Rounding Off After Multiplication and Division

When rounding off the results of multiplication or division you use the least number of significant digits rather than the least number of decimal places.

1.0 = .33
3.0

In the first of these, 1.0 and 3.0 both have two significant digits, so the value calculated from 1.0 divided by 3.0 is rounded off to two digits.

1.00000 = .33
3.0

In the second of these examples, going across, 1.00000 is very precise with six digits; but 3.0 only has two digits, so the answer is rounded off to two digits.

1 = .1
8

Now, this one you may not like because 1/8 is not equal to .1. It is supposed to be .125, and it is if you have exactly 1/8. But if the one and the eight are both measurements made with only one significant digit of precision, then the answer must also be rounded off to one digit. Now, if a person happened to know that there was more precision, then they should write down additional digits. (By the way, one significant digit is not very precise for any measurement.) If you want your results to come out to three digits of precision, then your measurements have to be three digits of precision also.

1.000 = 0.1250
8.00000

In the next case, 1.000 has four digits and 8.00000 has six digits, so the answer is rounded off to four digits. Notice here, even though the last digit is a zero, I put it there because it shows the degree of precision to which the original measurements were made. Calculators by the way won't do that for you. They will drop trailing zeros. Consequently you may have to put them in yourself when it is appropriate.

1.0 x 3.00 = 3.0

The same technique is used for multiplication as well as division. Some more examples are shown, 1.0 x 3.00 is equal to 3.0. Two digits because the 1.0 has two digits. The zero is there because it is the second significant digit and it shows the degree of precision to which the number is known.

4.321 x 1.2 = 5.2

Next, 4.321 x 1.2. The answer will be rounded off to two digits because the 1.2 only has two digits.

Let me comment about why we can usually use the least number of significant digits as the key to where to round off the answer. Remember, we are rounding off to indicate the degree of precision or uncertainty in a calculated value. Precision is the part of the number you do know and uncertainty is the part you don't know. You round off at the dividing line between them.

2 = .00428266
467

Let's take this calculation as our starting point.

2 = .00428266
467

2 = .0042735
468

The denominator is a three digit number. We are not completely certain about that last digit. It says 7 but it could be an 8. If it were, the calculated value would be different. That difference shows up in the third significant digit, matching the position of the last digit in these three digit numbers in the denominator.

2 = .00428266
467

3 = .00642398
467

The numerator is a one digit number. It says 2 but it might possibly be a 3. If it were, the calculated answer would again be different. In this case the difference shows up in the first digit, matching the position of the last digit in these one digit numbers.

Depending on the numbers used, the match-up of variation and significant digits will not always work so smoothly, but it will be close. So we will round off calculations to the least number of significant digits rather than calculate out the various possibilities, unless of course you want to.

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E-mail instructor: Eden Francis