# Class 6 – Number-System

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## Online Tests

Topic Sub Topic Online Practice Test
Number-System
• Hindu arabic and International Number System
• Roman and Binary systems
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Number-System
• Whole numbers
• Integers
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Number-System
• Fractions
• Addition and Subtraction of Fractions
• Multiplication & Division of Fractions
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Number-System
• Decimals
• Addition and Subtaction of Decimals
• Multiplication & Division of Decimals
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Number-System
• Simplification & Use of Brackets
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## Study Material

 Number: A number is a concept which exists only in mind. The earliest concept of number was a thought that allowed people to mentally picturize the size of some collection of objects. To write down a number we use a symbol called “Numeral”. Numeral: A group of digits denoting a number is called ‘Numeral’. Digits: The numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are called digits. The counting numbers were used by people in the Stone Age for counting things like property, cattle, enemies, etc. The very first stage of learning mathematics begins with counting of objects from ‘1’ onwards. Natural numbers: The counting numbers starting from 1 are called natural numbers. These are denoted by the letter ‘N’. N = {1, 2, 3, 4, 5, 6 …} (i) Representing a number in figures is called Notation. (ii) The act of reading numbers in words when expressed by means of numerals is called Numeration. (iii) In our number system we use 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 (10 digits) to represent the numerals. So it is called positional number system with base ‘10’ (base-10 system). There are two types of number systems. i) Hindu Arabic (or) Indian system. ii) International (or) English system. Hindu Arabic system: In the Hindu-Arabic numeration system, ten ones are replaced by one ten, ten tens are replaced by one hundred, ten hundreds are replaced by one thousand, 10 one thousands are replaced by 10 thousands, and so on. When numbers are composed of more than three digits, commas are used to separate the digits into groups called Periods. The periods in Hindu-Arabic numeration starting from extreme right are Units, Thousands, Lakhs, Crores, etc. i) In units period, there are 3 places which are ones, tens and hundreds. ii) In thousands period, there are 2 places which are thousands and ten thousands.

iii) In lakhs period, there are 2 places which are lakhs and ten lakhs.

iv) Crores period consists of any number of places.

So, in Hindu Arabic system commas are used after 3, 2, 2,…… from right to left with respect to periods.

 CRORES PERIOD LAKHS PERIOD THOUSANDS PERIOD ONES PERIOD Ten Crores Crores Ten lakhs Lakh Ten Thousands Thousands Hundreds Ten Units

Example: The number 345678909 is written in Hindu-Arabic system by using commas as 34, 56, 78, 909.

And read as Thirty four crores, fifty six lakhs, seventy eight thousands, nine hundred and nine.

International System: In this system of numeration, periods are made with commas after every three digits from the right. The periods in this system are made as follows:

 BILLIONS PERIOD MILLIONS PERIOD THOUSANDS PERIOD ONES PERIOD Hundred billions Ten billions Billions Hundred millions Ten millions Millions Hundred Thousands Ten Thousands Thousands Hundreds Ten Units

Conversion from Hindu-Arabic system to English System:

 Hindu Arabic system English system One lakh Hundred thousands Ten lakhs One million One crore Ten millions Ten crores Hundred millions

Example: The date of birth of the great Indian mathematician Srinivasa Ramanujan is 22nd December 1887, written as 22121887.

The number written in Hindu-Arabic and international system as:

i) In International system is twenty two millions, one hundred twenty one thousands, eight hundred and eighty seven.

ii) In Hindu Arabic system is two crores, twenty one lakhs, twenty one thousands, eight hundred and eighty seven.

Face-value of a digit in a number: The face-value of a digit in a numeral is the digit itself at whatever place it may be.

Example: In the number 869456

The face-value of 9 is 9

The face-value of 6 is 6

Place-value of a digit in a numeral: The place-value of a digit in a number depends on the place it occupies in the place value chart.

Example: In the number 32568

1. The place-value of 5 = 5 × 100 = 500 ( 5 in hundreds place)

2. The place-value of 3 = 3 × 10,000 = 30,000 ( 3 in ten thousands place)

Place value and face value: Every digit of a number has two values-the face value and the place value. The face value of a digit is the value of the digit itself and does not change, while the place value of the digit changes according to its position in the number. This can be shown as:

 Number Digit Face value Place value 67,923 6 6 60,000 7 7 7,000 9 9 900 2 2 20 3 3 3

Example: What is the difference between place value and face value of 8 in 478526?
Solution:

The place value of 8 = 8 × 1000 =8000

The face value of 8 = 8

The difference between place value and face value of 8 = 8000 – 8 = 7992

The place-value or face-value of ‘0’ is always ‘0’ wherever it may be.

Expanded form of a number: If a number is written as the sum of the place value of all the digits of the number, then the number is said tobe in its expanded form.

