1. Introduction to Number Systems

2. Storyline … Different number systems Why use different ones?
Binary / Octal / Hexadecimal
Conversions
Negative number representation
Binary arithmetic
Overflow / Underflow

3. Number Systems Four number system Decimal (10) Binary (2) Octal (8)
Hexadecimal (16)

4. Binary numbers? Computers work only on two states
Off
Basic memory elements hold only two states
Zero / One
Thus a number system with two elements {0,1}
A binary digit – bit !

5. Decimal numbers 1439 = 1 x 103 + 4 x 102 + 3 x 101 + 9 x 100
Thousands Hundreds Tens Ones
Radix = 10

6. Binary Decimal 1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20=
(1101)2 = (13)10
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ….

7. Decimal Binary 2 13
LSB 1 2 6 2 3 1 2 1 1
MSB
(13)10 = (1101)2

8. Octal Decimal 137 = 1 x 82 + 3 x 81 + 7 x 80 = 1 x 64 + 3 x 8 + 7 x 1=
(137)8 = (95)10
Digits used in Octal number system – 0 to 7

9. Decimal Octal 8 95
LSP 7 8 11 3 8 1 1
MSP
(95)10 = (137)8

10. Hex Decimal
BAD = 11 x x x 160
= 11 x x x 1 =
(BAD)16 = (2989)10
A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

11. Decimal Hex 16
2989
LSP 13 16
186 10 16 11 11
MSP
(2989)10 = (BAD)16

12. Why octal or hex? Ease of use and conversion
Three bits make one octal digit
=> in octal
Four bits make one hexadecimal digit
E B => EB5 in hex
4 bits = nibble

14. Negative numbers Three representations Signed magnitude 1’s complement

15. Sign magnitude Make MSB represent sign Positive = 0 Negative = 1
E.g. for a 3 bit set
“-2”
Sign
Bit 1
MSB
LSB

16. 1’s complement MSB as in sign magnitude Complement all the other bits
Given a positive number complement all bits to get negative equivalent
E.g. for a 3 bit set
“-2”
Sign
Bit 1

17. 2’s complement 1’s complement plus one E.g. for a 3 bit set “-2” Sign

18. Decimal number 3 011 2 010 1 001 000 -0 100 --- 111 -1 101 110 -2 -3
Signed magnitude
2’s complement
1’s complement 3
011 2
010 1
001
000 -0
100
---
111 -1
101
110 -2 -3 -4
No matter which scheme is used we get an even set of numbers but we need one less (odd: as we have a unique zero)

19. Binary Arithmetic Addition / subtraction Unsigned Signed
Using negative numbers

20. Unsigned: Addition Like normal decimal addition B A 0101 (5)
The carry out of the MSB is neglected + 1 10
0101 (5)
(9)
1110 (14)

21. Unsigned: Subtraction
Like normal decimal subtraction B A
A borrow (shown in red) from the MSB implies a negative - 1 11
1001 (9)
(5)
0100 (4)

22. Signed arithmetic Use a negative number representation scheme
Reduces subtraction to addition

23. 2’s complement Negative numbers in 2’s complement 001 ( 1)10
001 ( 1)10
101 (-3)10
110 (-2)10
The carry out of the MSB is lost

24. Overflow / Underflow Maximum value N bits can hold : 2n –1
When addition result is bigger than the biggest number of bits can hold.
Overflow
When addition result is smaller than the smallest number the bits can hold.
Underflow
Addition of a positive and a negative number cannot give an overflow or underflow.

25. Overflow example 011 (+3)10 110 (+6)10 ????
110 (+6) ????
1’s complement computer interprets it as –1 !!
(+6)10 = (0110)2 requires four bits !

26. Underflow examples Two’s complement addition 101 (-3)10
Carry (-6) ????
The computer sees it as +2.
(-6)10 = (1010)2 again requires four bits !