Numbers in Physics

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As a physicist, you have to deal with four different types of numbers:

Pure Numbers:

Physicists often deal with pure numbers, which are the numbers of the mathematician. Consider the formula for kinetic energy:

In this expression, the fraction "1/2" and the exponent "2" are pure, exact numbers. The exponent isn't approximately 2, it's 2, period. This is not a difficult concept for the beginning physicist, because they generally think (because of their previous training, unfortunately) that all numbers work that way. Big news for you: That is not the case.


Experimental physicists often need to count things. For instance, in an experiment on simple harmonic motion, you will generally need to count the vibrations of an object. Now, this is a measurement of sorts, but a measurement that does not (usually) need a measuring instrument. Also, its result is always a whole number.

Usually, we consider counts to be exact values in that there is no measurement uncertainty generally associated with a count. Certainly it is possible to make a mistake in counting, but good technique suggests that counts be checked - two people can count, instead of just one, for instance.


Looking back at the formula for kinetic energy, "m" represents mass. In order to "plug something in for m", the mass of an object has to be measured. It comes as quite a shock to many beginning physicists that no measurement is exact. All measurements have an associated uncertainty, and a good deal of the job of the experimental physicist is determining what that uncertainty is.

Calculated Values:

Looking back at the formula for kinetic energy once more, the symbols "K" and "v" represent still another type of number that the experimental physicist needs to deal with - they are calculated values. To calculate the speed, v, of an object, you need to divide the distance it traveled (a measurement) by the time required (another measurement). In order to calculate the kinetic energy of an object you multiply its mass (a measurement) times its speed (another measurement) to the second (a pure number) power, then divide the product by two (another pure number). If there are uncertainties associated with the distance and time measurements - and there are - then there will be an uncertainty in the speed of the object - but how much uncertainty? By the same token, how does the uncertainty in the measurement of the object's mass and speed affect what we know about its kinetic energy? This sort of analysis is called "propagation of errors", and you need to know something about error propagation in order to make intelligent analyses of experimental data.


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last update August 10, 2005 by JL Stanbrough