CONVERSION OF OTHER BASES TO DENARY SYSTEM

Conversion of other bases to denary system. To convert numbers in the other bases to denary system, expand the given number in power of its bases.

Examples:

1. Convert 354₁₆ to denary

354₁₆ = 3 x 16² + 5 X 16¹ + 4 x 16⁰

= 3 x 256 + 5 x 16 + 4 x 1

= 768 + 80 + 4

852_ten

2. Convert 255_eight to base ten

255₈   = 2 x 8² + 5 x 8¹ + 5 x 8⁰

= 2 x 64 + 5 x 8 + 5 x 1

= 128 + 40 + 5

= 173_ten

3. Convert 1011001₂ to base ten

1011001_two = 1 x 2⁶ + 0 x 2⁵ + 1 x 2⁴ + 1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 2⁰

= 1 x 64 + 0 x 32 + 1 x 16 + 1 x 8 + 0 X 4 + 0 x 2 + 1 x 1

= 64 + 0 + 16 + 8 + 0 + 0 +1

= 89_ten

 

4. Convert 3A7₁₆ to base ten

3A7₁₆ = 3 x 16² + 10 x 16¹ + 7 x 16⁰

                = 3 x 256 + 10 x 16 + 7 x 1

= 768 + 160 + 7

= 935_ten

ADDITION, SUBTRACTION AND MULTIPLICATION OF BINARY NUMBERS

There are four rule guiding binary addition

0 + 0 = 0

0 + 1 = 1   (Always remember you are working with binary digit not decimal)

1 + 0 = 1

1 + 1 = 10

 

1. Add 101₂ + 1000₂
2. Add 1001101₂ + 111001₂

 

1. 101₂

+ 1000₂

1101₂

 

2. 1001101₂

+    111001₂

10000110₂

Binary Subtraction

0 – 0 = 0; 0 – 1 = 1; 1 – 0 = 1; 1 – 1 = 0

(1). Subtract 101₂ from 1001₂     (2). 10001₂ – 1111₂

1. 1001₂

–       101₂

100₂

 

2. 10001

–         1111

0010₂

Binary Multiplication

The rules for binary multiplication are: 0 x0 = 0; 1 x 0 = 0; 0 x 1 = 0; 1 x 1 = 1

(1).1101₂ x 110₂                                          (2). 101₂ x 10₂

1. 1101₂

x       110₂

0000₂

+     1101₂

1101₂

1001110₂

 

2. 101₂

x     10₂

000₂

+  101₂

1010₂

 

Decimal	Binary	Octal	Hexadecimal
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F

EVALUATION

1. Define number base
2. Convert the following to base 10

• 10011_two
• 317_eight

3. Evaluate the following:

• 101₂ x 101₂
• 111001₂ + 1001₂
• 10111₂ – 100₂