# Significant Figures in Calculation

Significant figures in calculations are used to keep the precision of the calculation in check. The more significant figures the more precise the measurement is. It is understood that only the last digit is uncertain. However, let us say that we have two numbers that have different significant figures, or even containing different decimal places. When calculating the answers for any problem with significant figures, how does one know how many significant figures to use for the answer? One golden rule to follow is that the answer cannot be more precise, containing more significant figures, than any of the measurements used to find that answer. When carrying out computations involving significant figures, there are two sets of rules to abide by that cover the four basic math computations.

### Addition and subtraction Rule

In addition and subtraction the result can have no more decimal places than the measurement with the fewest number of decimal places. In other words, find the number with the lowest decimals and that will show you the number of decimals to use in your answer.

The sum in the addition problem was 49.
335 in normal calculations (Containing 3 decimal places). However, out of the set of numbers, 12 had the lowest numbers of decimal places (Containing 0 decimal places), the answer must also contain zero decimal places therefore, making the answer 49.

The difference of the subtraction problem was 7.
116 in normal calculations(Containing 3 decimal places). However with .10 having only two decimal places, the answer can only contain two decimal places. Due to the fact that the digit after the cut off point was a 6, which is more than the middle digit 5, one can round up the cut off number up, thus making the answer 7.12 (Containing 2 decimal places).

### Multiplication and Division Rule

In multiplication and division the result must be reported with the same number of significant figures as the measurement with the fewest significant figures. This is different compared to addition and subtraction because it is not decimal places that are used, but rather significant figures.
Example:

Calculate the density of a sample with mass 25.624 g and a volume of 25 mL.

D=M/V, so 25.624g/25mL=1.02496 g/mL, but we record the answer simply as 1.0 g/mL to meet the significant figure requirement.

The answer through simple calculations contained 6 significant figures, but the lowest significant figure in number set, 25 mL, contained 2 significant figures, thus the answer must also contain 2 significant figures.

### Using Scientific Notation:

If your calculation gives you a long number that needs to be condensed to show the correct number of significant figures, one can use scientific notation to display the correct number of significant figures.

Ex.1) 100,000 needs to be shown with 3 significant figures in a calculation. One can show 100,000 = 1.00 x 10^5. The 1.00 portion shows the 3 significant figures. The 10^5 does not count in the significant figures.

(See above diagram as well.)

### Rounding

When you are at the cut off number, you round it based on the next digit to the right of the cut off number. If it's 4 and below keep the cut off digit the same. If the number after the cut off digit is 5 or above, round it up.

Ex. 1.456 = 1.46

The cut off number is 5, and the number after the 5 is a 6, which is greater than 5, so the cut off digit is rounded up to a 6.

### Rounding to Even

Normally when the number after the cut off number is a 5, there is usually always another number there that will make it "weigh" more, which is why the number 5 usually makes things round up.

If by some chance the number is 4.655(000000000 etc.) and you needed 3 significant figures, the number after the cut off digit is 5, but all the digits after that are assumed as 0000 infinitely. The 5 in this case, doesn't have any digits behind it to make it "weigh" enough to round up. In this case we use the round to even rule, where we round the cut off number to the closest even number.

Ex. 4.655 = 4.66
The 5 is rounded to 6 because it is the closest even number. 4 is also just as close, but its a rule to round up.

Ex. 4.645 = 4.64
The 4 is rounded to 4 because it is the closest even number, since it is already an even number.

### Resources

Brown, LeMay, Bursten. Chemistry: The Central Science. Prentice Hall, 2003.

About.com. "Chemistry." http://chemistry.about.com/library/weekly/aa082701a.htm.

Pictures by: Steve Tran