Systems of Numeration
We know two systems of numeration.
(i) HinduArab 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 2digit based number system.
The numerals of numbers in the three systems are as follows.
10digitbased numbers: 358293, 934528
5digitbased numbers: XL, CDXXVI, MMC
2digitbased numbers: 1101011, 1101111
The twodigit based numbers and 10digit based numbers may be interchanged.
In 10digit based numbers, the place values from right to left are as follows:
In 2digit based numbers, the place values from right to left are given below.
Let there be a
2based number (1101011)_{2} and we have to write it as a
10based number:
The 2based number is written in its extended form (i.e., according to the place value) as shown here:
Therefore, Number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107= (2based number) 1101011 = (10based number) 107
In short 2based number 1101011 may be changed into 10based number
Therefore, Number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107
= (2based number) 1101011 = (10based number) 107
In short 2based number 1101011 may be changed into 10based number
(1101011)
_{2} = 1 x 2
^{6} + 1 x 2
^{5} + 0 x 2
^{4} + 1 x 2
^{3} + 0 x 2
^{2} + 1 x 2
^{1} + 1 x 2
^{0}
= (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 10based number may also be changed into 2based number.
Say, we have to change (107)
_{10} into 2based number.

one’s place
2’s place
4’s place
8’s place
16’s place
32’s place
64’s place

2based number = 11010111
(i) 107 is divided by 2, quotient is 53 the remainder is 1.
This remainder 1 is the digit of 2based number having place 2
^{0} = 1
(ii) 53 is divided by 2, quotient is 26 the remainder is 1, it is the digit of 2based number having place value 2
^{1} = 2
(iii) 26 is divided by 2, quotient = 13, R = 0.
Remainder 0 has the place value 2
^{2} = 4 where 0 x 4 = 0
(iv) 13 is divided by 2, quotient = 6, R = 1, place value = 2
^{3} = 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 2
^{5} = 1 x 32 = 32
(vii) 1 is divided by 2, quotient = 0, R = 1,place value = 1 x 2
^{6} = 1 x 64 = 64
Therefore, (107)
_{10} = (1101011)
_{2}
2based number = 1101011
10based number = 64 + 32 + 0 + 8 + 0 + 2 + 1 = (107)
_{10}
Say we have to change (119)_{10} into 2based number.

2^{0} = 1place
2^{1} = 2place
2^{2} = 4place
2^{3} = 0place
2^{4} = 16place
2^{5} = 32place
2^{6} = 64place

Verification
1 1 1 0 1 1 1
= 2
^{6} + 2
^{5} + 2
^{4} + 2
^{3} + 2
^{2} + 2
^{1} + 2
^{0}
= 64 + 32 + 16 + 0 + 4 + 2 + 1
= 119
2based number
= 1110111

10based 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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