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3. Representing Integer Data Chapt. 4 ITEC 1011 Introduction to Information Technologies Ranges for Data Formats No. of bits Binary BCD ASCII 1 0–1 2 0–3 3 0–7 4 0 – 15 0–9 5 0 – 31 6 0 – 63 7 0 – 127 8 0 – 255 0 – 99 0–9 9 0 – 511 16 0 - 65,535 0 – 9999 0 – 99 24 0 – 16,777,215 0 – 999999 0 – 999 Etc. ITEC 1011 Introduction to Information Technologies In General (binary) Binary No. of bits Min Max n 0 2n - 1 ITEC 1011 Introduction to Information Technologies Signed Integers • Previous examples were for “unsigned integers” (positive values only!) • Must also have a mechanism to represent “signed integers” (positive and negative values!) • E.g., -510 = ?2 • Two common schemes: sign-magnitude and twos complement ITEC 1011 Introduction to Information Technologies Sign-Magnitude • 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: 0 0 0 1 0 1 -5: 1 0 0 1 0 1 +ve 5 -ve 5 ITEC 1011 Introduction to Information Technologies Ranges (revisited) Binary Unsigned Sign-magnitude No. of bits Min Max Min Max 1 0 1 2 0 3 -1 1 3 0 7 -3 3 4 0 15 -7 7 5 0 31 -15 15 6 0 63 -31 31 Etc. ITEC 1011 Introduction to Information Technologies In General (revisited) Binary Unsigned Sign-magnitude No. of bits Min Max Min Max n n-1 n-1 n 0 2 - 1 -(2 - 1) 2 -1 ITEC 1011 Introduction to Information Technologies Difficulties with Sign-Magnitude • Two representations of zero • Using 6-bit sign-magnitude… • 0: 000000 • 0: 100000 • Arithmetic is awkward! pp. 95-96 ITEC 1011 Introduction to Information Technologies Complementary Representations • 9’s complement • 10’s complement • 1’s complement • Read sections 4.4 and 4.5 (pp. 102-113) ITEC 1011 Introduction to Information Technologies Exercises – Complementary Notations • What is the 3-digit 10’s complement of 247? • Answer: • What is the 3-digit 10’s complement of 17? • Answer: • 777 is a 10’s complement representation of what decimal value? • Answer: Skip answer Answer ITEC 1011 Introduction to Information Technologies Exercises – Complementary Notations Answer • What is the 3-digit 10’s complement of 247? • Answer: 753 • What is the 3-digit 10’s complement of 17? • Answer: 983 • 777 is a 10’s complement representation of what decimal value? • Answer: 223 See p. 110-111 ITEC 1011 Introduction to Information Technologies Twos 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… 1. Decide upon the number of bits (n) 2. Find the binary representation of the +ve value in n-bits 3. Flip all the bits (change 1’s to 0’s and vice versa) 4. Add 1 ITEC 1011 Introduction to Information Technologies Twos Complement Example • Represent -5 in binary using 2’s complement notation 1. Decide on the number of bits 6 (for example) 2. Find the binary representation of the +ve value in 6 bits 000101 +5 3. Flip all the bits 111010 4. Add 1 111010 + 1 -5 111011 ITEC 1011 Introduction to Information Technologies 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: 0 0 0 1 0 1 -5: 1 1 1 0 1 1 +ve 5 -ve ? (previous slide) ITEC 1011 Introduction to Information Technologies “Complementary” Notation • Conversions between positive and negative numbers are easy • For binary (base 2)… 2’s C +ve -ve 2’s C ITEC 1011 Introduction to Information Technologies Example +5 0 0 0 1 0 1 1 1 1 0 1 0 2’s C + 1 -5 1 1 1 0 1 1 0 0 0 1 0 0 2’s C + 1 +5 0 0 0 1 0 1 ITEC 1011 Introduction to Information Technologies Exercise – 2’s C conversions • What is -20 expressed as an 8-bit binary number in 2’s complement notation? • Answer: • 1100011 is a 7-bit binary number in 2’s complement notation. What is the decimal value? • Answer: Skip answer Answer ITEC 1011 Introduction to Information Technologies Exercise – 2’s C conversions Answer • What is -20 expressed as an 8-bit binary number in 2’s complement notation? • Answer: 1101100 • 1100011 is a 7-bit binary number in 2’s complement notation. What is the decimal value? • Answer: -29 ITEC 1011 Introduction to Information Technologies Range for 2’s Complement • For example, 6-bit 2’s complement notation 100000 100001 111111 000000 000001 011111 -32 -31 ... -1 0 1 ... 31 Negative, sign bit = 1 Zero or positive, sign bit = 0 ITEC 1011 Introduction to Information Technologies Ranges (revisited) Binary No. of Unsigned Sign-magnitude 2’s complement bits Min Max Min Max Min Max 1 0 1 2 0 3 -1 1 -2 1 3 0 7 -3 3 -4 3 4 0 15 -7 7 -8 7 5 0 31 -15 15 -16 15 6 0 63 -31 31 -32 31 Etc. ITEC 1011 Introduction to Information Technologies In General (revisited) Binary No. of Unsigned Sign-magnitude 2’s complement bits Min Max Min Max Min Max n n-1 n-1 n-1 n-1 n 0 2 - 1 -(2 - 1) 2 -1 -2 2 -1 ITEC 1011 Introduction to Information Technologies 2’s Complement Addition • Easy • No special rules • Just add ITEC 1011 Introduction to Information Technologies What is -5 plus +5? • Zero, of course, but let’s see Sign-magnitude Twos-complement 11 1 1 1 1 11 -5: 10000101 -5: 11111011 +5: +00000101 +5: +00000101 10001010 00000000 ITEC 1011 Introduction to Information Technologies 2’s Complement Subtraction • Easy • No special rules • Just subtract, well … actually … just add! A – B = A + (-B) add 2’s complement of B ITEC 1011 Introduction to Information Technologies What is 10 subtract 3? • 7, of course, but… • Let’s do it (we’ll use 6-bit values) 10 – 3 = 10 + (-3) = 7 +3: 000011 1s C: 111100 +1: 1 001010 -3: 111101 +111101 000111 ITEC 1011 Introduction to Information Technologies What is 10 subtract -3? (-(-3)) = 3 • 13, of course, but… • Let’s do it (we’ll use 6-bit values) 10 – (-3) = 10 + (-(-3)) = 13 -3: 111101 1s C: 000010 +1: 1 001010 +3: 000011 +000011 001101 ITEC 1011 Introduction to Information Technologies Thank You ITEC 1011 Introduction to Information Technologies

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posted: | 4/9/2010 |

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