Class 7 – Number System
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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, a symbol is used which is called as “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, 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, ………} There are two types of number systems. i) Hindu Arabic (or) Indian system. ii) International (or) English system. 
Hindu Arabic system: In the HinduArabic 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 HinduArabic 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. So, in Hindu Arabic system commas are used after 3, 2, 2, …… from right to left with respect of periods. iv) In crores period the periods of HinduArabic system are followed. 

CRORES PERIOD 
LAKHS PERIOD 
THOUSAND PERIOD 
ONES PERIOD 

Ten Crores 
Crores 
Ten lakhs 
Lakh 
Ten Thousands 
Thousands 
Hundreds 
Tens 
Units 
Example: The number 783679120 is written in HinduArabic system by using commas as 78, 36, 79, 120.
And read as seventy eight crores, thirty six lakhs, seventy nine thousands, one hundred and twenty.
International System: In this system of numeration, periods are formed 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 
Tens 
Units 
Example: The date of birth of the great Indian mathematician Shakuntala Devi is 4th November 1929, Written as 4111929.
This number can be written in words as,
i) In International system it is four millions, one hundred eleven thousands, nine hundred and twenty nine.
ii) In Hindu Arabic system it is forty one lakhs, eleven thousands, nine hundred and twenty nine.
Facevalue of a digit in a number: The facevalue of a digit in a number is the digit itself at whatever place it may be.
Placevalue of a digit in a number: The placevalue of a digit in a number depends up on the place it occupies in the place value chart.
Example: In the number 54329
1. Placevalue of 3 = 3 × 100 = 300 (∵ 3 in hundreds place)
Face value of 3 = 3
2. Placevalue of 4 = 4 × 1,000 = 4,000 (∵ 4 in thousands place)
Face value of 4 = 4
Example: What is the difference between place value and face value of 5 in 354928?
Solution: The place value of 5 = 5 × 10000 = 50000
The face value of 5 = 5
The difference between place value and face value is 50000 – 5 = 49995.
Note: The placevalue or facevalue of ‘0’ is always ‘0’ wherever it may be.
Ordering of numbers:
i) The arrangement of numbers from the smallest to the greatest is called ascending order.
Example: 468, 542, 603, 692, 986, 999 are in ascending order.
ii) The arrangement of numbers from the greatest to the smallest is called descending order.
Example: 942, 836, 789, 647, 531, 98 are in descending order.
Table of forming 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 
Example: How many 6digit numbers are there in all?
Aliter1Solution:
The smallest 7 – digit number = 1000000
The smallest 6 – digit number = 100000
Number of all 6 digit numbers = 100000 – 100000
= 900000 (nine lakhs)
Aliter:
The greatest 6 – digit number = 999999
The greatest 5 – digit number = 99999
Number of all 6 digit numbers = 999999 – 99999
= 900000 (nine lakhs)
Hence, there are in all nine lakhs of 6digit numbers.
Roman system of Numeration: The numeral system of ancient Rome or Roman numerals uses combination of letters from Latin alphabet to denote the numbers.
The basic ‘7’ Roman numerals are as follows.
I 
V 
X 
L 
C 
D 
M 
1 
5 
10 
50 
100 
500 
1000 
The numbers 1 to 9 can be expressed in Roman Numerals as follows.
I, II, III, IV, V, VI, VII, VIII, IX.
Example: The year 2015 in Roman numeral is written as ‘MMXV’.
The Roman numeral system does not include ‘Zero’.
If a bar is placed over a numeral, it is multiplied by 1000.
Thus, , , etc.
Rule 1: Repetition of a symbol in a Roman numeral means addition.
Cautions: (i) Only I, X, C, M can be repeated
(ii) V, L and D are never repeated
(iii) No symbol in a Roman numeral can be repeated more than 3 times.
