Binary Numeral System

Imagine all of us with one finger on each of two hands and one toe on each of two feet. Suppose further that all the numbers we use could be expressed with the digits 0 and 1. Since we are going to use only two symbols to write any number, we can call this a binary, or base-two, system. In a binary system the value of any place in a numeral is twice as large as the place to its right. Thus the place values—from right to left—in a binary system are ones, twos, fours, eights, and so on.

The number 1 expressed in the binary system would be written as 1two. Thus the number represented by 1ten and 1two is the same. (The subscript two indicates that we are expressing numbers in the binary system; the subscript ten indicates the decimal system.) How would we indicate 2ten as a binary numeral?

Suppose we have a set of 2 stars. If we draw a ring around them, we have 1 set of 2 stars, and no single stars remaining. We would write 2ten = 10two, which is read as “1 two plus 0 ones.” If we begin with a set of 3 stars and draw a ring around 2 of them, we would have 1 set of 2 stars plus 1 set of 1 star. This would be indicated as 3ten = 11two, read as “1 two plus 1 one.”

How would 4ten be expressed? Notice that we have been making pairs of equivalent sets wherever possible. Thus if we had a set of 4 stars, we could first form 2 sets of 2 stars each. Now draw a ring around these sets. This approach suggests that 4ten may be thought of as 1 four, no twos, and no ones, and written as 100two. In similar fashion, 5ten = 101two, which is “1 four, no twos, and 1 one.”

Note how 6ten would then be treated. We would first have 3 sets of 2 stars in each set. Then we could pair up 2 of these sets, and wind up with 1 set of four, 1 set of two, and no ones. We write 6ten as 110two.

Using the same development, it is clear that 7ten = 111two, interpreted as 1 set of four, 1 set of two, and 1 one. We can analyze 8ten in the same way. First we have 4 sets of 2 stars in each set. Then we have 2 sets with (2 × 2) or 4 stars in each. Finally, we have 1 set with (2 × 2 × 2) or 8 stars in it. This would be written as 1000two and interpreted to mean 1 eight, 0 fours, 0 twos, and 0 ones. We now summarize what we have learned so far about binary numeration in Table I, and see if we can express our ideas.

A decimal is a number written in base 10. These numbers are expressed by use of the decimal system, a base-and-place numeration system, with each place representing a power of 10 and zero used as a place holder. For example, the number 1,980 means (1 × 103) + (9 × 102) + (8 × 101) + (0 × 100), where 100=1. This system was invented by the Hindus and brought to Europe by the Arabs before 1200, although it was not universally adopted until the 17th century.