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

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Benjamin Garrett
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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 Each decimal digit individually converted to binary Requires 4 bits per digit 8-bit location hold 2 BCD digits 00 to BCD Hexa: 4 bits can hold 16 values, 0 to F A to F not used in BCD 1

2 Ranges for Data Formats In General (binary) No. of bits Etc , ,777,215 BCD ASCII No. of bits Min Max n 0 2 n -1 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!) Eg,-5 E.g., 10 =? 2 Two common schemes: 1) sign-magnitude 2) two s complement Extra bit on left to represent sign 0 = positive value 1 = negative value E.g., 6-bit sign-magnitude representation of +5 and 5: +5: :

3 Ranges (revisited) Unsigned Sign-magnitude No. of bits Min Max Min Max Etc. In General (revisited) Unsigned Sign-magnitude No. of bits Min Max Min Max n 0 2 n - 1 -(2 n-1-1) 2 n-1-1 Difficulties with Sign-Magnitude Two representations of zero 0: : Arithmetic is awkward! One s complement Principle: Invert bits (0 1 and 1 0) 6: Range -6:

4 Add / Sub in 1 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 ti (no special rules!) To represent a negative number in 2 s complement notation 1. 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 1 s to 0 s and vice versa) 4. Add 1 Learn! 4

5 Two s complement representation Two s Complement Example Represent 5 in binary using 2 s complement notation 1. 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 Add Sign Bit In 2 s complement notation, the MSB is the sign bit (as with sign-magnitude notation) 0 = positive value 1 = negative value +5: : ? (previous slide) Complementary Notation Conversions between positive and negative numbers are easy For binary (base 2) 2 s C s C 5

6 Example Detail for s C -5 2 sc 2s : Positive Vl Value = Flip : (One s complement) Add 1: Detail for 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 = 1 Zero or positive, sign bit = 0 kc 6

7 Ranges (revisited) No. of Unsigned Sign-magnitude 2 s complement bits Min Max Min Max Min Max Etc. No. of bits In General (revisited) Unsigned Sign-magnitude 2 s complement Min Max Min Max Min Max n 0 2 n - 1 -(2 n-1-1) 2 n n-1 2 n-1-1 To remember 2 s Complement Addition Easy No special rules Just add What is -5 plus +5? Zero, of course, but let s see Sign-magnitude Two s-complement : : : :

8 2 s Complement Subtraction Easy No special ilrules Just subtract, well actually just add! What is 10 subtract 3? 7, of course, but Let sdoit(we ll use 6-bit values) 10 3 = 10 + (-3) = 7 A B = A + (-B) add 2 s complement of B +3: s C: : 1-3: What is 10 subtract -3? 13, of course, but Let s do it (we ll use 6-bit values) (-(-3)) = 3 Overflows and Carries 10 (-3) = 10 + (-(-3)) = 13-3: s C: : 1 +3:

Positional notation Ch Conversions between Decimal and Binary. /continued. Binary to Decimal

Positional notation Ch Conversions between Decimal and Binary. /continued. Binary to Decimal Positional notation Ch.. /continued Conversions between Decimal and Binary Binary to Decimal - use the definition of a number in a positional number system with base - evaluate the definition formula using