CHEM 1053: Significant figures in calculations

When keeping track of significant figures in calculations, exact numbers do not affect the number of significant figures in your answer. You could consider them to have an infinite number of significant figures. There are three sources of exact numbers. The first is when you're counting discrete objects. I might say there are 25 students in a classroom. This number is exact because I couldn't have 25.5 students. The second source of an exact number is in defined quantities. In the conversion factor shown, 1 meter is exactly 1000 millimeters by definition. We'll use conversion factors in dimensional analysis. These types of defined quantities will not affect our number of significant figures in our answer. The third source is for integral numbers that are part of an equation. For example, the radius of a circle is equal to the diameter of the circle divided by 2. The number 2 does not affect our significant figures in the calculation. We have different rules for significant figures in multiplication or division and addition or subtraction. For multiplication or division, the number of significant figures in our answer can only have as many significant figures as the factor with the fewest number of significant figures. For example, if we were multiplying these three measured quantities together, our calculator would say that the answer is 75,107.142. A calculator does not keep track of significant figures. To determine how many significant figures we can keep in our answer, let's look at how many significant figures each factor has. 3.505 has four significant figures, 120.000 has six significant figures, and 0.0056 has two significant figures. Our answer can only keep as many significant figures as the factor with the fewest number of significant figures. So we can only keep two significant figures in our answer. This means we will round our answer down to 75,000. However, if we write out our answer in decimal form, the number of significant figure is ambiguous. So we'll need to report our answer in scientific notation to show that we have just two significant figures. In addition or subtraction, we don't keep track of the number of significant figures in each factor. We only look at how significant each number is with respect to the decimal point. For example, if we add these five numbers together, our calculator will report the answer as 1292.74339. Again, our calculator doesn't keep track of significant figures. Let's look at each number with respect to the decimal point. The number 3.5700 is significant to the ten thousandth's digit or four numbers after the decimal point. If this was a length, it would have been measured with an instrument that could tell the difference between 3.570 and 3.571. And the last digit was estimated so that the overall measurement was 3.5700. Our next measurement is 10.070, which is significant to the thousandths digit, so it was measured with a different instrument. We can continue down to the number 958, which is significant to the one's digit. This means the measurement was made with an instrument that could tell the difference between 950 and 960, and the last digit—the one's digit—is estimated. This is our least significant number with respect to the decimal point. So this limits our answer to the one's digit. We cannot keep numbers in our answer beyond that or else it will communicate that all of our measurements were made with more precise instruments. We'll round our answer to the one's digit so that it becomes 1293. Notice that our answer has four significant figures in it even though some of the numbers in our addition had only three significant figures. With addition and subtraction, we only look with respect to the decimal point. If we have mixed calculations containing both addition or subtraction and multiplication or division, we need to apply our rules for significant figures according to the order of operations in the problem. If we enter this example into our calculator, we'll get the answer 0.5. To determine how many significant figures we need, we should complete the addition in the numerator first, and then we can complete the division according to the order of operations. Whenever you have a number in scientific notation and you want to keep track of significant figures for addition or subtraction, you can convert the number to decimal form. We do this because the rules for addition and subtraction are with respect to the decimal point. If you add these numbers together in your calculator, you would just get the number 2. But calculators don't know how many significant figures we should keep. So with respect to the decimal point, the least significant number in our calculation is significant to the hundredths digit or two digits past the decimal point. So our answer should be significant to the hundreds digit. This means we have three significant figures after we complete the addition in the numerator. Now we can consider the division. For division and multiplication, we have to look at the number of significant figures in each factor. In the numerator, we've already said we have three significant figures. In the denominator, we have four significant figures in the number 4.000. So our answer can only have as many significant figures as the factor with the fewest significant figures. Our answer needs to have three significant figures, so we do need to add two additional zeroes so that our answer is 0.500.