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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. Some examples of this are shown in Example 24 in your workbook.
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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. |
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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. |
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Now, the next 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. |
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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. |
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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. |
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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.
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Let's take this calculation as our starting
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2 = .00428266
467 2 =
.0042735
468 |
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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 |
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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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