Significant Figures and Rounding





Table of Contents

Introduction
Rules for Counting Significant Figures
More About Trailing Zeros
Why Round Off Your Computations?
Some Rounding Rules

Online Rounding Practice
Tools for Practicing Numerical Skills





Introduction

Leopold uses a meter stick to measure the width of a wooden plank to the nearest millimeter or so.  He has many mathematically valid choices as to how to write his result:

0.208 m 20.8 cm 208000 µm 2.08 x 10-1 m 2.08 x 102  mm
.208 m 0020.8 cm 0.000208 km 2.08 x 100  dm 2.08 x 105  µm
2.08 dm 208 mm .000208 km 2.08 x 101  cm 2.08 x 10-4 km

The different versions have different numbers of digits and extend to different numbers of decimal places, but they all represent the same measurement with the same uncertainty.  We conclude from this that a number's precision -- its random error -- is not indicated by the number of digits or by the number of decimal places.

The versions in the fourth and fifth columns, expressed in scientific notation, make it clear that only the three digits "208" tell you anything about precision.  All other digits in the first three columns are unnecessary leading zeros (0.208 m), leading zeros which tell you where the decimal point goes (.000208 km), and trailing zeros which tell you where the ones place is (208000 µm).

Just what do the digits "208" tell us about the number's precision?  They tell us that the random uncertainty is a few parts in 208 -- that is, roughly a part in 100, or 1%.  The larger the number of "significant digits" (or "significant figures"), the finer the precision.  By using the correct number of significant figures, we let the world know just how reliable our measurements are.

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Rules for Counting Significant Figures


1.  Always count nonzero digits

Example:  21 has two significant figures, while 8.926 has four

2.  Never count leading zeros

Example:  021 and 0.021 both have two significant figures

3.  Always count zeros which fall somewhere between two nonzero digits

Example:  20.8 has three significant figures, while 0.00104009 has six

4.  Count trailing zeros if and only if the number contains a decimal point

Example:  210 and 210000 both have two significant figures, while 210. has three and 210.00 has five

Read why 210.00 is very different from 210

5.  For numbers expressed in scientific notation, ignore the exponent and apply Rules 1-4 to the mantissa

Example:  -4.2010 x 1028 has five significant figures

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More About Trailing Zeros


Rule 4 is the one which throws many students at first.  Let's compare three measured times:

210 s means "210 s, give or take a few tens of seconds"

210. s means "210 s, give or take a few seconds"

210.00 s means "exactly 210 s, give or take a few hundredths of seconds"

In other words, they're three different numbers.  The third has three more significant figures than the first, indicating that it's roughly 103 times more precise than the first.  If you record your measurement as 210 s, you're saying that the true time could very well be as low as 180 s or as high as 240 s.  If you call it 210.00 s, however, you're claiming that the true time is probably between about 209.97 s and 210.03 s.  Those are two radically different statements!

Your calculator doesn't understand Rule 4.  Your calculator is wrong.  Keep your trailing zeros straight, and keep the decimal point in numbers like "210." straight.

And what about "210000 s"?  Does that mean, "210000 s, give or take a few tens of seconds"?  Hundreds of seconds?  Thousands?  If you read a number like this, you must assume the worst: none of the trailing zeros are significant, so it's give or take a few tens of thousands of seconds (2.1 x 105 s).  If you write a number like this, shame on you: Rewrite it in scientific notation.  For instance, writing "2.100 x 105 s" clearly indicates that the uncertainty is a few hundreds of seconds.

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Why Round Off Your Computations?


Fine, you've measured a few things in the lab and have recorded each value to the proper precision.  Is that it?  Usually not: You'll want to use your measurements to compute various other quantities.  When you add measured values to each other, divide measured values by each other, or square measured values, how do you round off the result?

