Theoretical thoughts: extrapolation

A risky guess

When students continue their graph (or calculate) on down to find absolute zero, they are making a risky guess that the behaviour of a gas would stay the same. Explain that continuing beyond all measurements like that is called extrapolation.

You could say that:

Extrapolation is a risky business, trusting or pretending that what you have observed continues on and on.

Did the Sun rise in the East this morning? Did it rise in the East yesterday? Did it rise in the East many a morning before that? Are you willing to extrapolate these observations into the future and say that you are sure the Sun will rise in the East tomorrow? Are you quite sure?...

Suppose you were not a human being but were a butterfly who emerged from a chrysalis on a warm day in early summer and flew about from flower to flower, day after day. What would you observe about the weather? A fine day, another fine day, another fine day, ... If you were that insect, you might extrapolate and predict that every day in the future would be fine. You would not foresee the wintry day which would end your happy flights.

Extrapolation and interpolation

Extrapolation would be very risky if we trusted it for the molecules of a real gas like air or carbon dioxide. If we cool any gas enough, it fails to remain a gas, for example carbon dioxide becomes cold solid crystals of dry ice. Air that has been cooled down and pushed together so that its molecules (moving relatively slowly at that low temperature) hang together in a liquid.

You could say:

Yet we can safely make the extrapolation in imagination and find a useful absolute zero as a starting point for the grand Kelvin scale of temperature.

Interpolation means reading something off a graph between two measured points on it. (Or you can calculate an intermediate value between two values given in some table.) Interpolation is useful in science and engineering; and, if carefully done, it is safe. But in developing new science and technology, extrapolation is more important.

Although extrapolation is risky, it is the way in which some of the great discoveries have been made. Scientists guess what might happen if they continued our knowledge into an unknown region, then they try to test their guess by experiments. And sometimes, those experiments lead them in quite a new direction of knowledge.

Extrapolation is rash but sometimes very fruitful; interpolation is safe but dull.