# Why is Moore's Law Exponential?

A lot of people talk about Moore's Law but very few ever attempt to explain why it is so. This web page posits a very simple derivation of the formula, and why current technological development is exponential, not just in computers, but in all things technical and economic.

In the simplest of terms, one can say that many small (incremental) improvements in many fields add up to a large change. More precisely, it is embodied in the formula for the exponential:

ex = limn->oo (1+x/n)n

To give a specific example of this formula, imagine the many technologies that go into to making of a computer chip. Every year, there is a 5% or 10% or 20% increase in the size of a silicon die that a computer CPU is made out of. The change is not exponential, it is linear, more or less. There are also small, incremental improvements in all aspects of die- making technology. One is able to make purer silicon every year. One is able to etch slightly deeper than the year before, and one can improve doping concentration and uniformity a little bit at a time. One can planarize the die a little bit better; this allows more metal layers to be used. Small improvements in dielectric chemistry allow better control over capacitance in the circuits. Each of these small improvements add up to allow smaller features to be etched into the die, without jeopardizing the uniformity of the lot or the reliability of the circuit. But smaller features are also possible only because the machinery to image a pattern onto the die improves every year. This machinery in turn is the result of many small improvements. One need only to envision the product sheet for the new-generation equipment: just like any other product data advertisement, it will list a dozen or so key improvements: this new feature, that new improved thingy, a dozen things all smaller, faster, better. If in turn you looked at the product sheets for the stuff that the machine is made out of, these each also list small improvements: some motor is more powerful, smaller, or cheaper. Some screw has more uniform and precise threads. Some piece of plastic has a more accurate shape because it was molded of higher quality plastic, and the mold was of higher quality.

You get the drift. Although each improvement is small, the net effect is multiplicative. It is the multiplicative nature of these improvements that makes Moore's law happen, as well as the corresponding laws in many, many fields of endevour. As another example, Livingstone's law (Blewett, Fermi, Livingstone, Sessler) states that the energy of particle accelerators used by physicists will double in energy every two years. This law, first noticed in 1950, has been in play for over seventy years, much longer than Moore's law. There seem to be corresponding exponential growth curves in many, many areas.

It is worth exploring why advances in technology don't allow automobiles to go exponentially faster every year. I believe that the answer lies in whether the advances are diluted or are concentrated. In the case of the CPU chip, the advances are all focused on one goal: make the CPU faster. Every improvement aims at that. In cars, there is exponential improvement, but it is in exponentially many areas: they are quieter, use less gas, are more luxurious. They have better suspension and nicer interiors. The radio is better, as are the windshield wipers. Few of these improvements result in higher speed. The improvements that do help improve speed are better aerodynamics, better suspensions, and better roadways. But these in fact show up as improved safety figures: there are fewer highway accidents, and those that do occur are less fatal. Advances in automobile technology are dilutive: there are many things that improve on many fronts; the improvement in any one single number is not terribly dramatic. There is no single number, such as the CPU speed, that adequately sums up progress in automobile manufacture.

Draft of May 2003, Linas Vepstas linas@linas.org
Copyleft (c) 2003 Linas Vepstas