Physics I Significant Figures

Measurement necessary for science “I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be.” Lord Kelvin, 1883Lord Kelvin

Measurement To predict or postdict nature it is convenient to assign numbers that can be manipulated using mathematical techniques One way of doing that is via measurement –Analog –Digital –Definitions –Counting

Significant Figures A significant figure comprises –Certain digits Those read directly from the measuring device –One uncertain digit Interpolated in analog case Omitted in digital case The idea is to represent the uncertainty in the measurement Sig. Figs. are the coarsest way of doing so –Statistical measures are better

Analog measurement One quantity “stands for” another Continuously variable Uncertainty introduced because scales can only be finitely subdivided –Least count Smallest marked division Length examples –Interpolate visually, psychologically between least- count markers Uncertain digit (doubtful digit) is the interpolated digit

Digital Measurements Finite number of LCD “readouts” Uncertain digit (doubtful digit) is the most significant digit that does not appear in the readout

Definitions Some quantities are “exact” by definition Examples –The speed of light in vacuo m/s (exactly) –Conversion between inch and meter 1 inch = m (exactly) –Conversion between Joules and “thermochemical calorie” 1 cal th = J (exactly) –Conversion between Joules and International table calorie cal IT = J

Counting Exact (no uncertainty) Assumes no mistakes made

Precision of measured numbers Suppose we cite a measured number to the nth decimal place –E.g., –This true value x of this number presumably satisifies < x < The uncertainty of x,  x, is thus ±0.5 x The relative uncertainty is  x / x  0.5 x / = 1.6 x = 1.6 x % That is about 2 parts in a million, i.e., 2 ppm,

Significant figures relate to relative uncertainty The number of significant figures thus is connected with the relative uncertainty of our approximation to the true value When we use the methods of computing with significant figures, we are controlling the relative uncertainty of the result. The rule of thumb for multiplication and division states that the relative uncertainty of a product or quotient is not smaller than the largest relative uncertainty that goes into the computation

Decimal digits in addition and subtraction When adding or subtracting measured numbers the rule of thumb is that the result has no more decimal digits than the factor with the fewest decimal digits. We are not using significant figures here, but decimal digits

Square roots and sig. figs The relative uncertainty of the square root of a number is about half the relative uncertainty of the number. This is the “same order of magnitude” so for now we use the same number of significant figures in the square root function as we find in its argument –Later we may learn a more precise way of estimating uncertainty

Trigonometric functions and sig. fig.s Hecht cites a rule without justification! –The values of trigonometric functions have the same number of significant figures as their arguments

Hecht peculiarity Hecht uses a non-standard notation that may confuse you When he gives a number terminating in zeros, he assumes the zeros are all significant even when they are not following by a decimal point –This violates the policy of every other author I have seen, and that is quite a few

Guard digits Numerical analysts often include a few more digits than are significant in their computation and round off at the final computation.

Rules for counting sig. figs. All embedded or trailing zeros are significant –Hecht is unusual in counting trailing zeros that are not followed by a decimal point Leading zeros are never significant –They serve only to mark the location of the radix

A counter example We will stick slavishly to the rules of thumb BUT Here is a counter example –3.21 x comprises 3 sig. figs 8.4 comprises 2 sig. figs. –Calculator says product is –Rule of thumb says product is 27 (round to two sig. figs.)

But is that reasonable? Apply relative uncertainty rule –Relative uncertainty of 3.21 = /3.21=1.6 x –Relative uncertainty of 8.4 = 0.05 / 8.4 =6.0 x –Relative uncertainty of 27 = 0.5 / 27 = 1.9 x –Relative uncertainty of 27.0 = 0.05 /27= 2 x 10 -3

Observe The rule of thumb limits us to an answer of 27 The relative uncertainty rule lets us write 27.0 –One more significant digit than the rule of thumb.

Calculation comment Subtraction of two very nearly equal numbers is a dangerous operation because one reduces considerably the number of significant figures. – – = We have gone from 6 sig. figs. down to only 1.