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06_Positional_Fractions

# 06_Positional_Fractions - Number Systems Again(ch 2 CMPE12...

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Unformatted text preview: Number Systems Again (ch 2 +) CMPE12 1 Cyrus Cyrus Bazeghi Positional Fractions Mesopotamians used positional fractions Sqrt(2) = 1.24,51,1060 = 1 x 600 + 24 x 60-1 + 51 x 60-2 + 10 x 60-3 = 1.414222 Most accurate approximation until the Renaissance CMPE12 2 Cyrus Cyrus Bazeghi Positional Fractions Generalized Representation For a number “f” with “n” digits to the left and “m” to the right of the decimal place fn-1 fn-2 … f2 f1 f0 f-1 f-2 f-3 … fm-1 Decimal point Position is the power CMPE12 3 Cyrus Cyrus Bazeghi Fractional Representation • What is 3E.8F16? = 3 x 161 + E x 160 + 8 x 16-1 + F x 16-2 = 48 + 14 + 8/16 + 15/256 • How about 10.1012? = 1 x 21 + 0 x 20 + 1 x 2-1 + 0 x 2-2 + 1 x 2-3 = 2 + 0 + 1/2 + 1/8 CMPE12 4 Cyrus Cyrus Bazeghi Converting Decimal -> Binary fractions • Consider left and right of the decimal point separately. • The stuff to the left can be converted to binary as before. • Use the following table/algorithm to convert the fraction CMPE12 5 Cyrus Cyrus Bazeghi For 0.810 to binary Fraction 0.8 0.6 Fraction x 2 1.6 1.2 Digit left of decimal point 1 most significant (f-1) 1 0.2 0.4 0.8 0.4 0.8 (it must repeat from here!!) 0 0 •Different bases have different repeating fractions. •0.810 = 0.110011001100…2 = 0.11002 •Numbers can repeat in one base and not in another. CMPE12 6 Cyrus Cyrus Bazeghi What is 2.210 in: •Binary •Hex CMPE12 7 Cyrus Cyrus Bazeghi ...
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06_Positional_Fractions - Number Systems Again(ch 2 CMPE12...

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