(i) III = (1 + 1 + 1) = 3 (ii) MM = 1000 + 1000 = 2000
(iii) XX = (10 + 10) = 20 (iv) CCC = 100 + 100 + 100 = 300
Rule 2: A smaller numeral written to the right of a larger numeral is always added to the larger numeral.
Example: (i) VII = 5 + 1 + 1 = 7 (ii) XVI = 10 + 5 + 1 = 16
(iii) LVII = 50 + 5 + 1 + 1 = 57
Rule 3: A smaller numeral written to the left of a large numeral is always subtracted from the larger numeral.
Cautions: (i) V, L and D are never subtracted.
(ii) I can be subtracted from V and X only
(iii) X can be subtracted from L and C only
(iv) C can be subtracted from D and M only.
Rule 4: When a smaller numeral is placed between two larger numerals, it is always subtracted from the larger numeral immediately following it.
Examples: (i) XXIV = 20 + (5 – 1) = 24 (ii) LXIX = 50 + 10 + (10 – 1) = 69
(iii) CXXIV = 100 + 10 + 10 + (5 – 1) = 124
Note: Repetition of a symbol in a Roman numeral means addition.
Example: Express the following numbers as a Roman numerals.
(i) 345 (ii) 989
Solution:
(i) 345 = 300 + 40 + 5 = 300 + (50 – 10) + 5 = CCCXLV
(ii) 989 = (1000 – 100) + 80 + (10 – 1) = CMLXXXIX
Example: Write the following in HinduArabic numerals.
(i) MCLIX (ii) CMXLIX
Solution:
(i) MCLIX = 1000 + 100 + 50 + (10 – 1) = 1159
ii) CMXLIX = (1000 – 100) + (50 – 10) + (10 – 1) = 949.(iii)
Example: Show that each of the following is meaningless. Give reason in each case.
(i) IC (ii) XVV
Solution: (i) I can be subtracted from V and X only
∴ IC cannot be represented as a roman number.
(ii) V, L, D are never repeated
∴ XVV cannot be represented as a roman number.
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Binary system: The binary numeral system (or) base2 system represents numeric values using two symbols, 0 and 1. More specifically, the usual base2 system is a positional notation with a radix 2.
*[In mathematical numeral systems, the base or radix is the simplest case in the number of unique digits, including zero that a positional numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is 10, because it uses ten digits from 0 to 9.]
System 
Base 10 System 
Binary System or Base 2 System 
Base/Radix 
10 
2 
Digits used 
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 
0,1 
Counting Example 
After a digit reaches 9, an increment resets it to 0 but also caused an increment of the next digit to the left. 000, 001,..,009, 010, 011, ..099, 100,.. 
After a digit reaches 1, an increment resets it to 0 but also causes an increment of the next digit to the left. 000, 001, 010, 011, 100…. 
Conversion of numbers into Binary System: The given number is divided by 2
(Radix is 2) then the remainder is either ‘0’ or ‘1’. It can be written to the right of the dividend (given number) continue down words, dividing each new quotient by ‘2’ upto the quotient is ‘0’. Starting with bottom read the remainders upwards to the top.
Example: Convert 72 to binary system. Solution: 
Example: Convert 48 into binary system.

Conversion of Binary numbers into numbers: Since binary is a base2 system, each digit represents an increasing power of 2, with the rightmost digit representing 2^{0}, the next representing 2^{1}, then 2^{2}, and so on. To determine the decimal representation of a binary number simply take the sum of the products of the binary digits and the powers of 2 which they represent.
Example: Convert 101010_{(2)} into base10 system.
Solution: 101010_{(2)}
1 
0 
1 
0 
1 
0 
2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 
1 × 32 
0 × 16 
1 × 8 
0 × 4 
1 × 2 
0 × 1 
101010_{(2)} = [1 × 32] + [0 × 16] + [1 × 8] + [0 × 4] + [1 × 2] + [0 × 1]
= 32 + 0 + 8 + 0 + 2 + 0
=42
∴ 101010_{(2)} = 42_{(10)}.
Relation between HinduArabic system and International system
Hindu Arabic system 
International system 
One lakh 
Hundred thousands 
Ten lakhs 
One million 
One crore 
Ten millions 
Ten crores 
Hundred millions 