Page 30 - Math 7

Number Systems in Diff erent Bases

2.4 Binary number system
In denary number system, we use ten digits 0 to 9 to write any number. However, in
binary number system, we use only two digits 0 to 1 to express any number. Binary
number system has its broad applications in digital electronics.
Computers and hand held calculators actually use the binary system for their
internal calculations since the system consists of only two symbols, 0 and 1. All
numbers can then be represented by electronic “switches”, of one kind or another,
where “on” indicates 1 and “off” indicates 0.

2.5 Conversion of binary numbers to decimal numbers
To convert a binary number into decimal, it is expanded in the power of 2. Then, by
simplifying the expanded form of the binary number, we obtain a decimal number.
For example:
1
3
2
(i) 11011 = 1 u 2 + 1 u 2 + 0 u 2 + 1 u 2 + 1 u 1 0
4
2
= 1 u 16 + 1 u 8 + 0 u 4 + 1 u 2 + 1 u 1
= 16 + 8 + 0 + 2 + 1 = 27
? 11011 = 27
2
(ii) 1011001 = 1 u 2 + 0 u 2 + 1 u 2 + 1 u 2 + 0 u 2 + 0 u 2 + 1 u 2 0
1
2
4
3
6
5
2
= 1 u 64 + 0 u 32 + 1 u 16 + 1 u 8 + 0 u 4 + 0 u 2 + 1 u 1
= 64 + 0 + 16 + 8 + 0 + 0 + 1 = 89
? 1011001 = 89
2
2.6 Conversion of decimal numbers to binary numbers
Again, let’s take 15 blocks of cubes and regroup them into the group of 2 blocks.

7 pairs blocks of cube and 1 cube
Now, let’s arrange the groups of 2 blocks into the base of 2 with the maximum
possible powers.

8 4 2 1
2 2 2 2 1 2 0
3
So, 15 = 1 × 2 + 1 × 2 + 1 × 2 + 1 × 2° = 1111
2
3
1
2
In this way, the denary number 15 can be expressed in binary number as 1111 .
2
We can convert a decimal number into a binary number by using the place value
table of the binary system. For example:

Vedanta Excel in Mathematics - Book 7 28 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur