COMP201 Computer Systems Number Representation

1. COMP201 Computer SystemsNumber Representation

2. Number Representation • Introduction • Number Systems • Integer Representations • Examples • Englander Chapter 2 and Chapter 4.

3. Introduction • Data must be converted to Binary before it can be stored in Computer

4. Introduction • Format used will depend on data: • e.g. consider the number 9 • 9 may be represented as 1001 • -9 may be represented as 10111 • 9.0 may be represented as 0000010010000001 • the character 9 may be represented as 0111001 • And the intended purpose of the number within the computer

5. Considerations • When a coding format is being devised a number of considerations need to be made: • Ease of manipulation • Conciseness and machine efficiency • Accuracy sufficient for problem • Standardised data communication

6. Computers store and Manipulate numbers using the binary (base 2) number system Base 10 = 510 Roman = V Binary = 1012 Base3 = 123 Number Representation

7. Binary Representation • Each bit represents a value two times the value of the bit to it’s right. Dec Bin 0 000 = 0x22 + 0x21 + 0x20 1 001 = 0x22 + 0x21 + 1x20 2 010 = 0x22 + 1x21 + 0x20 3 011 = 0x22 + 1x21 + 1x20 4 100 = 1x22 + 0x21 + 0x20

8. Octal Representation • Each octal number can be perfectly represented by 3 binary digits • 0 000 • 001 • 010 • 011 • 100 • 101 • 110 • 111 010 110 101 001 = 26518 =??10

9. Hexadecimal Representation • Each Hex number can be perfectly represented by 4 binary digits • Two Hex digits can be used to represent a byte; four for a word. • 0 0000 • 0001 • 0010 • 0011 • 0100 • 0101 • 0110 • 0111 • 1000 • 1001 • A 1010 • B 1011 • C 1100 • D 1101 • E 1110 • F 1111 0010 1110 1010 0001 = 2EA116

10. Number Conversion methods • (You should have already studied this in a previous class. If you did not, or don’t remember, study Chapter 2 in detail.) • Base= number of characters in system Decimal Octal Hexadecimal Binary 10 8 16 2

11. Number Conversion methods • Place value: the value of a digit depends upon its placement relative to a reference, say a decimal point. • For instance, 2610 = 20 + 6 = 2 x 101 + 6 x 100 • This leads to methods for converting between bases. 26518= 2 x 83 + 6 x 82 + 5 x 81+ 1 x 80 • Substitute decimal values for powers of 8 to convert 26518 into decimal equivalent (1456)

12. Integer Representations • BCD (binary coded decimal) • Sign and Magnitude • Excess Notation • Two’s complement

13. BCD (Binary Coded Decimal) • Each Decimal digit is coded as a 4 bit binary code • Developed for early calculators • e.g. 35910 = 0011 0101 1001bcd • Easy for people to understand, Hard for computers to work with • Signed Magnitude • Extra bit added to the code to represent the sign • In most cases • a 0 represents a +ve • a 1 represents a –ve

14. BCD Example • Each Decimal digit is coded as a 4 bit binary code 410= 00000100 • And two digits, 4510, would be stored in two different memory locations! • 00000100 • 00000101

15. Packed BCD • Packed BCD simply makes use of the storage space normally wasted in storage of BCD, by using the leftmost 4-bits (nibble) for one digit, and the rightmost nibble for another. • 4510= 01000101 in packed BCD • BCD used in some old financial software and calculators, but now very rare.

16. Sign • Negative integers are often required • Computers do not have internal minus signs • There are several ways to represent negative and positive integers • Choice is often based on the ease of manipulation for the intended purpose

17. Sign and Magnitude • One of the simplest systems is to allocate one bit as the “sign” bit. The other bits are the “magnitude”. • E.g. +24 = 00011000 -24 = 10011000 • Simple to find the sign • Two values for zero • Must process sign separately

18. Excess Notation • fix number of bit positions used • smallest number with 1 in the MSB represents zero (e.g. 1000) • All bit strings greater than this represent +ve numbers • All bit strings less than this represent -ve numbers • Example is known as excess eight notation because 8 (1000) represents zero

19. Ones Complement • Negative numbers formed by inverting the positive representation (logical NOT) • E.g. +24 = 00011000 -24 = 11100111 • Two values for zero • Addition requires carry bit to be wrapped around

20. Two’s Complement • To negate a number invert the bits (logical NOT) then add one. • E.g. +24 = 00011000 invert 11100111 Add one -24 = 11100111 + 1 -24 = 11101000 • Works for converting negative to positive as well

21. Two’s Complement • Can represent numbers in the range -2n-1 to (2n-1-1) • Only one representation of zero Binary Negative Compliment 0 1 1 0 Positive +1 Result

22. Two’s Complement– another viewpoint: The number wheel (For 4 bits) Reference: Katz: Contemporary Logic Design, p243 -1 +0 +1 -2 1111 0000 1110 0001 -3 +2 1101 0010 -4 +3 1100 0011 -5 1011 +4 0100 1010 0101 -6 +5 1001 0110 +6 -7 1000 0111 +7 -8

23. Two’s Complement • Leftmost bit is always the sign bit • All zeroes is the only representation of zero • Simple to implement negation, addition and subtraction in hardware. • Can be sign extended • +510 = 00000101 = 0000000000000101 • -510 = 11111011 = 1111111111111011

24. Two’s Complement Examples • Express the following in 2’s complement notation (use 16 bit form): • 10000 • 100111100001001 • 0100111000100100 • Add the following 2’s complement numbers (they are 12 bits): • 011001101101 • 111010111011

25. Format Use • Ones complement common in older computer hardware • Two’s complement most common signed integer representation today. • Sign and magnitude used in some very early computers • Sign and magnitude and Excess notation both used in common floating point formats