### Calculation Methods

The calculating procedure with the adder and multiplier matrices, which we have mentioned previously, is called "calculation by co-ordination." In fact, the matrices do not calculate at all. It would be more correct to say that they look up the results as if in a table or a ready-reckoner. How much is eight times three? Of course! Twenty-four! The and-gate with the output lead "24" is "co-ordinated," without having to be forced by mathematical necessity, to the point of intersection of leads "8" and "3." Not only numbers but also facts of a general kind can be connected together by "coordinators," by matrices. But you will be reading about that later on.

There is yet another method of calculating electronically: the "binary" way. It is the historic method of the electronic computers which we first mentioned in chapter 2, and we must not fail to say a few further words about it. It is still used by many automatic calculators.

It is now time to familiarize yourself with a new code - a 35-channel code. Yes, you read it right. 35 channels are quite a lot. There are neither punched tapes nor magnetic tapes with so many. They would have to be more than 4 inches wide. So we take tapes with seven tracks on them and always treat five of them, joined together, as a unit.

35 channels-that means 35 bits for each coded number. That is not just megalomania. You will see below just what can be done with them:

0000000 0000000 0000000 0000000 0000000 = 0
0000000 0000000 0000000 0000000 0000001 = 1
0000000 0000000 0000000 0000000 0000010 = 2
0000000 0000000 0000000 0000000 0000011 = 3
0000000 0000000 0000000 0000000 0000100 = 4
0000000 0000000 0000000 0000000 0000101 = 5

For one-digit numbers this code is a bit complicated. But it is also capable of expressing larger numbers. The biggest is

1111111 1111111 1111111 1111111 1111111 = 34,359,738,367

The code cannot get any higher than that, but even so, this is quite a number. In our old five-channel code, this eleven-digit number would require eleven times five, in other words fifty-five, bits - that is twenty bits more than in the 35-channel code. So those 35 channels are by no means as unpractical as you might have supposed, for counting with large numbers.

35 channels are not absolutely necessary for binary counting, We have chosen this example simply because prominent binary computers work with a channel system of this magnitude. Other systems use 30, 33, 36 or 40 channels. In every case, however, there are far more than our five channels of Chapter 3. That is because in binary calculation it is not possible to code every digit in a number of several digits: the whole number must be converted into bits at one time.

In the examples we are going to show you we shall manage quite well with small numbers, however. So we shall restrict ourselves to a 4-channel code. It will be enough to express all numbers between 0 and 15; the coded numbers will be called - as they are by modern scientists - "binary numbers." Here is the table:

 0000 = 0 0001 = 1 0010 = 2 0011 = 3 0100 = 4 0101 = 5 0110 = 6 0111 = 7 1000 = 8 1001 = 9 1010 = 10 1011 = 11 1100 = 12 1101 = 13 1110 = 14 1111 = 15

As we have just pointed out, our 4-channel code is very similar to the 35-channel code-except that the many zeros in the first 31 digits are lacking. That is not just an accident. All codes used for binary calculation are completely equal. (Strictly speaking, they are not codes, artificial sign-languages, at all, but special number-systems which have rules that are just as strict as those of our decimal system. The numbers in the sequence 1, 2, 3, 4, 5, 6, etc. are just as un-interchangeable as those in binary sequence 1, 10, 11, 100, 101, 110, etc. But that is only a marginal note that will probably be of little interest to you, and we shall continuealthough not quite correctly - to speak of a "binary code.")