Prerequisite Material from 252 (Starts Here)
Number Systems
Decimal
Here's the decimal number system as an example:
digits (or symbols) allowed: 0-9
base (or radix): 10
the order of the digits is significant
345 is really
3 x 100 + 4 x 10 + 5 x 1
3 x 10^2 + 4 x 10^1 + 5 x 10^0
3 is the most significant symbol (it carries the most weight)
5 is the least significant symbol (it carries the least weight)
Binary
Here's a binary number system:
digits (symbols) allowed: 0, 1
base (radix): 2
each binary digit is called a BIT
the order of the digits is significant
numbering of the digits
msb lsb
n-1 0
where n is the number of digits in the number
msb stands for most significant bit
lsb stands for least significant bit
1001 (base 2) is really
1 x 2^3 + 0 x 2^2 + 0 x 2^1 + 1 x 2^0
9 (base 10)
11000 (base 2) is really
1 x 2^4 + 1 x 2^3 + 0 x 2^2 + 0 x 2^1 + 0 x 2^0
24 (base 10)
Of interest: most assembly languages have no way to represent
binary values!
Octal
Here's an octal number system:
digits (symbols) allowed: 0-7
base (radix): 8
the order of the digits is significant
345 (base 8) is really
3 x 8^2 + 4 x 8^1 + 5 x 8^0
192 + 32 + 5
229 (base 10)
1001 (base 8) is really
1 x 8^3 + 0 x 8^2 + 0 x 8^1 + 1 x 8^0
512 + 0 + 0 + 1
513 (base 10)
Hexadecimal
here's a hexadecimal number system:
digits (symbols) allowed: 0-9, a-f
base (radix): 16
the order of the digits is significant
hex decimal binary
0 0 0000
1 1 0001
.
.
.
9 9 1001
a 10 1010
b 11 1011
c 12 1100
d 13 1101
e 14 1110
f 15 1111
a3 (base 16) is really
a x 16^1 + 3 x 16^0
160 + 3
163 (base 10)
A common syntax used to represent hexadecimal values (in code)
is to place the symbols "0x" as a prefix to the value.
MIPS assembly languge does this.
Example: 0x8d is the hexadecimal value 8d (10001101 binary)
A second common syntax is to place a suffix of 'h' onto a value,
indicating that it is hexadecimal.
Example: 8dh (same example as just given)
Note that h is not a symbol used in hexadecimal, so it can indicate
the representation used. The Intel architectures do this in their
assembly languages. This representation is actually more time
consuming (meaning the execution time of the code) to interpret,
since the entire number must be read before it can be decided
what number system is being used.
In General
Given all these examples, here's a set of formulas for the
general case.
Given an n-digit number (in weighted positional notation):
S S . . . S S S
n-1 n-2 2 1 0
the subscript gives us a numbering of the digits
given a base b, this is the decimal value
the summation (from i=0 to i=n-1) of S * b^i
i
Transformations Between Bases
any base --> decimal
just use the definition (summation) given above.
134 (base 5)
1 x 5^2 + 3 x 5^1 + 4 x 5^0
25 + 15 + 4
44 (base 10)
decimal --> another base
one algorithm that works:
Divide decimal value by the base until the quotient is 0.
The remainders give the digits of the value.
Note: this algorithm works for decimal to ANY base. Just
change the base.
examples:
36 (base 10) to base 2 (binary)
36/2 = 18 r=0 <-- lsb
18/2 = 9 r=0
9/2 = 4 r=1
4/2 = 2 r=0
2/2 = 1 r=0
1/2 = 0 r=1 <-- msb
36 (base 10) == 100100 (base 2)
14 (base 10) to base 2 (binary)
14/2 = 7 r=0 <-- lsb
7/2 = 3 r=1
3/2 = 1 r=1
1/2 = 0 r=1 <-- msb
14 (base 10) == 1110 (base 2)
38 (base 10) to base 3
38/3 = 12 r=2 <-- ls digit
12/3 = 4 r=0
4/3 = 1 r=1
1/3 = 0 r=1 <-- ms digit
38 (base 10) == 1102 (base 3)
100 (base 10) to base 5
100/5 = 20 r=0
20/5 = 4 r=0
4/5 = 0 r=4
100 (base 10) = 400 (base 5)
binary --> octal
1. group into 3's starting at least significant symbol
(if the number of bits is not evenly divisible by 3, then
add 0's at the most significant end)
2. write 1 octal digit for each group
example:
100 010 111 (binary)
4 2 7 (octal)
10 101 110 (binary)
2 5 6 (octal)
binary --> hex
(just like binary to octal!)
1. group into 4's starting at least significant symbol
(if the number of bits is not evenly divisible by 4, then
add 0's at the most significant end)
2. write 1 hex digit for each group
example:
1001 1110 0111 0000
9 e 7 0
1 1111 1010 0011
1 f a 3
hex --> binary
(trivial!) just write down the 4 bit binary code for
each hexadecimal digit
example:
3 9 c 8
0011 1001 1100 1000
octal --> binary
(just like hex to binary!)
