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COMS 161 Introduction to Computing Title: Numeric Processing Date: October 20, 2004 Lecture Number: 23 1 Announcements • Exam 2 – Monday 10/25/2004 – Covers • LANs • The Internet • HTTP ands HTML Chapter 4 Chapter 17 Chapter 18 • Today’s Material – Chapter 6 2 Review • LAN’s • The Internet • HTML and HTTP 3 Outline • Numeric Processing 4 Digital Number Representations • Integers – Infinite discrete subset of the number line – Represented with a limited range • Decimal numbers (real numbers) – Infinite and continuous – Represented with limited range and limited precision 5 Integer Storage • Integer values can be exactly represented base10 conversion base2 6 Integer Storage • Integer values can be exactly represented base10 conversion base2 1 1 = 20 0000 0001 7 Integer Storage • Integer values can be exactly represented base10 conversion base2 1 1 = 20 0000 0001 2 1 = 21 0000 0010 8 Integer Storage • Integer values can be exactly represented base10 conversion base2 1 1 = 20 0000 0001 2 2 = 21 0000 0010 4 4 = 22 0000 0100 9 Integer Storage • Integer values can be exactly represented base10 conversion base2 1 1 = 20 0000 0001 2 2 = 21 0000 0010 4 4 = 22 0000 0100 8 8 = 23 0000 1000 10 Integer Storage • Integer values can be exactly represented base10 conversion base2 1 1 = 20 0000 0001 2 2 = 21 0000 0010 4 4 = 22 0000 0100 8 8 = 23 0000 1000 9 9 = 8 + 1 = 23+20 0000 1001 11 Integer Storage • Integer values can be exactly represented base10 conversion base2 1 1 = 20 0000 0001 2 2 = 21 0000 0010 4 4 = 22 0000 0100 8 8 = 23 0000 1000 9 9 = 8 + 1 = 23+20 0000 1001 10 10 = 8 + 2 = 23 + 21 0000 1010 12 Integer Storage • Integer values can be exactly represented base10 conversion base2 1 1 = 20 0000 0001 2 2 = 21 0000 0010 4 4 = 22 0000 0100 8 8 = 23 0000 1000 9 9 = 8 + 1 = 23+20 0000 1001 10 10 = 8 + 2 = 23 + 21 0000 1010 27 27 = 16+8+2+1 = 24+23+21+20 0001 1011 most significant bit least significant bit 13 Integer Storage • Integers are typically 32 bits (word size) • Number of unique items that can be represented with 32 bits 232 = 4,294,967,296 • One-half of the symbols – Represent positive numbers – Represent negative numbers – Sign bit distinguishes between + and - numbers 14 Integer Storage • Positive numbers 0, 1, 2, …, 231 - 1 • Negative numbers -231 + 1, -231 + 2, …, -2, -1, 0 • Two representations of zero – Get rid of one of them – Gives us one more number – Add it to the negative numbers 15 Integer Storage • Range of integer numbers -231, -231 + 1, …, -2, -1 0, 1, 2, …, 231 - 1 -2,147,483,648 … 2,147,483,647 • Integer overflow error – Trying to represent an integer that is larger than the most positive allowable integer or more negative than most negative integer – Frequently occurs during math operations 16 Integer Overflow • 3 bits – Can represent 23 = 8 values – { 0, 1, 2, 3, 4, 5, 6, 7 } 410 = 1002 + 310 = 0112 410 = + 510 = 1002 1012 How do I add two binary numbers? 111 710 = 910 = 1 0012 2 carry out overflow 17 Negative Numbers • The range of integer numbers is -231, -231 + 1, …, -2, -1 0, 1, 2, …, 231 - 1 -2,147,483,648 … 2,147,483,647 – How do we represent negative numbers? • Could use one bit as a sign bit, but … – Two’s complement representation solves • The problem of two zeros • Mathematical operations giving incorrect results 18 Two’s Complement Numbers • Two steps in determining a two’s complement representation of a number – Positive numbers are the same as the positive sign-magnitude representation – Negative numbers • Invert the bits of the unsigned quantity • Add 1 to the result 19 Two’s Complement Numbers – Negative numbers • Invert the bits of the unsigned quantity • Add 1 to the result Decimal number -4 -7 Binary magnitude Bit inverse Add one Two’s complement representation 0100 1011 1011 + 1 1100 1000 1011 + 1 1001 0111 11 1011 1 11 00 20