Significant Digits: Additional Considerations

Purplemath

Rounding to a certain number of significant digits, or to a named "place", is fairly straightforward. The real question comes in how to round answers to the "appropriate" number of significant digits. If you've been given some values (lengths, say, or masses) and told to work with them, arriving at an answer which is rounded to the "appropriate" number of sig-digs, what does this mean?

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.

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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 measure 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.

As a result, 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.

Rounding Addition

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.

• Round to the appropriate number of significant digits:

The requirement to round your answer, when adding values, to the same "place" as the largest (that is, furthest to the left, with respect to the decimal point) last accurate place of the input values, might make a little more sense if you view the addition vertically:

 13.214
234.6
  7.0350
+ 6.38
--------
261.2290

When you look at the columns of digits, it kinda makes sense that, yeah, you can't claim any accuracy past the tenths place, because that's the last column that all the input numbers share.

Here's another example:

• Simplify, rounding 1247 + 134.4 + 450 + 78 to the appropriate number of significant digits.

Rounding Multiplication

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.

• Simplify, and round to the appropriate number of significant digits:

• Find the product of 0.00435 and 4.6, simplifying to the appropriate number of digits.

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 that trailing zero is therefore a necessary part of the answer.

Just remember the difference: