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Chapter 2
Number Systems
Consists of a set of symbols
called digits and a set of relations
such as +, -, x, /.
RADIX OR BASE
The total number of digits is the radix, or base of the system. Each position in the number
is a power of the base. The power starts at 0 and increases by one as we move each
position to the left of the radix point, and decreases by one as we move to the right of the
radix point.
Example: decimal number 2351.2318
---------------------------------------------------2 3 5
1 . 2
3
1
8
---------------------------------------------------10^3 10^2 10^1 10^0
10^-1 10^-2 10^-3 10^-4
powers increase by one powers decrease by one
10^3
10^2
10^1
10^0
X
X
X
X
2 = 2000
3 = 300
5 = 50
1= 1
10^-1
10^-2
10^-3
10^-4
X 2 = .2
X 3 = .03
X 1 = .001
X 8 = .0008
= 2351.2318
BINARY NUMBER SYSTEM
In the binary number system the radix is 2. The BInary digiTS (bits) can take on the
values 0 or 1
--------------------------------------------------1
1 0 1 0 . 1 1
0
1
--------------------------------------------------2^4 2^3 2^2 2^1 2^0
2^4
2^3
2^2
2^1
2^0
X 1 = 16
X1= 8
X0= 0
X1= 2
X0= 0
2^-1 2^-2 2^-3 2^-4
2^-1 X 1 = .5
2^-2 X 1 = .25
2^-3 X 0 = 0
2^-4 X 1 = .0625
= 26.8125
Since each bit can have the values of 0 or 1, with two bit positions we can derive four
(2^2) distinct patterns. In general, with n bits it is possible to have 2^n combinations of
0's and 1's, these combinations can take on the decimal values of 0 through (2^n - 1).
BINARY/DECIMAL
Binary value Decimal value
--------------------------------------2^0
2^1
2^2
2^3
2^4
2^5
2^6
2^7
2^8
2^9
2^10
2^12
2^16
2^20
2^24
2^30
2^32
2^40
1
2
4
8
16
32
64
128
256
512
1024 ----------> 1 K
4096
65536 ----------> 64 K
1,048,576 ----------> 1 Megabyte
16,777,216
1.0737 E9 -------- > 1 Gigabyte 1,000 M
4.095 E9
1.0995 E12 --------- > 1 Terabyte 1,000 G
OCTAL NUMBER SYSTEM
In the octal number system the radix is 8. The digits can take the values of 0,1,2,3,4,5,6
or 7.
----------------------------------------------------4 7 5 .
3
4
----------------------------------------------------8^2 8^1 8^0
8^2 X 4 = 256
8^1 X 7 = 56
8^0 X 5 = 5
8^-1 8^-2
8^-1 X 3 = .375
8^-2 X 4 = .0625
= 317.4375
Using the octal number system, with an n digit number there are 8 different combinations
of digits possible. Which means 8 different numbers can be represented in the range of 0
through (8^n - 1).
HEXIDECIMAL SYSTEM
This number system has a radix of 16, so 16 digits are needed. For this system the
numbers 0 - 9 are used and the letters A,B,C,D,E and F are used to represent the numbers
10 - 15.
--------------------------------------------4 B E 1 . C
3
--------------------------------------------16^3 16^2 16^1 16^0
16^3
16^2
16^1
16^0
16^-1 16^-2
X 4 = 16384
X B(11) = 2816
X E(14) = 224
X1 = 1
16^-1 X C(12) = .75
16^-2 X 3 = .0117188
= 19425.7617188
Since each digit can have 16 different values with n digits there can be 16^n different
numbers in the range 0 through (16^n - 1).
In general, for any number system of radix (base) R, having n digits, there will be R^n
different combinations having the values 0 through (R^n - 1). Also, the maximum value
that a number can have is R^n - 1.
Integers
Binary -> Decimal
(1 0 1 1)2 = ( ( (1 x 2 + 0) x 2 + 1) x 2) + 1
= (1x22 + 0x2 + 1) x 2 + 1
= (1x23 + 0x22 + 1x2) + 1
= 1x23 + 0x22 + 1x21 + 1x20
= 1x8 + 0x4 + 1x2 + 1x1
=8+0+2+1
= 11
Integers
Base -> Decimal
Result <- Most Significant Digit
 Multiply Result by Base and add
next digit to right
 Repeat step 2 until least
significant digit has been added
Example:
(672)8 == (442)10

Integers
Decimal -> Base
Result <- Decimal Number
 Divide Result by Base and save
remainder
 Result <- Quotient
 Repeat step 2 until no more
quotients
Example:
(442)10 == (672)8

Fractions
Decimal -> Base
Result <- Decimal Number
 Multiply Result by Base and save
whole number
 Result <- Fraction
 Repeat step 2 until no fractions or
significance exceeded
Example:
(0.62)10 == (0.4753)8

Fractions
Base -> Decimal
Divisor <- Denominator of LSD
 Treat Fraction as a whole number
(that is, ignore decimal point) and
convert
 Divide the result by the Divisor of
Step 1
Example:
(0.4753)8 == (0.619873)10
