Factional Values
What is 0.75 in binary?
How could we represent fractions?
• In decimal:
– As fractions : 1/5
How could we represent fractions?
• In decimal:
– As fractions : 1/5
– As decimals : 0.2
hundreds
102
tens
101
ones
100
tenths
10-1
hundredths
10-2
0
2
0
Column Pattern
• What goes to the right of 1’s column?
23
8
22
4
21
2
20
1
Column Pattern
• Negative powers of two:
23
8
-3
2
22
4
=
1
23
21
2
=
20
1
1
8
2-1
0.5
= 0.125
2-2
0.25
2-3
0.125
2-4
0.0625
Number
• Binary decimal:
23
8
22
4
21
2
20
1
2-1
0.5
2-2
0.25
1
0
1
1
2-3
0.125
2-4
0.0625
10.112 = 2 + 0.5 + 0.25 = 2.7510
Number
• Binary decimal:
23
8
22
4
21
2
20
1
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625
0
1
0
0
1
0.10012 = 0.5 + 0.0625 = 0.562510
Number
• Binary decimal:
23
8
22
4
21
2
20
1
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625
0
1
0
1
0
0.10102 = 0.5 + 0.125 = 0.62510
Missing Numbers
• Where is 0.6?
0.10012 = 0.5 + 0.0625 = 0.562510
0.10102 = 0.5 + 0.125 = 0.62510
More Digits
• An attempt with 11 decimal digits:
Binary Decimal
Place
2-1
2-2
2-3
2-4
2-5
2-6
2-7
2-8
2-9
2-10
2-11
Bit
1
0
0
1
1
0
0
1
1
0
1
Value
0.5
0
0
0.0625
0.03125
0
0
0
0.000488
Total
• Can only approximate
0.600098
0.003906 0.001953
Decimals are an Approximation
• Decimal has same issue:
– What is 1/3 ??
– What is 2/7 ??
More Digits
• Given limited digits, must allocate them
– To left of decimal:
Bigger values
25
32
24
16
23
8
22
4
21
2
20
1
2-1
0.5
2-2
0.25
– To right of decimal:
More accurate values
21
2
20
1
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625
2-5
0.03125
2-6
0.015625
Fixed Decimal Problems
• Fixed decimal points waste space:
400,000,000,000,000,000
0.000000000000025
Fixed Decimal Problems
• Fixed decimal points waste space:
400,000,000,000,000,000 vs 4.0 x 1017
0.000000000000025
vs 2.5 x 10-14
• In computers, space is precious
– Computers use a floating decimal point
(Like scientific notation)
Floating Point
Sign
0
Exponent
1
0
Mantissa
0
1
0
• Bits used to represent 3 parts:
– Sign
– Exponent
– Fraction (or Mantissa)
0
0
Sign
Sign
0
Exponent
1
0
Mantissa
0
1
• 0 = positive, 1 = negative
0
0
0
Exponent
Sign
0
Exponent
1
0
Mantissa
0
1
0
0
0
• Binary integer in excess notation
– Gives power of 2 to multiply by
100 = 0
So 20
Binary
000
001
010
011
100
101
110
111
Value
-4
-3
-2
-1
0
1
2
3
Mantissa
Sign
0
Exponent
1
0
Mantissa
0
1
0
0
0
• Fractional Value
– Always a decimal
1000 = 0.5
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625
1
0
0
0
Examples
Sign
Exponent
0
+
1 0 0
0 so 20
+ 20 x 0.5
=
+ 1 x 0.5
=
+ 0.5
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625
1
0
0
0
Mantissa
1
0
0
0.5
0
Examples
Sign
Exponent
1
-
1 1 1
3 so 23
- 23 x 0.5625
=
=
- 8 x 0.5625
-4.5
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625
1
0
0
1
Mantissa
1
0 0
0.5625
1
Examples
Sign
0
+
+ 2-4
2-1
0.5
2-2
0.25
2-3
0.125
2-4
0.0625
0
1
0
0
Exponent
0
0
-4 so
2-4
x 0.25
= + 0.0625 x 0.25
= +0.015625
Mantissa
0
or
1
24
0
1
0
0.25
0
Floating Point Math is HARD
Sign
0
Exponent
1
0
Mantissa
0
1
0
0
• XOR the signs
• Calculate new exponent
– Excess Notation – different math rules!
• Calculate new mantissa
– Normal binary multiplication
0
Choices, choices
• Choices
– More bits to exponent
– More bits to mantissa
– How we represent each
• What is our excess start point?
• Do mantissas have to start with 1?
32 Bit Floating Point
• PC processors are 32 or 64 bit
– Size of data they can work on at one time
• IEEE specifies conventions for floating points:
Remember…
• Decimals are approximate
• 32 bit float gives 6-7 significant decimal digits
• 64 bit float gives 15-16 significant decimal
digits