Basic Definition REPRESENTING INTEGER DATA. Unsigned Binary and Binary-Coded Decimal. BCD: Binary-Coded Decimal

Transcription

1 Basic Definition REPRESENTING INTEGER DATA Englander Ch. 4 An integer is a number which has no fractional part. Examples: Unsigned and -Coded Decimal BCD: -Coded Decimal Each decimal digit individually converted to binary Requires 4 bits per digit 8-bit location hold 2 BCD digits to BCD Hexa: 4 bits can hold 6 values, to F A to F not used in BCD

2 Ranges for Data Formats In General (binary) No. of bits BCD ASCII No. of bits Min Max n 2 n , Etc. 6,777, Remember!! Signed Integers Sign-Magnitude Previous examples were for unsigned integers (positive values only!) Must also have a mechanism to represent signed integers (positive and negative values!) E.g., -5 =? 2 Two common schemes: ) sign-magnitude 2) two s complement Extra bit on left to represent sign = positive value = negative value E.g., 6-bit sign-magnitude representation of +5 and 5: +5: + 5-5: - 5 2

3 Ranges (revisited) In General (revisited) No. of bits 2 3 Unsigned 3 7 Sign-magnitude No. of bits Unsigned Sign-magnitude n 2 n - -(2 n- - ) 2 n Etc. Difficulties with Sign-Magnitude Two representations of zero : : Arithmetic is awkward! One s complement Principle: Invert bits ( and ) 6: -6: Range 3

4 Add / Sub in s complement Overflow Overflow sign of result sign both operands Two s Complement Most common scheme of representing negative numbers in computers Affords natural arithmetic (no special rules!) To represent a negative number in 2 s complement notation. Decide upon the number of bits (n) 2. Find the binary representation of the positive value in n-bits 3. Flip all the bits (change s to s and vice versa) 4. Add Learn! 4

5 Two s complement representation Two s Complement Example Represent 5 in binary using 2 s complement notation. Decide on the number of bits, for example: 6 2. Find the binary representation of the positive (5) value in 6 bits Flip all the bits 4. Add + -5 Sign Bit In 2 s complement notation, the MSB is the sign bit (as with sign-magnitude notation) = positive value = negative value +5: -5: + 5 -? (previous slide) Complementary Notation Conversions between positive and negative numbers are easy For binary (base 2) 2 s C s C 5

6 +5 2 s C -5 2 s C Example + + Detail for -2-2 : Positive Value = Flip : (One s complement) Add : + +5 Detail for - 29 Range for 2 s Complement 2 s Complement: Flip : (One s complement) Add One: + Converts to: = - 29 For example, 6-bit 2 s complement notation Negative, sign bit = Zero or positive, sign bit = kc 6

7 Ranges (revisited) In General (revisited) No. of bits Etc. Unsigned Sign-magnitude s complement No. of bits n Unsigned 2 n - Sign-magnitude -(2 n- - ) 2 n- - To remember 2 s complement Min -2 n- Max 2 n- - 2 s Complement Addition Easy No special rules Just add What is -5 plus +5? Zero, of course, but let s see Sign-magnitude -5: +5: + Two s-complement -5: +5: + 7

8 2 s Complement Subtraction Easy No special rules Just subtract, well actually just add! What is subtract 3? 7, of course, but Let s do it (we ll use 6-bit values) 3 = + (-3) = 7 A B = A + (-B) add 2 s complement of B +3: s C: +: -3: + What is subtract -3? 3, of course, but Let s do it (we ll use 6-bit values) (-(-3)) = 3 Overflows and Carries (-3) = + (-(-3)) = 3-3: s C: +: +3: + 8