Number
Systems
Positional number systems
Whole numbers
Fractions
Accuracy
Copyright  1993-95, Thomas P. Sturm
Fundamentals of Computer Science
1
Number Systems
Concept of a Positional Number System
What does 4,752 (base 10) represent (in base 10)?
4 x 1000
+ 7 x 100
+ 5 x 10
+ 2x1
4 x 103
7 x 102
5 x 101
2 x 100
4000
700
50
2
So what does 1011 (base 2) represent (in base 10)?
1x8
0x4
1x2
1x1
1 x 23
0 x 22
1 x 21
1 x 20
8
0
2
1
While this makes a nice definition, it is far too much work for
"practical" use - binary numbers get to be long
Fundamentals of Computer Science
2
Number Systems
Counting Revisited
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Fundamentals of Computer Science
Binary
0
1
10
11
100
101
110
111
1000
1001
1010
1011
1100
1101
1110
1111
10000
10001
10010
10011
10100
10101
10110
10111
11000
11001
11010
11011
11100
11101
11110
11111
100000
3
Octal
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
20
21
22
23
24
25
26
27
30
31
32
33
34
35
36
37
40
Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
10
11
12
13
14
15
16
17
18
19
1A
1B
1C
1D
1E
1F
20
Number Systems
Faster Conversion of Binary to Decimal
Dibble-Dabble, Horner's Method, Algorithm 1.1
1. Start at the left, start with 0
2. Add in leading digit
3. If there's another digit, multiply total by 2, add in next digit,
repeat this step until you reach the radix (binary) point.
Ex: 11010 (binary) converted to base 10 is:
0
+1
1
x2
2
+1
3
x2
6
+0
6
x2
12
+1
13
x2
26
+0
26
Fundamentals of Computer Science
4
Number Systems
Even Faster in Your Head
Ex: 11010010110 (binary) converted to base 10 is:
(in your head 1,3,6,13,26,52,105,210,421,843,1686)
Which has just got to be a lot quicker than
1024
+ 512
+ 128
+ 16
+4
+2
(not even counting the time it takes to generate the powers of
2)
Fundamentals of Computer Science
5
Number Systems
Fast Decimal to Binary Conversion
(Dibble-Dabble, Horner's Method, Algorithm 1.2)
Since all even numbers end in 0 (binary), and all odd numbers end
in 1 (binary), and since division by 2 yields a 0 remainder
for even numbers and a 1 remainder for odd numbers, the
remainder after division by 2 gives you the LSB (least
significant bit) of the corresponding binary number.
Just repeat this process until you get a 0 quotient
Ex: Convert 75 (decimal) to binary
2 | 75
2 | 37
+
1
2 | 18
+
1
2|9
+
0
2|4
+
1
2|2
+
0
2|1
+
0
0
+
1
Now, the trick to remember, the LSB is ON TOP, so read the
answer from the bottom up:
75 (decimal) = 1001011 (binary)
Fundamentals of Computer Science
6
Number Systems
Converting Other Bases
Ex: Convert 17 (base 9) to base 6.
Solution: First convert 17 (base 9) to base 10, using the first fast
conversion algorithm:
1
x9
9
+7
16
Next, convert 16 (base 10) to base 6, using the second fast
conversion algorithm:
6 | 16
6|2
+ 4
0
+ 2
Reading up, 17 (base 9) equals 24 (base 6), which both equal 16
(base 10)
Fundamentals of Computer Science
7
Number Systems
Base 8 has a Special Relationship to Base 2
Conversion between octal and binary is particularly fast and trivial,
if you are careful. Since 23 = 8, every 3 binary digits
converts to 1 octal digit, but you have to start in the correct
place - the radix point
The conversion is
000 0
001 1
010 2
011 3
100 4
101 5
110 6
111 7
Ex. Convert 10110011001111101000 to octal
Solution: starting at the RIGHT, block off into groups of 3
10|110|011|001|111|101|000
Now convert group-by-group to
2631750 (octal)
Conversion from octal to binary is even easier.
Ex. Convert 307125 to binary
Solution: 011|000|111|001|010|101, or, removing the dividing bars
and the (useless for now) preceding 0,
11000111001010101
Fundamentals of Computer Science
8
Number Systems
Hexadecimal to Binary
Hex to binary is just as quick and trivial, except, since 16 = 24, the
binary numbers are grouped by 4's, and a longer
conversion table needs to be "remembered"
Binary
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Hexadecimal
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Ex: Convert 1011010100001110101101010110011 to hex
Solution: 101|1010|1000|0111|0101|1010|1011|0011 converts to
5A875AB3 (hex)
Ex: Convert ACD01BF to binary
Solution: 1010110011010000000110111111
Fundamentals of Computer Science
9
Number Systems
Binary Numbers in Real Machines
Real computers have a fixed length space for the storage of
positive whole numbers in binary.
First question: How do we store numbers that are "too short to
fit"?
Solution: Add preceding zeroes.
