Once convinced that the binary method does have its merits, it may be a little easier to pursue a mastery of representing numbers in binary notation, difficult as it may seem at the outset. The usual way to convert is to remember, or list, the powers of 2, and start at the left side with the largest power that can be divided into the decimal number we want to convert. Suppose we want to change the number 500 into binary. First we make a chart of the positions:
| Power of 2 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| Decimal Number | 256 | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| Binary Number | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
Since 256 is the largest number that will go into 500, we start there, knowing that there will be nine binary digits, or “bits” in our answer. We place a 1 in that space to indicate that there is indeed an eighth power of 2 included in 500. Since 128 will go into the remainder, we put a 1 in that space also. Continuing in this manner, we find that we need 1’s until we reach the “8” space which we must skip since our remainder does not contain an 8. We mark a 1 in the 4 space, but skip the 2 and the 1. Our answer, then, in binary notation is 111110100. This number is called “pure binary.” It can also lead to pure torture for human programmers whose eyes begin to bug with this “bit chasing,” as it has come to be called. Everything is of course relative, and the ancient Roman might gladly have changed DCCCLXXXVIII to binary 1101111000, which is two digits shorter.
There is a simpler way of converting that might be interesting to try out. We’ll start with our same 500. Since it is an even number, we put a 0 beneath it. Moving to the left, we divide by two and get 250. This also is an even number, so we mark down a 0 in our binary equivalent. The next division gives 125, an odd number, so we put down a 1. We continue to divide successively, marking a zero for each even remainder, and a 1 for the odd. Although it may not be obvious right away, we are merely arriving at powers of two by a process called mediation, or halving.
| Decimal | 1 | 3 | 7 | 15 | 31 | 62 | 125 | 250 | 500 |
| Binary | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 |
Obviously we can reverse this procedure to convert binary numbers to their decimal equivalents.
There is an interesting extension of this process called duplication by which multiplication can be done quite simply. Let us multiply 95 times 36. We will halve our 95 as we did in the earlier example, while doubling the 36. This time when we have an even number in the left column, we will simply cancel out the corresponding number in the right column.
| 95 | 36 |
| 47 | 72 |
| 23 | 144 |
| 11 | 288 |
| 5 | 576 |
| 2 | |
| 1 | 2304 |
| —— | |
| 3420 |
This clever bit of mathematics is called Russian peasant multiplication, although it was also known to early Egyptians and many others. It permits unschooled people, with only the ability to add and divide, to do fairly complex multiplication problems. Actually it is based on our old stand-by, the binary system. What we have done is to represent the 95 “dyadically,” or by twos, and to multiply 36 successively by each of these powers as applicable. We will not digress further, but leave this as an example of the tricks possible with the seemingly simple binary system.
Even after we have learned to convert from the decimal numbers we are familiar with into binary notation almost by inspection, the results are admittedly unwieldy for human handling. An employee who is used to getting $105 a week would be understandably confused if the computer printed out a check for him reading $1101001. For this reason the computer programmer has reached a compromise with the machine. He speaks decimal, it speaks binary; they meet each other halfway with something called binary-coded decimal. Here’s the way it works.