The Purplemath Forums
Significant Digits: Additional Details (page 3 of 3)
Sections: General rounding, Rounding and significant digits
The real question comes in how to round answers to the "appropriate" number of significant digits.
The idea is this: Suppose you measure a block of
wood. The length is 5.6 inches, the width is 4.4 inches, and the
thickness is 1.7 inches, at least as best you can tell from your tape measure. To find
the volume, you would multiply these three dimensions, to get 41.888 cubic inches.
But can you really, with a straight face, claim to have measured the volume of that block of wood
to the nearest thousandth of a cubic inch?!? Not hardly! Each of your measurements was accurate
(as far as you can tell) to two significant digits: your tape was marked off in tenths of inches,
and you wrote down the closest tenth of an inch that you could see. So you cannot claim five decimal
places of accuracy, because none of your measurements exceeded two digits of accuracy. You can
only claim two significant digits in your answer. In other words, the "appropriate" number
of significant digits is two, and you would report (in your physics lab report, for instance) that
the volume of the block is 42 cubic inches, approximately.
How do you round when they give you a bunch of numbers to add? You would add (or subtract) the numbers as usual, but then you would round the answer to the same decimal place as the least-accurate number.
13.214 + 234.6 + 7.0350 + 6.38
Looking at the numbers, I see that the second number, 234.6, is only accurate to the tenths place; all the other numbers are accurate to a greater number of decimal places. So my answer will have to be rounded to the tenths place:
13.214 + 234.6 + 7.0350 + 6.38 = 261.2290
Rounding to the tenths place, I get: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
13.214 + 234.6 + 7.0350 + 6.38 = 261.2
Here's another example:
Looking at each number, I see that I will have to round the final answer to the nearest tens place, because 450 is only accurate to the tens place. First, I add in the usual way:
1247 + 134.5 + 450 + 78 = 1909.5
...and then I round my result to the tens place:
1247 + 134.5 + 450 + 78 = 1910
How do you round, when they give you numbers to multiply (or divide)? You would multiply (or divide) the numbers as usual, but then you would round the answer to the same number of significant digits as the least-accurate number.
16.235 × 0.217 × 5
First, I note that 5 has only one significant digit, so I will have to round my final answer to one significant digit. The product is:
16.235 × 0.217 × 5 = 17.614975
...but since I can only claim one accurate significant digit, I will need to round 17.614975 to 20, which is accurate to one significant digit.
16.235 × 0.217 × 5 = 20
First I multiply:
0.00435 × 4.6 = 0.02001
Looking at the original numbers, I see that 4.6 has only two significant digits, so I will have to round 0.02001 to two significant digits. The 2 is the first significant digit, so the 0 following it will have to be the second significant digits. In other words, I must report the answer as being:
0.00435 × 4.6 = 0.020
The answer should not be 0.02, because 0.02 has only one significant digit; namely, the "2". The trailing zero in 0.020 indicates that "this is accurate to the thousandths place, or two significant digits", and is therefore a necessary part of the answer.
Just remember the difference:
For adding, use "least accurate place".
For multiplying, use "least number of significant digits".