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Lecture3 - 1 LECTURE 1.1 Overview(Covers pages 13 27 Binary...

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1 LECTURE 1.1 Dr. Vyas Overview: (Covers pages 13 – 27) Binary Integer and Fraction representation Definition of Error and significant digits Examples/programs Machine Computation (Appendix I, left to students as self study material) I. Base Numbers 1. The arithmetic that we commonly use to communicate the process of scientific computation is in base 10 (decimal form ). The following is implied by base 10 : Let N be any positive integer, then 1 2 2 1 0 k k k N a a a a a a - - = K , where {0,1,2,3,4,5,6,7,8,9} k a .The base 10 is clearly seen via the equivalent expression, 1 1 0 1 1 0 10 10 10 10 k k k k N a a a a - - = × + × + × + × K 2. For example, 3 2 1 0 1257 1000 200 50 7 1 10 2 10 5 10 7 10 = + + + = × + × + × + × 3. The digital computer uses the base 2 (binary form ) system to compute the required computations. However, our interaction with the computer is via the decimal form which computer converts into equivalent binary form , computes what is required by the user in the binary form , and returns the computed results to the user after re-converting into the decimal form . Therefore, it is important to refresh the fundamental aspects of the binary calculations and approximations. 4. The following is implied by base 2 : Let N be any positive integer, then 1 2 2 1 0 k k k N b b b b b b - - = K , where {0,1} k b .The base 2 is clearly seen via an equivalent expression, 1 1 0 1 1 0 2 2 2 2 k k k k N b b b b - - = × + × + × + × K Example 1: Convert 1257 to an equivalent binary form The solution is best described in a tabular form, 10 9 8 7 6 5 4 3 2 1 0 10011101001 10011101001 two N b b b b b b b b b b b = = = 1257 2(628) 1 = + 0 1 b = 0628 2(314) 0 = + 1 0 b = 0314 2(157) 0 = + 2 0 b = 0157 2(078) 1 = + 3 1 b = 0078 2(039) 0 = + 4 0 b = 0039 2(019) 1 = + 5 1 b = 0019 2(009) 1 = + 6 1 b = 0009 2(004) 1 = + 7 1 b = 0004 2(002) 0 = + 8 0 b = 0002 2(001) 0 = + 9 0 b = 0001 2(0) 1 = + 10 1 b =

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2 How do we know our answer is correct? The easiest way to check is to convert the answer obtained back to the decimal form. This is shown next. 10 9 8 7 6 5 4 3 2 1 0 10011101001 1 2 0 2 0 2 1 2 1 2 1 2 0 2 1 2 0 2 0 2 1 2 1024 128 64 32 8 1 1257 two N = = × + × + × + × + × + × + × + × + × + × + × = + + + + + = 5. The geometric series appearing on page 17 and associated example problems is a separate topic and is left as the revision type homework. You must have studied the sequences and series in MATH 242. 6. Binary fractions can be expressed in similar way as for the binary integers. The major difference being that the powers associated with the base 2 are negative . 7. 1 2 1 2 1 2 2 2 ... 2 ..., {0,1} 0. ... k k k k R d d d d R d d d - - - = × + × + + × + = Example 2: Convert 1 10 into binary fraction form .
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Lecture3 - 1 LECTURE 1.1 Overview(Covers pages 13 27 Binary...

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