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There's even a bug in my multiplication program. If you run it twice, the second time through it will multiply A7h by 256 and add that result to the result already calculated. This is because after you run the program once, the number at address 1003h will be 0. When you run it the second time, FFh will be added to that value. The result won't be 0, so the program will keep running until it is.

We've seen that this machine can do multiplication, and in a similar way it can also do division. I've also asserted that this machine can use these primitive functions to do square roots, logarithms, and trigonometric functions. All a machine needs is the hardware to add and subtract and some way to use conditional jump instructions to execute the proper code. As a programmer might say, "I can do the rest in software."

Of course, this software might be quite complex. Many whole books have been written that describe the algorithms that programmers use to solve specific problems. We're not yet ready for that. We've been thinking about whole numbers and haven't taken a crack at how to represent decimal fractions in the computer. I'll get to that in Chapter 23.

I've mentioned several times that all the hardware to build these devices was available over a hundred years ago. But it's unlikely that the computer shown in this chapter could have been built at that time. Many of the concepts implicit in its design weren't apparent when relay computers were first built in the mid-1930s and only started to be understood around 1945 or so. Until that time, for example, people were still trying to build computers that internally used decimal numbers rather than binary. And computer programs weren't always stored in memory but instead were sometimes coded on paper tape. In particular, in the early days of computers, memory was expensive and bulky. Building a 64-KB RAM array from five million telegraph relays would have been as absurd one hundred years ago as it is now.

It's time to put what we've done in perspective and to review the history of calculation and computing devices and machines. Perhaps we shall find that we don't have to build this elaborate relay computer after all. As I mentioned in Chapter 12, relays were eventually replaced with electronic devices such as vacuum tubes and transistors. Perhaps we shall also find that someone else has built something that's equivalent to the processor and the memory we designed but that can fit in the palm of your hand.

Chapter 18. From Abaci to Chips

Throughout recorded history, people have invented numerous clever gadgets and machines in a universal quest to make mathematical calculations just a little bit easier. While the human species seemingly has an innate numerical ability, we also require frequent assistance. We can often conceive of problems that we can't easily solve ourselves.

The development of number systems can be seen as an early tool to help people keep track of commodities and property. Many cultures, including the ancient Greeks and native Americans, seem to have counted with the assistance also of pebbles or kernels of grain. In Europe, this led to counting boards, and in the Middle East to the familiar frame-and-bead abacus:

Although commonly associated with Asian cultures, the abacus seems to have been introduced to China by traders around 1200 CE.

No one has ever really enjoyed multiplication and division, but few people have done anything about it. The Scottish mathematician John Napier (1550–1617) was one of those few. He invented logarithms for the specific purpose of simplifying these operations. The product of two numbers is simply the sum of their logarithms. So if you need to multiply two numbers, you look them up in a table of logarithms, add the numbers from the table, and then use the table in reverse to find the actual product.

The construction of tables of logarithms occupied some of the greatest minds of the subsequent 400 years while others designed little gadgets to use in place