Your calculator probably gives you answers to twelve or fourteen digits. But those are usually not all reliable.
Every measurment has some uncertainty associated with it.
How long is this block?
With the equipment (the ruler) and your skill, you may decide that it is
L1 = 1.31 m
long. With better equipment, you might be able to measure it more accurately. But you know it is between 1.3 m and 1.4 m in length and you decide it is 1.31 m long. To indicate that digits beyond this are uncertain, we might write
L1 = 1.31????? m
Now measure the length of another block,
This time we have better equipment--perhaps just a ruler with better markings. Now we decide that the length of this block is
L2 = 0.548 m
That means we know its length is between 0.54 m and 0.55 m. To indicate that digits beyond this are uncertain, we might write
L2 = 0.548???? m
What is the length of these two block placed together?
Of course, we "know" the total length is just the sum of the two lengths,
Ltot = L1 + L2
But how do we handle the fact that the two blocks' lengths are known with different uncertainties or with different accuracies? Write the addition with the trailing question marks to understand this better,
While the final digit 8 or .008 in L2 = 0.548 m may be quite well known, the corresponding part of L1's measurement is not well known at all. Only the final 1 or 0.01 in L1 = 1.31 m is known. That means the third digit beyond the decimal in Ltot is uncertain and, therefore, meaningless. So we write it only as Ltot = 1.85 m.
This prototype addition problem provides the generalization we need for significant figures in adding and subtracting.
In addition and subtraction, the number of decimal places in the answer is equal to the smallest number of decimal places in any of the terms being added or subtracted.
What about multiplication and division?
Again, let us start with block number one,
With the equipment (the ruler) and your skill, you decide, as before, that it is
L = 1.31 m
long. You know it is between 1.3 m and 1.4 m in length and you decide it is 1.31 m long. To indicate that digits beyond this are uncertain, we might write
L = 1.31????? m
We are going to find the area of this block, so measure its width.
And we might determine that its width is
W = 0.2345 m
with our better equipment, skill, and care.
Now what is the area?
We know that area is length multiplied by width,
A = L W
So, if we just enter that into a calculater we find,
A = L W = (1.31 m)(0.2345 m)
A = 0.307 195 m2
But is that reasonable? Do we really know the area that accurately? Carry out the multiplication by hand,
and we find that the calculator is, indeed, correct. Now, as before, let's add question marks to these number to indicate which digits are uncertain and again carry out the mathematical operation by hand,
You might think that we should have A = 0.307 1?? m2. But the mathematics is even "worse" than that. Because we don't know what 4 + 5 + ? is in the long-hand multiplication above, we do not know that the next digit, from 3 + 3 + 5 is really going to be the 1 from 11. So that digit, too, is uncertain, as we have written above.
Again, we will generalize from this prototype for multiplication (and division).
When multiplying (or dividing) quantities, the number of significant figures in the answer is equal to the smallest number of significant figures in the quantities.
In our example, the length L has three significant figures while the width W has four significant figures. The answer, the area A, therefore, is known only to three significant figures.
Calculators provide a vast array of digits. Do not be misled. Remember, GIGO (Garbage In, Garbage Out). You must determine which of those figures from your calculator is significant.
Order of Magnitude Scientific Notation Return to Table of Contents, Ch 1 Introduction
(c) 2005, Doug Davis; all rights reserved