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multiplier of 1000. A 1K resistor, for example, is 1000 ohms, and lOOK dollars is $100,000.
However, in the computer world, K means "multiply by 1024." So a 16bit-wide word can have
65,536 possible values, or 64K (65,536/1024 = 64).
A similar rule applies to the term meg, or million. A I-meg resistor is 1,000,000 ohms. In
computer lingo, a meg is 1024 x 1024, or 1,048,576.

Floating Point

A limitation on any integer number scheme, regardless of the number of bits, is the
difficulty in representing fractional numbers such as 2.54 or 3.3. When we looked at decimal
numbers at the beginning of this appendix, we saw that they increase in powers of 10 as you
move from right to left across the digits. As you move to the right of the decimal point,
decimal numbers increase in negative powers of 10:
10' 10' 10" . . . lo-' lo-? 1o-'i 1 o-4
Or 100 10 1 .... .01 .001 ,0001
1
Binary numbers work the same way:
22 2' 20 , . . 2-1 2-9 2-'3 2-+
Decimal 4 2 1 ... .5 .25 .125 .0625

And hex numbers as well:
16' 16' 16" . . . 16-' 1 6-' 1
Decimal 256 16 1 . . . .0625 .0039 .000244

So we can write a decimal number, such as 2.54, in binary and hex:
2 54 = 010.100010100011~ = 2.8A316
Note that in binary and hex, the number is a repeating value. Just like 1/3 is a repeating
decimal in base 10 (but not in base 3), some fractional numbers cannot be exactly converted
between bases.
We could represent fractional binary numbers in a computer by defining a 16bit number
as ranging from zero to 4095 instead of zero to 65,535. 4096 values can be represented by
the upper 12 bits of the l6bit word. This leaves the lower 4 bits available to represent
fractional values. For instance, the hex value 1002 would be interpreted as 100.2, or
+ 2 x 16-', or 256.125 in decimal.
Such an arrangement makes calculations fairly easy and keeps everything in an integer
format. However, the resolution of the fractional part of the number is limited, and there is
a tradeoff between the accuracy of the fractional part and the maximum size of the number.
The more bits that are allocated to the fractional part, the smaller the maximum number
can be. The fewer bits allocated to the fractional part, the less precision we have to repre-
sent numbers with.
A better means of representing fractional values would emulate the decimal system that
we are already familiar with. If you have 4 decimal digits, you can write ,0001, 10, or 1000.
All these numbers use 4 digits, but the decimal point can move, or float, to represent

Appendix B 31 1