The basic principle is simple to grasp.  If Fran carefully measures out 72.39 g of some powder and places it into a beaker, and then Frank grabs a handful of the stuff with a mass of 200 g or so and throws it in, how precisely known is the total mass?  Just to the nearest 100 g or so, since the second mass was so poorly measured:

72.39 g  +  2 x 102 g  =  3 x 102 g

If Harvey carefully measures the length of a table to be 239.7 cm, and Hortense glances at the table and figures that the width is 90 cm (give or take a few tens of cm), how should we round off the table's area when we multiply these two numbers together?  This area isn't well known because the width isn't well known, regardless of how carefully the length was measured:

239.7 cm  x  90 cm  =  2 x 104 cm2

MORAL: It's always the least precise operand which determines the precision of the result.

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Some Rounding Rules


1.  When adding or subtracting numbers, find the number which is known to the fewest decimal places, then round the result to that decimal place.

Example:  21.398  +  405  -  2.9  =  423

(The operand 405 is only known to the ones place, so the result must be rounded to the ones place.)

2a.  When multiplying or dividing numbers, find the number with the fewest significant figures, then round the result to that many significant figures.

Example:  0.049623  x  32.0  /  478.8  =  0.00332

(The operand 32.0 is only known to three significant figures, so the result must be rounded to three significant figures.)

2b.  If either the unrounded result or the result rounded according to Rule 2a has 1 as its leading significant digit, and none of the operands has 1 as the leading significant digit, keep an extra significant figure in the result (while making sure that the leading digit remains 1).

Example A:  3.7  x  2.8  =  10.4

(Following Rule 2a would give us 10. as our result.  This is only precise to a few parts in ten -- a few tens of percent -- which is substantially less precise than either of the two operands.  We instead err on the side of extra precision: 10.4.)

Example B:  3.7  x  2.8  x  1.6  =  17

(This time, the operand 1.6 is only known to a few tens of percent, so the result should be rounded to 17 rather than 16.6.)

Example C:  38  x  5.22  =  198

(Rule 2a would give us 2.0 x 102, but since the unrounded result is 198.36, Rule 2b says to keep an extra significant figure.)

Example D:  7.81  /  80  =  0.10

(Rule 2a says to round 0.097625 to 0.1, at which point Rule 2b tells us to keep a second significant figure.  Writing 0.098 would imply uncertainty of a few parts in 98 -- that is, a few percent.  This is much too optimistic, since the operand 80 is uncertain by a few tens of percent.  So we keep 1 as the leading digit and write 0.10.)

3a.  When raising a number to some power which isn't very large or very small -- say, squaring it (power = 2) or taking the square root (power = ½) -- count the number's significant figures, then round the result to that many significant figures.

Example:  (5.8)2  =  34

(The operand 5.8 is known to two significant figures, so the result must be rounded to two significant figures.)

3b.  If either the unrounded result or the result rounded according to Rule 3a has 1 as its leading significant digit, and the operand's leading significant digit isn't 1, keep an extra significant figure in the result.

Example A:  (3.9)2  =  15.2

Example B:  square root of 0.0144  =  0.120

Example C:  (40)2  =  1.6 x 103

(Rule 3a would yield 2 x 103, but the unrounded result has 1 as its leading digit, so Rule 3b says to keep an extra significant figure.)

4.  Mathematically exact numbers like ½ and pi are infinitely precise, so they don't influence the precision of any computation.

Example A:  If ten marbles together have a mass of 265.7 g, the mean mass per marble is (265.7 g)  /  10  =  26.57 g.

(Assuming you can count, the number 10 has zero uncertainty!)

Example B:  A circle with measured radius 2.86 m has circumference C  =  2 pi (2.86 m)  =  17.97 m.

(The factors 2 and pi are exact, so we invoke Rules 2a and 2b to obtain a result with four significant figures.)

5.  In order to avoid "roundoff error" during multistep calculations, keep an extra significant figure for intermediate results, then round properly when you reach the final result.

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What's that?  You're not entirely sure that you've mastered those rounding rules?  Not to worry: You can practice online to your heart's content.

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