The Counting Base
Theradix or base of a numeration system is the number of single-digit symbols it has. The
radix-ten system, also called base-ten or the decimal numeration system, therefore has ten sym-
bols, not counting commas (or decimal points, which we’ll get into later). But there are sys-
tems that use bases other than ten, and that have more or less than ten symbols to represent
the digits. In this section we’ll look at some of them. You can get a good “mental workout”
by playing with these! But they’re more than mind games. The base-two and base-sixteen
systems, in particular, are commonly used in computer science.A subtle distinction
Doesn’t 5 always mean the quantity five, 8 always mean the quantity eight, 10 always mean
the quantity ten, and 16 always mean the quantity sixteen? Not necessarily! It’s true in base-
ten, but it is not necessarily true in other bases.- Here are five pound signs: #####
- Here are eight pound signs: ########
- Here are ten pound signs: ##########
- Here are twelve pound signs: ############
- Here are sixteen pound signs: ################
In the base-eight numeration system, the total number of pound signs in the second line in
the above list would be written as 10, the third line as 12, the fourth line as 14, and the last
line as 20. In the base-sixteen numeration system, the total number of pound signs in the third
line would be written as the letter A, the fourth line as C, and the last line as 10. (If you’re
confused right now, just hold on a couple of minutes!)
When the expression for a number is a spelled-out word like “eighteen” or “forty-five”
or “three hundred twenty-one,” we mean the actual quantity, regardless of the radix. If I
write, “There are forty-five apples in this basket,” it is absolutely clear what I mean. But if
I write, “There are 45 apples in this basket,” you must know the radix to be sure of how
many apples the basket contains.The decimal system
As you count upward from zero in the base-ten system, imagine proceeding clockwise around
the face of a ten-hour clock as shown in Fig. 1-5A. When you have completed the first revolu-
tion, place a digit 1 to the left of the 0 and then go around again, keeping the 1 in the tens place.
When you have completed the second revolution, change the tens digit to 2. You can keep going
this way until you have completed the tenth revolution in which you have a 9 in the tens place.
Then you must change the tens digit back to 0 and place a 1 in the hundreds place.The Roman system
The Roman numeration scheme can be considered as a base-five system, at least when you
start counting in it. Imagine a five-hour clock such as the one shown in Fig. 1-5B. You start
with I (which stands for the number one), not with 0. You can complete one revolution and
go through part of the second and the system works well.The Counting Base 11