Adding and SubtractingReference > Science > Significant Figures
A couple summers ago, we had contractors digging a foundation hole for our new home. We had marked a point on an existing building as the top of the foundation wall, and the contractors worked down from there to determine how deep to dig the hole. Later, when we had doubts about whether they'd done it correctly, we had to combine two measurements to find an actual depth: the height from the ground to the mark on the existing building, and the height from the bottom of the hole to the ground. Because of the positioning of the mark on the wall (22 feet from the actual hole), we could measure the depth below ground more accurately than the height above ground.
Let's say that we measured the height from the mark to the ground as 1.2 meters, and the height from the ground to the bottom of the hole as 1.135 meters. That gives us a total of 2.335 meters. But let's talk for just a second what the numbers mean:
1.2 meters means that we measured in meters, to a tenth of a meter.
1.135 meters means that we measured in meters, to a thousandth of a meter.
But if we only measured 1.2 meters to the tenth of a meter, is our sum really precise to a thousandth of a meter? No, it's not! Remember that the last place in the 1.2 is actually an estimate; it could be 1.1 or 1.3, which means that number could be off by as much as 10 centimeters in either direction! And if we could be off by 10 centimeters, does it really make sense to give an answer to a tenth of a centimeter? It really doesn't.
So this is what we do. We look at the number which is least precise, and its least significant digit has to be the last significant digit in our sum. So we do it like this:
1.2 + 1.135 -------- 2.335
1.2 is our least precise number, and the tenths place is its last significant digit. Therefore the tenths place of our sum (that's the first three) is the last significant digit. This means that everything after that place needs to go away. So we look at the decimal place to the right of our last sig fig, and we round. 2.335 rounds to 2.3 meters.
Example: Calculate 10200 + 121.1 + 35.
First, we add all of these together: 10200 + 121.1 + 35 = 10356.1
However, not all of these are significant digits. Which of our three numbers is least precise? 10200 has its last significant digit in the tens place, so this must be the last significant digit in our answer. So we round our answer to the nearest ten: 10360.
Example: Calculate 32500 + 1424 + 120.
Again, we begin by adding these three numbers together: 32500 + 1424 + 120 = 34044.
Now we ask which of our values is least precise. 32500's last significant digit is in the hundreds place, so we must round our result to the hundreds place: 34000. But wait a minute! If we leave our answer like that, we're saying its last significant digit is in the thousands place, not the hundreds. Why? Because the hundreds place is a zero, which, by our rules, is not a significant digit, unless we put an overbar on it. So we write our answer like this: 34000, to indicate that the first zero is a significant digit.
Example: Calculate 1520 + 0.1 - 0.001.
Do the addition and subtraction operations: 1520 + 0.1 - 0.001 = 1520.099.
Which of our values is least precise? 1520, which has its last significant digit in the tens place. Thus, our answer must be rounded to the tens place: 1520. Isn't that strange? We ended up with the same number we started with! It might seem counter-intuitive that this can happen, but since the number 0.1 and -0.001 are much smaller than our least precise measurement, it does make sense that they would not affect our result.