## Representing Prescision in Physics

Whenever we measure some value (like a length), we can only find it to a finite accuracy (perhaps a millimeter on a typical ruler). We need to acknowledge the limited accuracy of our numbers, by considering how many digits are "significant". These "Significant Figures" allow us to represent the prescision of our measurements and results calculated from the in an umambigous fashion.

If we tried to measure the distance from home to school with a meter long stick, it would not be possible to say, reliably, that the distance was 343.643m. You could say 344m or maybe even 343.5m but trying to imply that you can accurately judge tenths of millimeters on an unmarked meter long stick is dishonest.

In this case we would say that there are four significant figures (in 343.5m) These are the parts of the value that are reasonably reliable. Not necessarily 100% rock solid as there is rounding off and the possibility of other errors, but reasonably reliable. The final two digits of 343.643m are completely unreliable and should be ignored.

Typically, you can assume that (given no other information) any number given is accurate to within plus or minus one in the smallest place written out. Eg: 406 means a number somewhere between 405 and 407, 1.11 means 1.10 to 1.12 etc.

Rules for Significant Figures:

1) Counting from the left and ignoring all the leading zeros, keep only the digits up to the first doubtful one. As written. 22.568 has five significant figures and 7002 has four, but 0.0072 has only two, the zeros in this case are merely place holders and don’t really describe any part of the number.

It is not always clear, in traditional notation, how many significant figures some numbers have. For example, 80km might have two (if the distance was known to be between 79 and 81) or only one (if the distance was known to be between 70 and 90km) with the zero acting as a placeholder. In these cases, the use of scientific notation allows us to remove the ambiguity: the distinction between 8x101km and 8.0x101km is clear. Conversely, if we knew the distance was somewhere between 79.9km and 80.1km, we would be well justified in using three significant figures, writing the distance as 80.0km.

2) When multiplying or dividing, you should only keep as many significant figures as in the least precise of the numbers used in the calculation. (Only go to significant figures as a final step – rounding off in intermediate steps can introduce errors in your calculations.) For example, say we want the product of 11.3 and 6.8

11.3 x 6.8 = 76.84.

But, if we look at the extremes possible
11.2 x 6.7 = 75.04

11.4 x 6.9 = 78.66

So, the range of possible answers spanned by the possible values of the input, belies the four significant figures given in our answer above. However, if we round off to two significant figures, our answer becomes 77, which is within one or two units of our possible answers.

Notice that if we had rounded the 11.3 off to two significant figures before we started, we would have ended up with 11 x 6.8 = 75, an incorrect answer!

3) When adding or subtracting, the least significant digit of the answer occupies the same position as the least significant digit in the least precise number used in the calculation.

For example: 2.3 + 3.145 = 3.4, not 3.445.