Binary Numbers. The Number Bases of Integers Textbook Chapter 3

Transcription

1 Binary Numbers The Number Bases of Integers Textbook Chapter 3

2 The families of binary numbers Integers Unsigned Represented as unsigned binary Signed Sign-and-magnitude 1 s complement 2 s complement Biased notation Fractional Numbers IEEE Standard for Binary Floating Point Arithmetic 03b-2

3 Unsigned binary It s a binary number that is always positive ( 0) Unsigned numbers have no sign So we assume they are positive (or zero) 03b-3

4 Unsigned binary addition Ripple carry Same principle as in decimal (or any base b) Add down the columns starting with least significant bit Carry over to the next column as needed 03b-4

5 Unsigned binary addition Example 1 simple addition Example 2 a problem occurs b-5

6 Overflow in unsigned binary Not enough bits to represent the digit Depends on the number of bits allowed for the representation Overflow = roll-over 03b-6

7 Unsigned binary subtraction Ripple-borrow Same principle as in decimal (or any base b) Subtract down the columns starting with least significant bit Borrow from previous column as needed 03b-7

8 Unsigned binary subtraction Example b-8

9 Unsigned binary multiplication Multiply by one bit at a time (partial products) Add partial products for answer Same principle as in decimal (or any base b) Multiply n-bit numbers for a 2n-bit result 03b-9

10 Unsigned binary multiplication Example 1010 x b-10

11 Signed numbers in real life Most humans precede negative number with E.g., 2000 Accountants, however, use parentheses E.g., (2000) or color 2000 Example: in hex? = e = 3e b-11

12 Signed numbers in binary Four main signed representations Sign-and-magnitude One s complement Two s complement Biased notation 03b-12

13 Sign-and-magnitude The first bit carries information about the sign 0 is positive 1 is negative The rest of the number is the magnitude (it s bigness) 03b-13

14 Sign-and-magnitude How does it work? Pros Intuitive Easy to invert the sign Cons Add and sub must compare sign and magnitude of the operands Two representations of zero x y 03b-14

15 Sign-and-magnitude: arithmetic Depends on sign of operands Same sign Add the magnitudes, keep the sign Different sign Subtract smaller from larger, keep sign of larger Overflow when 03b-15

16 Sign-and-magnitude: arithmetic Example 1 Perform A+B on 5 bits Where A = 2, B = 10 Example 2 Perform A+B on 5 bits Where A = 6, B = 10 03b-16

17 Sign-and-magnitude mapping Number represented Number encoded Range on n bits: b-17

18 One s complement It is called one s complement Converting to one s complement: For positive numbers Converting to one s complement: For negative numbers Flip the bits of the magnitude 03b-18

19 One s complement How does it work? Pros Still intuitive Easy to invert the sign Cons Add and subtract require a special operation (end-around carry) Two representations of zero x y 03b-19

20 One s complement: arithmetic Add or subtract normally Same sign or different sign Doesn t matter If there is a carry-out, add it back in Overflow when 03b-20

21 One s complement: arithmetic Example 1 A+B with A= 2, B=1 on 3 bits Example 2 A+B with A=3, B=3 on 3 bits 03b-21

22 One s complement mapping Number encoded Range on n bits: b-22

23 Two s complement It is called two s complement Converting to two s complement: For positive numbers Converting to two s complement: For negative numbers Method 1 Flip the bits and add one Method 2 Keep all the leftmost zeros Keep first leftmost one Flip all other bits 03b-23

24 Two s complement How does it work? Pros Add/sub are simple Easy to implement in hardware Subtraction only requires one adder Single zero Cons Asymmetric range Logical negation arithmetic negation x y 03b-24

25 Two s complement: arithmetic Addition As normal binary addition Carry-out is ignored Subtraction A B = A+( B) Take 2 s complement of B to get B Add B to A Overflow when 03b-25

26 Two s complement: arithmetic Example A+B, with A=2, B=1 on 3 bits Example A-B, with A=1, B=2 on 3 bits 03b-26

27 Two s complement: Sign extension The most significant bit carries the sign MSb Example A B, with A=12, B=5 on 8 bits 03b-27

28 Two s complement: overflow Overflow happens when adding two operands of the same sign and the sign of the sum is different Depends on the number of bits Positive + positive = negative: OVERFLOW Negative + negative = positive: OVERFLOW Positive + negative = anything (no overflow) Why? Ex: 10+8 on 5 bits 03b-28

29 Two s complement mapping Number represented Number encoded Range on n bits: b-29

30 Biased notation A way of remapping the numbers Add the bias to the number to get into the biased notation Subtract the bias from the number to get out of biased notation 03b-30

31 Biased binary notation How does it work? Pros Preserves lexical order Single zero Cons Add and sub require one additional operation to adjust the bias x y 03b-31

32 Biased notation: arithmetic Add or subtract normally Then subtract the bias to compensate C = A (+ bias) + B (+ bias) bias Overflow when 03b-32

33 Biased notation: arithmetic Example A+B, with A = 2, B = 1, on 3 bits (bias-3) 03b-33

34 Biased notation mapping Number represented Number encoded Range on n bits: b-34

35 Binary representations summary Binary Representation Range Quirk Negative numbers Overflow Unsigned 0 to 2 n 1 No negatives None Carry-out from MSb Sign-and-Magnitude (2 n-1 1) to 2 n-1 1 Has two zeros First bit is sign; rest is unsigned Carry-out into MSb One s complement (2 n-1 1) to 2 n-1 1 Has two zeros Flip bits of magnitude Sign discrepancy Two s complement 2 n-1 to 2 n-1 1 Range is asymmetric (more negative numbers) Add one to one s complement Sign discrepancy Biased (2 n-1 1) to 2 n-1 Add bias to get into it??? Exceeds largest 03b-35

36 Integer data types Byte: ordered set of 8 bits Nybble: ordered set of 4 bits Word: ordered set of 03b-36

37 7-bit ASCII table 03b-37

38 Recommended Exercises Ex 2.1 to 2.27 (especially important are 2.12, 2.15, 2.20, and 2.25) Ex 2.35, 2.36, 2.37, 2.38 Ex 2.45 to 2.50 Ex 2.53 and b-38