The binary system obeys all the laws of the decimal. It is positional; it can represent any finite number; it can be used for addition, subtraction, multiplication, division, exponential functions, and any other arithmetical process known to man or to UNIVAC. The only difference is that it is to the base 2 instead of the base 10. It lops off eight of the ten basic decimal digits-0, 1, 2, 3, 4, 5, 6, 7, 8, and 9-retaining only 0 and 1.
You can count with it, of course. 1 is one; 10 is two; 11 is three; 100 is four; 101 is five; 110 is six; 111 is seven; 1000 is eight; 1001 is nine; 1010 is ten, and so on. You can subtract or add with it:
Four 100
Plus three 11
===
Is seven 111
You can multiply or divide with it:
Six 110
Divided by three 11
===
Is two 10
And you can do all of these things rather simply, without the necessity of memorizing multiplication tables, thus freeing your preadolescent evenings for baseball and doorbell-ringing.
Look back at Ivan's system of Russian multiplication; let us do it over again in a slightly different way. Let's halve both columns, the right as well as the left. And instead of striking out any number, let us write a "1" next to the odd numbers and a "0" next to the even ones. As follows:
87 1 93 1
43 1 46 0
21 1 23 1
10 0 11 1
5 1 5 1
2 0 2 0
1 1 1 1
Now, you might not know what you have just accomplished- and Ivan certainly wouldn't-but you have translated two decimal numbers into their binary equivalents. Reading from bottom to top, 1010111 is binary for 87; 1011101 is binary for 93.
To see what these mean, remember how we dissected a decimal number. A binary number comes apart in the same sort of pieces; the only difference is that the pieces are multiples of powers of 2, not of powers of 10. 1010111, then, is a shorthand way of saying:
1 x 2^6 = 64
0 x 2^5 = 0
1 x 2^4 = 16
0 x 2^3 = 0
1 x 2^2 = 4
1 x 2^1 = 2
1 x 2^0 = 1
==
87
which is what we said it was in the first place.
When you feed numbers like 87 and 93 into UNIVAC, its digestion gets upset-in fact, it won't accept them until they are predigested. So you must convert them into binary digits ("bidgets" or "bits"), just as we did above. Such binary numbers as 1010111 and 1011101 UNIVAC handles very well indeed. Multiply them? No trouble at all. UNIVAC, in its electronic way, does something like this:
1010111
X 1011101
=========
1010111
0
1010111
1010111
1010111
0
1010111
=============
1111110011011
That may look frightening, because it is unfamiliar; but it is still the same old product of 87 x 93; it is shorthand for:
1 X 2^12 4096
1 X 2^11 2048
1 X 2^10 1024
1 X 2^9 512
1 X 2^8 256
1 X 2^7 128
0 X 2^6 0
0 X 2^5 0
1 X 2^4 16
1 X 2^3 8
0 X 2^2 0
1 X 2^1 2
1 X 2^0 1
====
8091
Observe the simplicity! True, the number is long; but see how simple manipulating it becomes. Addition, for instance, is reduced to simple counting. (Binary counting, of course - 1, 10, 11, 100, and so on. You can call it "one," "ten," "eleven," and "one hundred," and so on. if you like, with no great harm.) To add a column of figures, like
101
100
110
111
=====
10110
you simply count the ones in the right-hand column (1, 10; write down 0 and 1 to carry); then count the ones in the middle column, starting, of course, with the one you carried (1, 10, 11; write down 1 and 1 to carry); then count the ones in the left-hand column, again remembering the one you carried (1, 10, 11, 100, 101; write down 1 and 10 to carry; write down the 10).
That is, I submit, about as simple as an arithmetical operation can get, and multiplication is nearly as much so. Multiplication becomes merely a matter of writing down the number, moved an appropriate number of places to the left, or not writing down the number at all (depending on whether the digit you are multiplying by is "1" or "0"). Thereafter it is addition, and addition, as we have seen, is merely counting. No multiplication tables! No tedious memorizing! No wonder UNIVAC and Ivan like it!
If binary arithmetic has a fault, it is that it is so excessively easy that it becomes boring.
But the world's work is full of boring operations that get done anyhow. We have found two good ways to handle them-either to turn them over to machines (like UNIVAC), which do not have the capacity for boredom, or to learn to do them as a matter of mechanical routine.
My wife observes (as most wives sometimes observe) that it doesn't much matter what sort of change she suggests, I can usually find a dozen splendid reasons for keeping things just as they are. Since the human animal is conservative, most of us can find objections to any sort of change. ("Better the devil you know.") Since the human animal is also educable, we often, however, overcome our objections when the change promises rewards.
Let us see what the drawbacks and rewards of changeover to binary notation may be. Not that the case is really arguable, since the silent vote of the computers constitutes a carrying majority over our human veto, but let us see if there are any advantages for us borable, error-prone humans.
The drawbacks stand out immediately, starting with the sheer physical size of a binary number as contrasted with its decimal equivalent. Still, a binary number isn't so very much longer than a decimal (about three times) as to be ipso facto out of the question. As a matter of fact, really large numbers are hopelessly unwieldy in any notation at all. In the prevailing decimal system scientific people express large numbers either as approximations (3 x *~ for instance) or in terms of their prime factors and exponents (193 x 641°* x 1861) or in other factored or shorthand ways. Even the headlines in our daily papers are more likely to read $6.5 billion than $6,500,000,000.
For "household-sized" numbers-oh, up to a million, let's say-it doesn't seem as though the mere matter of length ought to be a prevailing count against binary notation. You might use 20 binary digits to write a number that big (as against seven in decimal notation) and a number such as-to take one at random-101001111001011000010 is pretty hideous. But is 1372866, its decimal equivalent, utterly lovely?
Perhaps the number itself isn't so bad; perhaps the way we are reading it could stand some improvement. Look at the number 1111110011011, for instance. You just came across it a couple of pages ago (our old friend, the product of 87 and 93), and yet you almost certainly failed to recognize it. Is it because its recognition value is intrinsically low? Or because we lack practice in reading (and in establishing conventions of writing) that sort of number?
In decimal notation, remember, we simplify the reading of such large numbers by setting off groups of three. 5000000000000 is pretty hard to read by itself, though 5,000,000,000,000 reveals itself to be five trillion rather conveniently. Why should we not adopt a similar convention for binary numbers? There is no reason to stick to groups of three; let's make it groups of five, and thus write the expression for the product of 87 x 93 - that is, 8091 - as follows: 111,11100,11011.