(trivial!) just write down the 8 bit binary code for
each octal digit
example:
5 0 1
101 000 001
hex --> octal
do it in 2 steps, 1. hex --> binary
2. binary --> octal
Where the bases are powers of a common value, this transformation
is easy (like binary, base 4, octal, hexadecimal)
Examples:
base 3 to base 9
2100122 (base 3)
One base 9 digit is substituted for each 2 base 3 digits.
Why 2? Answer: 3^2=9
base
3 9
-------
00 0
01 1
02 2
10 3
11 4
12 5
20 6
21 7
22 8
2 10 01 22 (base 3)
2 3 1 8 (base 9)
On Representing Nonintegers
what range of values is needed for calculations
very large: Avogadro's number 6.022 x 10 ^ 23 atoms/mole
mass of the earth 5.98 x 10 ^ 24 kilograms
speed of light 3.0 x 10 ^ 8 meters/sec
very small: charge on an electron -1.60 x 10 ^ (-19)
scientific notation
a way of representing rational numbers using integers
(used commonly to represent nonintegers in computers)
exponent
number = mantissa x base
mantissa == fraction == significand
base == radix
point is really called a radix point, for a number with
a decimal base, it is called a decimal point.
all the constants given above are in scientific notation
normalization
to keep a unique form for every representable noninteger, they
are kept in NORMALIZED form. A normalized number will follow the
following rule:
1 <= mantissa < base
In this form, the radix point is always placed one place to
the right of the first significant (non-zero) symbol (as above).
On Precision, Accuracy, and Significant Digits
These terms are often used incorrectly or ignored. They are
important!
A measurement (in a scientific experiment) implies a certain
amount of error, depending on equipment used. Significant
digits tell about that error.
For example, a number given as
3.6 really implies that this number is in the range of
3.6 +- .05, which is 3.55 to 3.65
This is 2 significant digits.
3.60 really implies that this number is in the range of
3.6 +- .005, which is 3.595 to 3.605
This is 3 significant digits.
So, the number of significant digits given in a number tells
about how accurately the number is known. The larger the number
of significant digits, the better the accuracy.
Computers (or calculators, a more familiar machine) have a fixed
precision. No matter what accuracy of a number is known, they
give lots of digits in a number. They ignore how many significant
digits are involved.
For example, if you do the operation 1.2 x 2.2. given that
each number has 2 significant digits, a correct answer is
1.2
x 2.2
-----
24
+ 24
-----
264 --> 2.64 --> 2.6 or 2.6 +- .05
a calculator will most likely give an answer of 2.640000000,
which implies an accuracy much higher than possible. The
result given is just the highest precision that the calculator
has. It has no knowledge of accuracy -- only precision.
Binary Fractions
f f . . . f f f . f f f . . .
n-1 n-2 2 1 0 -1 -2 -3
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binary point
The decimal value is calculated in the same way as for
non-fractional numbers, the exponents are now negative.
example:
101.001 (binary)
1 x 2^2 + 1 x 2^0 + 1 x 2^-3
4 + 1 + 1/8
5 1/8 = 5.125 (decimal)
2^-1 = .5
2^-2 = .25
2^-3 = .125
2^-4 = .0625 etc.
converting decimal to 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 algorithm to convert the fraction:
fraction fraction x 2 digit left of point
.8 1.6 1 <-- most significant (f )
.6 1.2 1 -1
.2 0.4 0
.4 0.8 0
.8 (it must repeat from here!)
----
.8 is .1100
Non-Binary Fractions
same as with binary, only the base changes!
f f . . . f f f . f f f . . .
n-1 n-2 2 1 0 -1 -2 -3
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radix point
The decimal value is calculated in the same way as for
non-fractional numbers, the exponents are now negative.
examples:
101.001 (octal) to decimal
1 x 8^2 + 1 x 8^0 + 1 x 8^-3
64 + 1 + 1/512
65 1/512 = 65.0019 (approx)
13.a6 (hexadecimal) to decimal
1 x 16^1 + 3 x 16^0 + a x 16^-1 + 6 x 16^-2
16 + 3 + 10/16 + 6/256
19 166/256 = 19.64 (approx)
EXAMPLE: give 102.3 (base 5) in base 3
102 (base 5) to decimal:
1 * 5^2 + 0 * 5^1 + 2 * 5^0
25 + 0 + 2
27 (base 10) = 102 (base 5)
.3 (base 5) to decimal:
.3 * 5^(-1)
3/5, or .6 (decimal)
.6 (base 10) = .3 (base 5)
So, 102.3 (base 5) is 27.6 (decimal)
27 (decimal) to base 3
27/3 = 9 r=0 <-- ls digit
9/3 = 3 0
3/3 = 1 0
1/3 = 0 1 27 (base 10) = 1000 (base 3)
.6 x 3 = 1.8 1 (ms fractional digit)
.8 x 3 = 2.4 2
.4 x 3 = 1.2 1
.2 x 3 = 0.6 0
.6 x 3 this repeats the 4 digits
____
.6 (base 10) = .1210
A bar over the top of the digits that repeat is a commonly
used notation.
____
102.3 (base 5) = 1000.1210 (base 3)
Prerequisite Material from 252 (Ends Here)
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Copyright © Karen Miller, 2006
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