Java Reference
In-Depth Information
B
Computer Arithmetic
In the chapters of this topic, I have deliberately kept discussion of binary arithmetic to a minimum. However,
it is important overall, and fundamental to understanding how some operators work, so I'm including a sum-
mary of the subject in this appendix. If you feel confident about your math knowledge, this is all old hat to
you and you need read no farther. If you find the math parts tough, then this appendix should show you how
easy it really is.
BINARY NUMBERS
First let's consider what you mean when you write a common everyday number such as 321 or 747. Put more
precisely you mean
321 is:
3 × (10 × 10) + 2 × (10) + 1
and 747 is:
7 × (10 × 10) + 4 × (10) + 7
Because it is built around powers of ten, you call this the decimal system (derived from the Latin
decimalis
,
meaning
of tithes
, which was a tax of 10 percent — ah, those were the days . . .).
Representing numbers in this way is very handy for people with ten fingers and ten toes, or creatures with
ten of any kind of appendage for that matter. However, your PC is quite unhandy in this context, being built
mainly of switches that are either on or off. This is okay for counting up to two, but not spectacular at count-
ing to ten. For this reason your computer represents numbers to base 2 rather than base 10. This is called the
binary
system of counting, analogous to the
bi
cycle (two wheels), but nothing whatever to do with bibacity,
which means an inclination to drink a lot. With the decimal system, to base 10, the digits used can be from 0
to 9. In the binary system, to base 2, the digits can only be 0 or 1, ideal when you have only on/off switches
to represent them; off is usually 0, and on is 1 — simple. Each digit in the binary system is called a
bit
, which
is a contraction of binary digit. In an exact analogy to the usual base 10 counting, the binary number 1101 is
therefore:
1 × (2 × 2 × 2) + 1 × (2 × 2) + 0 × (2) + 1
which, if you work it out, amounts to 13 in the decimal system. In
Table B-1
,
you can see the decimal equi-
valents of 8-bit binary numbers illustrated.
TABLE B-1:
Decimal Equivalents of 8-bit Binary Numbers
