Control Data 3100 Reference Manual page 168

Appendix B
SUPPLEMENTARY ARITHMETIC INFORMATION
NUM BER SYSTEMS
Any number system may be defined by two characteristics, the radix or base and the
modulus. The radix or base is the number of unique symbols used in the system. The
decimal system has ten symbols, 0 through 9. Modulus is the number of unique quanti-
ties or magnitudes a given system can distinguish. For example, an adding machine
with ten digits, or counting wheels, would have a modulus of 1010-1. The decimal system
has no modulus because an infinite number of digits can be written, but the adding
machine has a modulus because the highest number which can be expressed is 9,999,999,999.
Most number systems are positional; that is, the relative position of a symbol determines
its magnitude. In the decimal system, a 5 in the units column represents a different quan-
tity than a 5 in the tens column. Quantities equal to or greater than 1 may be represented
by using the 10 symbols as coefficients of ascending powers of the base 10. The number 98410 is:
9 x 1 0 2 = 9 x 1 00 900
+8 x 10 1
=
8 x 10 = 80
+4 x 10
0
= 4 x 4
98410
Quantities less than 1 may be represented by using the 10 symbols as coefficients of
ascending negative powers of the base 10. The number 0.59310 may be represented as:
5 x 10-
1
= 5 x .1 =.5
+9 x 10-
2
= 9 x .01 = .09
+3 x 10-
3
= 3 x .001 = .003
0.59310
BINARY NUMBER SYSTEM
Computers operate faster and more efficiently by using the binary number system. There
are only two symbols, 0 and 1; the base
=
2. The following shows the positional value:
24
16
22
4
The binary number 0 1 1 0 1 0 represents:
'0 x 2
5
= 0 x 32 = 0
+ 1
X
24 = 1 x 16 = 16
+1x2 3 =1x8 8
+Ox2
2
= Ox4 = 0
+1x21=1x2 = 2
+Ox2° = Ox 1 =_0 __
2610
B-1
Binary point