Example:4235 = 4000 + 200 + 30 + 5 (expanded form).

Short form of a number: The expanded form of the number can be written as a single number (numeral) then it is called the short form of the number.

Example:70000 + 3000 + 500 + 00 + 7 = 73507 (short form).

Comparison of Numbers: In order to compare two numbers use the following rules:

Rule 1: If the two numbers with different number of digits are given, then the number which has more digits is the greater of the two.

Example:Find the greater among 278654 and 76543671.

Solution: In the given two numbers 278654has 6 digits, while 76543671 has 8 digits.

∴ 76543671 > 278654

Rule 2: If the two numbers to be compared have the same number of digits.

i) First compare the digits in the extreme left place in both the numbers.

ii) If they are of equal value, compare the second digit from the left in both the numbers.

iii) If they are of equal value, compare the third digit from the left in both the numbers.

iv) Continue until you come to unequal digits at the corresponding places.

v) Now the number with such digit is the greater of the two.

Example: Which is greater: 8569734 or 8561867?

Solution: Arrange the two given numbers in the place value chart.

 T-L L T-Th Th H T U 8 5 6 9 7 3 4 8 5 6 1 8 6 7

Both the numbers have 7 digits. At the ten-lakhs place, both have the same digit 8.Also at lakhs, and ten-thousands both the given numbers have same digits 5 and 6 respectively. But at thousands place the first number has 9 while the second one has 1, Since 9 > 1

∴ 8569734> 8561867.

Ordering of numbers:

i) The arrangement of numbers from the smallest to the greatest is called ascending order.
Example: 560, 567, 662, 665, 786

ii) The arrangement of numbers from the greatest to the smallest is called descending order.
Example: 789, 665, 662, 567, 560 are in descending order.

Rules for formation of the largest and least numbers with required number of digits:

To form the largest number with required number of digits fill the places with the digit 9.

To form the smallest number with required number of digits fill the places with the first digit in the number from left with 1 and rest with zeroes.

Table of forming the greatest and least numbers with required digits:

 Number of digits Largest Number Least Number 1 9 1 2 99 10 3 999 100 4 9999 1000 5 99999 10000

The smallest number using the given digits, each only once without repetition:

Case I. If none of the given digits is zero.

In this case, arrange the given digits in ascending order.

Examples:

(i) The smallest 3-digit number formed by using the digits 7, 3 and 9 is 379.

(ii) The smallest 4-digit number formed by using the digits 7, 2, 5 and 9 is 2579.

Case II. If one of the given digits is zero.

In this case, put 0 at second place from the left. Then the remaining places are filled from left to right by the remaining digits in ascending order.

Examples:

(i) The smallest 3-digit number formed by using the digits 0, 2 and 5 is 205.

(ii) The smallest 4-digit number formed by using the digits 8, 6, 1 and 0 is 1068.

The greatest number using the given digits, each only once without repetition: To form the greatest number, the given digits are arranged in descending order.

Examples:

(i) The greatest 3-digit number formed by using the digits 3, 6and 7 is 763.

(ii) The greatest 4-digit number formed by using the digits 0, 5,8 and 9 is 9850.

The smallest or greatest number, using the given digits, when repetition of digits is allowed:

In this case, first form the smallest or greatest number using the given digits each only once without repetition. Then, in the number so formed,replace the digit whose repetition is allowed.

Example: Find the greatest and least number of 5 digits using digits 8, 3, 6 with 6 as repetition.

Solution: The least 5 digit number is 36668.

The greatest 5 digit number is 86663.

Exception: If the smallest number to be formed with one of the digits as zero, then in the number, zero has to be in the second place from the left.Thus, if the digit at the left
most places is to be repeated twice then the repeated digits lie on either side of the zero in the second place.

Examples:

(i) The smallest 3-digit number formed by using the digits 0 and 7 with ‘7’ as repetition is 707.

(ii) The smallest 4-digit number formed by using the digits 0, 5 and 6 with 5 as repetition is 5056.

Example:How many 8-digit numbers are there in all?

Solution: The largest 8-digit number =99999999

The smallest 8-digit number = 10000000.

Number of all 8-digit numbers
= (99999999) – (10000000) + 1[1 is for the beginning number]
= (89999999 + 1) = 90000000

= nine crores.

Aliter 1:

The smallest 9 – digit number = 100000000
The smallest 8 – digit number = 10000000

Number of all 8 digit numbers = 100000000 – 10000000

= 90000000 (nine crores)

Aliter 2:
The greatest 8 – digit number = 99999999
The greatest 7 – digit number = 9999999

Number of all 8 digit numbers = 99999999 – 9999999

= 90000000 (nine crores)

Hence, there are in all nine crores of 8-digit numbers.