1. Decimal / Binary Conversions
By Dr. JJ Shepherd

2. Number Systems A set of symbols to represent values
Each number system has a set of symbols (digits or characters) representing a value
Each system has a base which represents the base symbols in the system
The symbol’s position indicates its order of magnitude
Values are the summation of symbols multiplied by their base to the power of an exponent corresponding to their position

3. Decimal System Most commonly used system It is Base 10
Symbols are {0,1,2,3,4,5,6,7,8,9}
Each value in this system can be expressed as a sum of each value multiplied by an exponent of 10
Example:
365 = 3x x x100

4. Binary System Commonly used in computing It is Base 2
Symbols are {0,1}
Each value in this system can be expressed as a sum of each value multiplied by an exponent of 2
Example:
101 = 1x22 + 0x21 + 1x20

5. Binary to Decimal
Using the same idea from the previous slide, expand the number by its factors
Then sum together that expression into the decimal system
Example
101 = 1x22 + 0x21 + 1x20 = 5
1101 = 1x23 + 1x22 + 0x21 + 1x20 = 13

6. Decimal to Binary 11 / 2 = 5 R 1 5 / 2 = 2 R 1 2 / 2 = 1 R 0
Going in the other direction requires division of the base
Integer divide the value by 2
Record the remainder
Repeat this until the value is 0
Reversing the remainders give the binary number
Examples
11 / 2 = 5 R 1
5 / 2 = 2 R 1
2 / 2 = 1 R 0
1 / 2 = 0 R 1
HALT!
11 = 1011
13 / 2 = 6 R 1
6 / 2 = 3 R 0
3 / 2 = 1 R 1
1 / 2 = 0 R 1
HALT!
13 = 1101
365 / 2 = 182 R 1
182 / 2 = 91 R 0
91 / 2 = 45 R 1
45 / 2 = 22 R 1
22/ 2 = 11 R 0
11 / 2 = 5 R 1
5 / 2 = 2 R 1
2 / 2 = 1 R 0
1 / 2 = 0 R 1
HALT!
365 =
32 / 2 = 16 R 0
16 / 2 = 8 R 0
8 / 2 = 4 R 0
4 / 2 = 2 R 0
2 / 2 = 1 R 0
1 / 2 = 0 R 1
HALT!
32 =
42 / 2 = 21 R 0
21 / 2 = 10 R 1
10 / 2 = 5 R 0
5 / 2 = 2 R 1
2 / 2 = 1 R 0
1 / 2 = 0 R 1
HALT!
32 =