand Thousands Hundred Tens Ones 2 9 8 7 3 5

The number two lakh ninety-eight thousand seven hundred and thirty-five is written by placing 2 at the place of "lakhs', 9 at the place of "Ten thousands', 8 at "Thousands', 7 at "Hundreds', 3 at "Tens' and 5 at "Ones',

Place Value

If a number contains more than one digit then the place occupies by each digit is its place value. In the number 732 the number 7 occupies the place of hundreds, therefore, the place value of 7 is seven hundred.

Face Value

The face value of a number does not change regardless of the place it occupies. Therefore, the face value of a number is the number itself.

International Place Value Chart

 Billions Millions Thousand Ten billion Billion Hundred million Ten million Million Hundred thousand Ten Thousand Hundred tens Ones 5 6 8 4 3 2 5 4 3 1

The above chart is the international place value chart. The number 5, 684, 325, 431 is read as five billion, six hundred eighty-four million, three hundred twenty-five thousand, four hundred thirty-one.

Comparison between Indian and International Number System

 International Hundred billion Ten billion Billion Hundred million Ten million Million Hundred thousand Ten  Thousand Thousand Hundred Ten Ones Indian Kharab Ten arab Arab Ten crore Crore Ten lakes Lakes Ten thousand Thousand Hundred Ten Ones

•             Example:

According to the place value of International number system, which one is six million five hundred fifteen thousand two hundred twenty-one?

(a) 60515221                                                     (b) 6515221

(c) 65150221                                                      (d) 600515221

(e) None of these

•             Example:

What is the reduced form of the following expanded from:

$7\text{ }\times \text{ }10000\text{ }+\text{ }5\text{ }\times \text{ }100\text{ }+\text{ }4\text{ }\times \text{ }10\text{ }+\text{ }6$.

(a) 70546                                                           (b) 7546

(c) 75460                                                           (d) 7546

(e) None of these

Explanation: $7\text{ }\times \text{ }10000\text{ }+\text{ }5\text{ }\times \text{ }100\text{ }+\text{ }4\text{ }\times \text{ }10\text{ }-\text{ }6\text{ }=\text{ }70000\text{ }+\text{ }500\text{ }+\text{ }40\text{ }+\text{ }6\text{ }=7546$

Roman Numerals

This numeric system is represented by Roman numerals which are the combinations of letters from the latin alphabet. Roman numerals are based on seven symbols as. given below:

 Symbol I V X L C D M Value 1 5 10 50 100 500 1000

The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are expressed in Roman numerals as I, II, III, IV, V, VI VII, VIII, IX, X

The rules for this system are:

•            Repeating a numeral upto three times represents addition of the numbers. Eg. III represents 1+1+1 = 3.

Note: Only I, X, C and M can be repeated. But the symbols V, L and D are never repeated.

•             Numerals that decrease from left to right denotes addition of the numbers, eg. XVI represents 10 + 5 + 1 = 16.
•             Writing a smaller numeral to the left of a larger numeral denotes subtraction, eg, IX represents 10 - 1 = 9 and XC = 100 - 10 = 90

Note:

(a) The symbols V, L and D are never written to the left of a symbol of greater value ie. V, L and D are never subtracted.

(b) The symbol I can be subtracted from V and X only.

(c) The symbol x can be subtracted from L, M and C only.

•            To represent larger numbers, a bar over a numeral means to multiply the number by 1000.
 5000 10000 50000 100000 500000 1000000 Value $\overline{V}$ $\overline{X}$ $\overline{L}$ $\overline{C}$ $\overline{D}$ $\overline{M}$

•              Example

Arrange the following Roman numerals in descending order.

MD, MDCCC, DCCCXC, CM

(a) MDCCC > MD> DCCCXC >CM

(b) MDCCC > DCCCXC > MD >CM

(c) MDCCC > CM> DCCCXC > MD

(d) MDCCC > MD> CM>DCCCXC

(e) None of these

Explanation:

Clearly, MDCCC = 1800, MD = 1500, CM = 900 and DCCCXC = 890

$\therefore$ Option (d) is correct.

TYPES OF NUMBERS

Natural Numbers

Every counting number is called a natural number. 1, 2, 3, 4, 5, etc. are natural numbers. Zero is excluded from the natural numbers.

Natural numbers are represented by N (First capital letter of its name).

