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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 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
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
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
Numbers
DIGITAL DESIGN 101, University of California
Floating-Point Review

General form
+/- mantissa × (radix)exponent

32-bit standard
0
1
Sign
9
Excess-127
characteristic
31
Normalized
Fraction
Implied binary point
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)
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

0
1
Sign
9
Excess-127
characteristic
31
Normalized
Fraction
Implied binary point

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)
Numbers
DIGITAL DESIGN 101, University of California