1. 1. Number Systems

2. Common Number Systems System Base Symbols Used by humans?
Used in computers?
Decimal 10
0, 1, … 9
Yes No
Binary 2
0, 1

3. Quantities/Counting Decimal Binary 1 2 10 3 11 4 100 5 101 6 110 7 1111 2 10 3 11 4
100 5
101 6
110 7
111
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4. Quantities/Counting Decimal Binary 8 1000 9 1001 10 1010 11 1011 12
1100 13
1101 14
1110 15
1111

5. Quantities/Counting Decimal Binary 16 10000 17 10001 18 10010 19 1001120
10100 21
10101 22
10110 23
10111
Etc.

6. Conversion Among Bases
The possibilities:
Decimal
Binary pp

7. Quick Example
2510 =
Base
Base

8. Weight
12510 => 5 x 100 = x 101 = x 102 =
Base

9. Binary to Decimal
Decimal
Binary

10. Binary to Decimal Technique
Multiply each bit by 2n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results

11. Example
Bit “0”
=> 1 x 20 = x 21 = x 22 = x 23 = x 24 = x 25 = 32
4310

12. Decimal to Binary
Decimal
Binary

13. Decimal to Binary Technique Divide by two, keep track of the remainder
First remainder is bit 0 (LSB, least-significant bit)
Second remainder is bit 1
Etc.

14. Example
12510 = ?2
12510 =

15. Binary Addition
Two 1-bit values A B
A + B 1 10
“two” pp

16. Binary Addition Two n-bit values Add individual bits Propagate carries
E.g., 1 1

17. Multiplication
Decimal (just for fun)
35 x
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18. Multiplication
Binary, two 1-bit values A B
A  B 1

19. Multiplication Binary, two n-bit values As with decimal values E.g.,x

20. Fractions Conversions
Decimal to decimal
3.14 => 4 x 10-2 = x 10-1 = x 100 = pp

21. Fractions Conversions
Binary to decimal
=> 1 x 2-4 = x 2-3 = x 2-2 = x 2-1 = x 20 = x 21 = pp

22. Fractions Conversions
Decimal to binary
x x x x x x
etc.
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