Whole Numbers

When 0 is included with counting numbers, it becomes whole number. Whole number is represented by W (First letter of its name). Whole numbers (W) = {0, 1, 2, 3, 4 .....}.

Prime Numbers

The numbers which have only two factors, 1 and the number itself are called prime numbers. The numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, etc are prime numbers.

Composite Numbers

The numbers which have more than two factors are called composite numbers. actors of 4 and 6 are: 1, 2, 4 and 1, 2, 3, 6 respectively. Hence, these are composite numbers.

Co-prime

If H.C.R (Highest Common Factor) of two numbers is 1 then the numbers are called co-prime. The numbers 21 & 22 are co-prime as both have no common factors other than 1.

Perfect Numbers

The natural numbers whose sum of positive divisors (excluding the number itself) is equal to the number itself are called perfect numbers. The first perfect number Is 6.

Note:

Euclid proved that ${{2}^{p-1}}\,({{2}^{p}}-1)$ is an even perfect number whenever ${{2}^{p}}-1$ is prime and o is a natural number

Successor

Successor of every number comes just after the number Successor of 25 = 25 + 1 = 26. Successor of 4573 = 4573 + 1 = 4574.

Predecessor

Predecessor of every number comes just before the number Predecessor of 25 is obtained by subtracting 1 from the number Predecessor of 23 = 23 - 1 = 22,

•             Example:

Choose the composite numbers from the following numbers:

87, 67, 45, 34, 23, 27, 33.

(a) 45, 87, 34, 27, 33                                          (b) 45, 87, 67, 33

(c) 33, 27, 23, 34                                                (d) Ali the above

(e) None of these

Explanation: In the given numbers 23 and 67 are prime numbers and rest are composite numbers.

•             Example:

How many prime numbers are there between 10 to 50?

(a) 10                                                                (b) 11

(c) 13                                                                (d) 18

(e) None of these

Explanation: Prime numbers between 10 and 50 are:

11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.

Closure Property

The sum of two whole numbers is always a whole number. If a and b are two whole numbers, then their addition (a + b) is also a whole number.

Commutative Property

The sum of two and more whole numbers remains same even if the order of the numbers are changed.

Associative Property

The addition of a set of numbers is same regardless of how the numbers are grouped. The associative property involves 3 or more numbers.

Zero (0) is called the additive identity of every whole number. When 0 is added to the whole number its identity does not change or number remains unchanged.

Additive inverse of a is - a and additive inverse of - a is a, therefore, the sum of number with its additive inverse is always zero.

PROPERTIES OF SUBTRACTION

•             Subtraction is inverse process of addition, and subtraction of two whole numbers is always a whole number.
•              If a < b then their subtraction, a - b is not a whole number.
•             Subtraction is neither commutative nor associative, i.e.$8-(6-5)\,\,\ne \,\,(8-6)-5$.

PROPERTIES OF MULTIPLICATION

Closure Property

If a and b are whole numbers, then their product $a\times b$ is also a whole number. Let us consider two whole numbers 3 and 4, their product is 12, which is also a whole number.

Commutative Property

The product of the whole numbers remains same even if the order of the multiplication s changed. In other words if a and b are whole numbers/ their product $a\text{ }\times \text{ }b\text{ }=\text{ }b\text{ }\times \text{ }a$

For example: $5\text{ }\times \text{ }6\text{ }=\text{ }30$, on changing their order, $6\text{ }\times \text{ }5\text{ }=\text{ }30$,. Thus$5\,\,\times \,\,6\,\,\times \,\,=\,\,6\,\,\times \,\,5$.

Similarly, $10\text{ }\times \text{ }15\text{ }=\text{ }150$and$15\text{ }\times \text{ }10\text{ }=\text{ }150$, thus, $10\text{ }\times \text{ }15\text{ }=\text{ 15 }\times \text{ }10$,

Associative Property

The product of more than two numbers remains same by changing the groups of the numbers. If a, b and c are three numbers then their product$(a\times b)\times c=a\times (b\times c)$.

For example: $(4\,\,\times \,\,5)\,\,\times \,\,6=20\,\,\times \,\,\,6-20$and$4\,\,\times \,\,(5\,\,\times \,\,6)\,\,\times \,\,30=120$. Thus$(4\,\,\times \,\,5)\,\times 6\,\,=4\times \,\,(5\,\,\times \,\,6)$.

Multiplicative Identity

The product of every whole number with 1 is the number itself.

If a is a whole number then, $a\,\,\times \,\,1=1\,\,\times \,\,a=a$

Multiplication of a number by 1 is the number itself, therefore, the identity of the whole number does not change thus 1 is called multiplicative identity of the whole number, i.e.$5\,\,\,\times \,\,\,1=1\,\,\times \,\,5=5\text{ }and\text{ }10\text{ }\times \text{ }1\text{ }=\text{ 1}\,\,\times \text{ }10\text{ }=\text{ }10$.

Multiplication of Whole Numbers with 0:

When a whole number is multiplied by 0, it becomes equal to zero.

In other words$O\,\,\times \,\,\,a=a\,\,\times \text{ }0\text{ }=\text{ }0.\text{ i}\text{.}e.\text{ }O\,\,\times \text{ }11\text{ }=\text{ }0\text{ }and\text{ }11\,\,\,\times \,\,\,0\text{ }=\text{ }0$.

Multiplicative Inverse

Multiplicative inverse of a number a is $\frac{1}{a}$ and multiplication of a and $\frac{1}{a}$  is 1,

Let us consider a number 4 and its multiplicative inverse $\frac{1}{4}$

Hence, its multiplication $=4\,\,\times \frac{1}{4}=1$

PROPERTIES OF DIVISION

•        If a and b are two whole numbers in the form of $a\text{ }\div \text{ }b$then it can be expressed by .The division of a by $\frac{a}{b}$ may or may not be a whole number, i.e. $10\div 5\text{ }=\text{ }\frac{10}{5}\text{ }=\text{ }2$ is a whole number. The division of $25\div 15\text{ }=\text{ }\frac{25}{15}\text{ }=\,\,\,\frac{5}{3}$ = 1.666 is not a whole number.
•           For every number a, $a\div 1=\frac{a}{1}=a$and $a\div \text{ }a=\text{ }\frac{a}{a}=l$, therefore, the division of every number by 1 is the number itself.
•       If a is a number whereas, $a\ne 0$then $0\div a\text{ }=\text{ 0}$ and $a\div 0$cannot be defined. The division of 0 by 4 or $0\div 4\text{ }=0\text{ }$= 0 and $4\div 0$ cannot be defined.
•        If a, b and c are whole numbers then $(a\div b)\div c\ne a\div (b\div c)$ therefore, division is not associative, i.e. $(16\div 4)\div 2\ne 6\div (4\div 2)$
•        If a, b, c, d are whole numbers in the form of a $a\overset{c}{\overline{\left){\frac{b}{d}}\right.}}$ then a is called divisor, b is dividend, c is quotient and d is remainder. Whereas, Dividend = divisor x quotient + remainder.

BODMAS Rule

When a single expression contains many mathematical operations then BODMAS rule is used for the simplification of the expression. The word BODMAS has been arranged according to the priority of the operations.

The letters of BODMAS express the following operations:

 B Stands for Bracket O Stands for of D Stands for Division M Stands for Multiplication A Stands for Addition and S Stands for Subtraction

•             Example:

Evaluate: $\mathbf{[}\left\{ \left( \mathbf{563 \times 8} \right)\mathbf{ + 70} \right\}\,\,\mathbf{-}\,\,\mathbf{4384] \div 10}$.

(a) 60                                                                (b) 45

(c) 19                                                                (d) 24

(e) None of these

Explanation: $[\left\{ \left( 563\text{ }\times \text{ }8 \right)\text{ }+\text{ }70 \right\}\,\,\text{-}\,\,\text{4384 }\!\!]\!\!\text{ }\div \text{ }10$.

$=\text{ }\left( 4504\text{ }+\text{ }70\text{ }-\text{ }4384 \right)\text{ }\div \text{ }10\text{ }=\text{ }190\text{ }\div \text{ }10\text{ }=\text{ }19$

Example:

Product of 23 and 13 is added with 56 then subtracted from 675. Which one of the following is correct for the arrangement of given statement using brackets?

(a) $675\text{ }-\text{ }\left\{ \left( 23\text{ }\times \text{ }13 \right)\text{ }+\text{ }56 \right\}$                           (b) $\left\{ \left( 23\text{ }\times \text{ }13 \right)\text{ }+\text{ }56 \right\}\text{ }-\text{ }675$

(c) $\left( 675\text{ }+\text{ }56 \right)\text{ }-\text{ }\left( 23\text{ }\times \text{ }13 \right)$                               (d) All the above

(e) None of these