Systems of Numeration

We know two systems of numeration.

(i) Hindu-Arab System of numbers based on 10 digits, i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

(ii) Roman System of numbers based on 5 digits, i.e., I, V, X, L, C, D and M.

The numbers based on 10 digits and 5 digits may be interchanged.

(iii) There is a third system of numbers named as computer system. This system of numbers is based on two digits, i.e., 0 and 1.

This system is also called 2-digit based number system.

The numerals of numbers in the three systems are as follows.

10-digit-based numbers: 358293, 934528

5-digit-based numbers: XL, CDXXVI, MMC

2-digit-based numbers: 1101011, 1101111

The two-digit based numbers and 10-digit based numbers may be interchanged.

In 10-digit based numbers, the place values from right to left are as follows:

Place Values


In 2-digit based numbers, the place values from right to left are given below.

Place Values Chart

Let there be a 2-based number (1101011)2 and we have to write it as a 10-based number:

The 2-based number is written in its extended form (i.e., according to the place value) as shown here:

Place Value Number

Therefore, Number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107= (2-based number) 1101011 = (10-based number) 107

In short 2-based number 1101011 may be changed into 10-based number



Therefore, Number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107 = (2-based number) 1101011 = (10-based number) 107

In short 2-based number 1101011 may be changed into 10-based number

(1101011)2 = 1 x 26 + 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20

          = (1 x 64) + (1 x 32) + (0 x 16) + (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1)

          = 64 + 32 + 0 + 8 + 0 + 2 + 1

          = (107)10

The 10-based number may also be changed into 2-based number.



Say, we have to change (107)10 into 2-based number.

10-based Numbers


one’s place

2’s place

4’s place

8’s place

16’s place

32’s place

64’s place



2-based number = 11010111

(i) 107 is divided by 2, quotient is 53 the remainder is 1.
This remainder 1 is the digit of 2-based number having place 20 = 1

(ii) 53 is divided by 2, quotient is 26 the remainder is 1, it is the digit of 2-based number having place value 21 = 2

(iii) 26 is divided by 2, quotient = 13, R = 0.
Remainder 0 has the place value 22 = 4 where 0 x 4 = 0

(iv) 13 is divided by 2, quotient = 6, R = 1, place value = 23 = 8, 1 x 8 = 8

(v) 6 is divided by 2, Quotient = 3, R = 0, place value of 0 = 24, 0 x 16 = 0

(vi) 3 is divided by 2, quotient = 1, R = 1, place value of 1 x 25 = 1 x 32 = 32

(vii) 1 is divided by 2, quotient = 0, R = 1,place value = 1 x 26 = 1 x 64 = 64

Therefore, (107)10 = (1101011)2

2-based number = 1101011

10-based number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = (107)10

Say we have to change (119)10 into 2-based number.

2-based number


20 = 1-place

21 = 2-place

22 = 4-place

23 = 0-place

24 = 16-place

25 = 32-place

26 = 64-place



Verification

    1     1     1     0     1     1     1

= 26 + 25 + 24 + 23 + 22 + 21 + 20

= 64 + 32 + 16 + 0 + 4 + 2 + 1

= 119

2-based number

= 1110111

10-based number

= 64 + 32 + 16 + 0 + 4 + 2 + 1

= 119

Related Concept

Patterns and Mental Mathematics

Counting Numbers in Proper Pattern

Odd Numbers Patterns

Three Consecutive Numbers

Number Formed by Any Power

Product of The Number

Magic Square

Square of a Number

Difference of The Squares

Multiplied by Itself

Puzzle

Patterns









4th Grade Math Activities

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