Significant Figures for Physics Help

Introduction

When multiplication or division is done using power-of-10 notation, the number of significant figures in the result cannot legitimately be greater than the number of significant figures in the least-exact expression. You may wonder why, in some of the preceding examples, we come up with answers that have more digits than any of the numbers in the original problem. In pure mathematics, this is not an issue, and up to this point we haven’t been concerned with it. In physics, however, things are not so clear-cut.

Consider the two numbers x = 2.453 × 10 ⁴ and y = 7.2 × 10 ⁷ . The following is a perfectly valid statement in arithmetic:

xy = 2.453 × 10 ⁴ × 7.2 × 10 ⁷

= 2.453 × 7.2 × 10 ¹¹

= 17.6616 × 10 ¹¹

= 1.76616 × 10 ¹²

However, if x and y represent measured quantities, as they would in experimental physics, the preceding statement needs qualification. We must pay close attention to how much accuracy we claim.

How Accurate Are We?

When you see a product or quotient containing a bunch of numbers in scientific notation, count the number of single digits in the decimal portions of each number. Then take the smallest number of digits. This is the number of significant figures you can claim in the final answer or solution. In the preceding example, there are four single digits in the decimal part of x , and two single digits in the decimal part of y . Thus we must round off the answer, which appears to contain six significant figures, to two. It is important to use rounding and not truncation! We should conclude that

xy = 2.453 × 10 ⁴ × 7.2 × 10 ⁷

= 1.8 × 10 ¹²

In situations of this sort, if you insist on being 100 percent rigorous, you should use squiggly equals signs throughout because you are always dealing with approximate values. However, most experimentalists are content to use ordinary equals signs. It is universally understood that physical measurements are inexact, and writing squiggly lines can get tiresome.

Suppose that we want to find the quotient x/y instead of the product xy? Proceed as follows:

x/y = (2.453 × 10 ⁴ )/(7.2 × 10 ⁷ )

= (2.453/7.2) × 10 ⁻³

= 0.3406944444 ··· × 10 ⁻³

= 3.406944444 ··· × 10 ⁻⁴

= 3.4 × 10 ⁻⁴

What About Zeros?

Sometimes, when you make a calculation, you’ll get an answer that lands on a neat, seemingly whole-number value. Consider x = 1.41421 and y = 1.41422. Both of these have six significant figures. The product, taking significant figures into account, is

xy = 1.41421 × 1.41422

= 2.0000040662

= 2.00000

This looks like it’s exactly equal to 2. In pure mathematics, 2.00000 = 2. However, not in physics! (This is the sort of thing that drove the famous mathematician G. H. Hardy to write that mathematicians are in better contact with reality than are physicists.) There is always some error in physics. Those five zeros are important. They indicate how near the exact number 2 we believe the resultant to be. We know that the answer is very close to a mathematician’s idea of the number 2, but there is an uncertainty of up to ±0.000005. If we chop off the zeros and say simply that xy = 2, we allow for an uncertainty of up to ±0.5, and in this case we are entitled to better than this. When we claim a certain number of significant figures, zero gets as much consideration as any other digit.