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©2004 Brooks/Cole FIGURES FOR CHAPTER 1 INTRODUCTION NUMBER SYSTEMS AND CONVERSION Click the mouse to move to the next page. Use the ESC key to exit this chapter. This chapter in the book includes: Objectives Study Guide 1.1Digital Systems and Switching Circuits 1.2Number Systems and Conversion 1.3Binary Arithmetic 1.4Representation of Negative Numbers 1.5Binary Codes Problems

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©2004 Brooks/Cole Figure 1-1: Switching circuit

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©2004 Brooks/Cole EXAMPLE: Convert to binary. Conversion (a)

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©2004 Brooks/Cole Conversion (b) EXAMPLE:Convert to binary.

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©2004 Brooks/Cole Conversion (c) EXAMPLE: Convert to binary.

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©2004 Brooks/Cole Conversion (d) EXAMPLE: Convert to base 7.

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©2004 Brooks/Cole Equation (1-1)

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©2004 Brooks/Cole Addition Add and in binary.

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©2004 Brooks/Cole Subtraction (a) The subtraction table for binary numbers is 0 – 0 = 0 0 – 1 = 1and borrow 1 from the next column 1 – 0 = 1 1 – 1 = 0 Borrowing 1 from a column is equivalent to subtracting 1 from that column.

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©2004 Brooks/Cole Subtraction (b) EXAMPLES OF BINARY SUBTRACTION:

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©2004 Brooks/Cole Subtraction (c) A detailed analysis of the borrowing process for this example, indicating first a borrow of 1 from column 1 and then a borrow of 1 from column 2, is as follows:

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©2004 Brooks/Cole Multiplication (a) The multiplication table for binary numbers is 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1

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©2004 Brooks/Cole Multiplication (b) The following example illustrates multiplication of by in binary:

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©2004 Brooks/Cole Multiplication (c) When doing binary multiplication, a common way to avoid carries greater than 1 is to add in the partial products one at a time as illustrated by the following example: 1111multiplicand 1101multiplier 11111st partial product nd partial product (01111)sum of first two partial products rd partial product ( )sum after adding 3rd partial product th partial product final product (sum after adding 4th partial product)

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©2004 Brooks/Cole Binary Division The following example illustrates division of by in binary:

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©2004 Brooks/Cole Table 1-1: Signed Binary Integers (word length n = 4)

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©2004 Brooks/Cole 1. Addition of 2 positive numbers, sum < 2 n –1. 2. Addition of 2 positive numbers, sum ≥ 2 n –1 2’s Complement Addition (a)

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©2004 Brooks/Cole 2’s Complement Addition (b) 3. Addition of positive and negative numbers (negative number has greater magnitude). 4. Same as case 3 except positive number has greater magnitude.

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©2004 Brooks/Cole 2’s Complement Addition (c) 5. Addition of two negative numbers, |sum| ≤ 2 n –1. 6. Addition of two negative numbers, | sum | > 2 n –1.

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©2004 Brooks/Cole 1’s Complement Addition (b) 3. Addition of positive and negative numbers (negative number with greater magnitude). 4. Same as case 3 except positive number has greater magnitude.

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©2004 Brooks/Cole 1’s Complement Addition (c) 5. Addition of two negative numbers, | sum | < 2 n –1. 6. Addition of two negative numbers, | sum | ≥ 2 n –1.

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©2004 Brooks/Cole 1. Add –11 and –20 in 1's complement. +11 = = taking the bit-by-bit complement, –11 is represented by and – 20 by ’s Complement Addition (d)

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©2004 Brooks/Cole 2’s Complement Addition (d) 2. Add – 8 and + 19 in 2's complement + 8 = complementing all bits to the left of the first 1, – 8, is represented by

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©2004 Brooks/Cole Table 1–2. Binary Codes for Decimal Digits

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©2004 Brooks/Cole Table 1-3 ASCII code (incomplete)

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