Class 7 – Number System

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Number System
  • Review of number systems
  • Review of numbers
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Number System
  • Fractions
  • Decimals
  • Introduction to rational numbers
  • Operations on rational numbers
  • Simplification of brackets.
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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 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.

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 Hindu-Arabic 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 Hindu-Arabic 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

 

http://upload.wikimedia.org/wikipedia/en/f/f6/Shakuntala_Devi.jpgExample: 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.

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

 

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

Example: In the number 54329

1. Place-value of 3 = 3 × 100 = 300                 (∵ 3 in hundreds place)

    Face value of 3 = 3

2. Place-value 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 place-value or face-value 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 6-digit numbers are there in all?

Aliter-1Solution:

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 6-digit 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’.

http://images.clipartpanda.com/bloc-clipart-note-md.pngThe 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 Hindu-Arabic 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) base-2 system represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 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 base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, 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 base-10 system.

Solution: 101010(2)

1

0

1

0

1

0

25

24

23

22

21

20

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).

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Relation between Hindu-Arabic 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