Binary Numbers

Class: Basic 5

Subject: Mathematics

Topic: Binary Numbers

Converting from binary to decimal.

A number in binary, that is base two can be changed to decimal (base ten).

To covert from base two to base ten, the following steps are required:

1. First assign position to the binary numbers, starting from right to left.

2. The position begins with zero upward. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

3. Then multiply each binary value with the base (2 raise to power of the position it represent). 2^2, 2^0, 2^3

4. Sum them up.

The following are values of numbers in base 2 :
2^0 = 1 2^1 = 2 2^2 = 4

2^3 = 8 2^4 = 16 2^5 = 32

2^6=64 2^7=128 2^8=256

2^9 = 512

For example:

1) Change 11011 base two to base ten.

Solution:
Position: 4 3 2 1 0
Binary digits: 1 1 0 1 1
Then the expansion is given below:
=(1× 2^4) + ( 1× 2^3) + (0×2^2) + (1×2^1) + (1× 2^0)

=(1×16 ) + (1× 8) + ( 0×4) + (1×2) + (1× 1)

= 16 + 8 + 0 + 2 + 1
= 27 base ten

2) Convert 11001 base 2 to base 10.

Solution:
Position: 4 3 2 1 0
Binary numbers: 1 1 0 0 1
Then the expansion is :
= (1×2^4) + (1×2^3) + (0×2^2) + (0×2^1) + (1×2^0)

= (1×16) + (1×8) + (0×4) + (0×2) + (1×1)

= 16 + 8 + 0 + 0 + 1
=25 base ten

3) Change 100111 base 2 to base 10.

Solution:
Position: 5 4 3 2 1 0
Binary no. : 1 0 0 1 1 1
Then the expanded form is given below:

=(1×2^5)+(0×2^4)+(0×2^3)+ (1×2^2)+(1×2^1)+(1×2^0)

= (1×32) + (0×16) + (0×8) + (1×4) + (1×2) + (1×1)

= 32 + 0 + 0 + 4 + 2 + 1
= 39 base 10

Exercise:

Convert the following binary numbers to base 10.

1) 10001 base two

2) 101010 base 2

3) 10101 base two