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Principles Of Digital Design Discussion: Numbers Binary to Decimal Conversion Decimal to Binary Conversion Floating-Point Conversion Positional Number System Each number is represented by a string of digits, in which the position of each

SLIDE 1

Principles Of

Digital Design

Discussion: Numbers

Binary to Decimal Conversion Decimal to Binary Conversion Floating-Point Conversion

SLIDE 2

Numbers DIGITAL DESIGN 101, University of California

Positional Number System

 Each number is represented by a string of digits, in

which the position of each digit has an associated weight

1234.5610 = 1 ∙ 103 + 2 ∙ 102 + 3 ∙ 101 + 4 ∙ 100 + 5 ∙ 10-1 + 6 ∙ 10-2

1101.112 = 1 ∙ 23 + 1 ∙ 22 + 0 ∙ 21 + 1 ∙ 20 + 1 ∙ 2-1 + 1 ∙ 2-2

SLIDE 3

Numbers DIGITAL DESIGN 101, University of California

Binary to Decimal Conversion

 Multiply and Add

1101.112 = 1 ∙ 23 + 1 ∙ 22 + 0 ∙ 21 + 1 ∙ 20 + 1 ∙ 2-1 + 1 ∙ 2-2

= 8 + 4 + 0 + 1 + ½ + ¼ = 13.75

SLIDE 4

Numbers DIGITAL DESIGN 101, University of California

Decimal to Binary Conversion

 Divide/multiply by 2 and concatenate remainders

13.7510 = 1101.112 = (1 ∙ 23 + 1 ∙ 22 + 0 ∙ 21 + 1 ∙ 20 ) + (1 ∙ 2-1 + 1 ∙ 2-2) = (1 ∙ 23 + 1 ∙ 22 + 0 ∙ 21 + 1 ∙ 20 ) + (1 ∙ 2-1 + 1 ∙ 2-2) 13.7510 : 2 = 6 + 1

6.75 : 2 = 3 + 0

3.75 : 2 = 1 + 1 1.75 : 2 = 0 + 1

0.75 x 2 = 1 + 0.5

0.5 x 2 = 1 + 0.0

SLIDE 5

Numbers DIGITAL DESIGN 101, University of California

Floating-Point Review

 General form

+/- mantissa × (radix)exponent

 32-bit standard

Sign: 0 for + and 1 for – Exponent = characteristic – bias

 where bias = (radixs/2 ) – 1  therefore, bias is (27-1=127) for 32-bit floating point binary number

Mantissa = 1.(normalized fraction)

Sign Excess-127 characteristic Normalized Fraction

1 9 31 Implied binary point

SLIDE 6

Numbers DIGITAL DESIGN 101, University of California

Floating-Point Binary to Decimal

 Problem: Convert a 32-bit floating-point number to

decimal

 1 01111100 10110000000000000000000  Procedure:

1. Determine sign, mantissa and exponent
Sign = Negative
Exponent = characteristic – bias = 011111002 -127 = 124 – 127 = -3
Mantissa = 1.10112 = 1.687510
2. Result = mantissa × (radix)exponent = -1.10112 ×2-3 (-1.6875 ×2-3)

Sign Excess-127 characteristic Normalized Fraction

0 1 9 31 Implied binary point