Ex: Show how 5 will be represented in 8 bits
Solution: 101 is only 3 bits long, so add 5 preceding zeroes to
obtain 00000101
Ex: Show how 5 will be represented in 32 bits
Solution: 00000000000000000000000000000101
Second question: What is the largest number you can store this
way in a fixed number of bits?
Solution: In binary it is all 1's, which is 1 less than 2 to the power
of the number of bits!
Ex: What's the largest number that can be stored in 8 bits
Solution: 11111111 = 100000000 - 1 = 255
Third question: How many different numbers can be stored in a
fixed number of bits?
Solution: 2 to the number of bits (all the combinations)
Ex: How many values can be stored in 8 bits
Solution: 256
Fundamentals of Computer Science
10
Number Systems
Estimating Large Powers of 2
Most real computers frequently deal with larger collections of bits
than 8. The VAX routinely handles 32 bits at a time (it
prefers it to anything shorter), and can handle collections
of 64, 128, and, on rare occasion, numbers than occupy
256 bits in length. How many combinations of these
numbers are there?
Repeatedly multiplying by 2 to get the answer will take a long time
- and is a waste of time.
Note: 210 = 1024 = 1K  1,000
Note: 220 = 210 x 210 = 1024 x 1024 = 1 Meg  1,000,000
Note: 230 = 210 x 210 x 210 = 1 Gig  1,000,000,000
250 would have 5 commas
2120 would have 12 commas
Now, all you need to do is get the LEADING DIGITS of the
answer by looking at the power of two of the TRAILING
DIGIT of the exponent
Ex: Estimate 236
Solution: 64,000,000,000
Ex: Estimate 245
Solution: 32,000,000,000,000
Ex: Estimate 2123
Solution: 8,000,000,000,000,000,000,000,000,000,000,000,000
Fundamentals of Computer Science
11
Number Systems
Positional Number System for Values Less than 1
What does .475 (base 10) represent (in base 10)?
4 x 10-1
4 x .1
.4
7 x 10-2
7 x .01
.07
-3
5 x 10
5 x .001
.005
So what does .1101 (base 2) represent (in base 10)?
1 x 2-1
1 x .5
.5
1 x 2-2
1 x .25
.25
0 x 2-3
0 x .125
.0
1 x 2-4
1 x .0625
.0625
This is a totally unreasonable way of doing this conversion - and
there is a way just as fast for fractionals as there is for
whole numbers
Fundamentals of Computer Science
12
Number Systems
Faster Conversion of Binary Fractions to Decimal
(Dibble-Dabble, Horner's Method, Extension of Algorithms 1.1
and 1.2)
1.
2.
3.
4.
Start at the right, start with 0
Add in the digit
Divide by 2
If there are more digits to the left, (but to the right of the binary
point), go back to step 2
Ex: .1101 (binary) converted to base 10 is:
+
2|
+
2|
+
2|
+
2|
0
1
1
0.5
0
0.5
0.25
1
1.25
0.625
1
1.625
0.8125
Fundamentals of Computer Science
13
Number Systems
Fast Decimal Fraction to Binary Conversion
(Horner's Method, Dibble-Dabble, Extension of Algorithms 1.1
and 1.2)
Use the "inverse" of the method used for integers
Note:
- if a number is between 1/2 and 1, then the first binary digit
after the binary point is a 1
- otherwise , if the number is less than 1/2, the first binary
digit is a 0.
Note:
- if we multiply the decimal number by 2, the whole number
we get is the first binary digit after the binary point.]
Ex: Convert .40625 (decimal) to binary
.40625
x2
0.8125
x2
1.625
x2
1.25
x2
0.5
x2
1.0
Now, the trick to remember is that the MSB is ON TOP, so read
the answer from the top down:
.40625(decimal) = .01101 (binary)
Fundamentals of Computer Science
14
Number Systems
Accuracy between Binary and Decimal
Note that 0.1 (decimal) does not come out evenly when converted
to binary
.1
x2
0.2
x2
0.4
x2
0.8
x2
1.6
x2
1.2
x2
0.4
x2
0.8
x2
1.6
x2
1.2
x2
0.4
...
.1 (decimal) = .0001100110011001100110011... (binary)
Also, .2, .3, .4, .6, .7, .8, and .9 are not exact
This poses a problem when trying to represent decimal
fractions in a binary computer - always need to worry
about accuracy
Fundamentals of Computer Science
15
Number Systems
Accuracy Conversion
How many binary digits are necessary to represent a given decimal
accuracy?
The proper relationship to examine is
n
10  2
m
If we know n, solve for m
n
m
log 10  log 2
10
10
n log
10
10  m log
n  m log
10
10
2
2
n .3m
m 
n
.3
So, to obtain 8 decimal place accuracy, we need 8/.3 = 26.7 bits of
binary accuracy (in practice, we need at least 27 bits).
Remember quick power of 2 estimation:
3 decimal digits = 10 binary digits
So, going the other way, 60 binary digits will yield 20 decimal
place accuracy.
Fundamentals of Computer Science
16
Number Systems