Significant Figures

Significant Figures, also called Significant Digits, are a way to limit how much accuracy is given in an answer, so that you don't give an extremely exact measurement when your measuring tool's accuracy is mediocre.

Determining Significant Figures

 * Pure Numbers: Pure numbers have infinite significant figures, meaning they are exact. If you have 12 eggs, then 12 is a pure number.  You don't have 12.01 eggs, or 11.9 eggs.
 * Physical Numbers: Physical numbers have finite significant figures. For example, the measured length of a pencil.
 * Estimating Digit: The estimating digit is digit in a measurement where unsurety starts. If you are measuring the length of an object with a normal ruler, you can be pretty sure that the measurement is 13.1 cm, but you are not as sure that it is 13.12 cm.  Thus, the estimating digit is 2.
 * Reading the number of significant figures in other people's measurements: The following rules can be used to determine the number of significant figures in another person's measurement:
 * All non-zero numbers count as significant figures.
 * Zeroes placed between significant figures are always significant; 4009 kg has four significant figures.
 * Zeroes placed after a decimal point and a non-zero number are significant; 4.00 kg has three significant figures and 0.0400 has three significant figures.
 * Digits with bars over them are always significant. Keep in mind that since they are significant, zeroes between them are significant.
 * All other digits are insignificant.
 * Information source.

Addition and Subtraction
When adding or subtracting two physical numbers, you first line up the numbers by their decimal points. You then draw a vertical line to the right of the estimating digit the shortest distance from the decimal point. That line is where you round off your answer.

1.01| +

2.01|3

=

3.02|3

Answer: 3.02.

Multiplication and Division
When multiplying or dividing two physical numbers, the answer will have the same number of significant figures as the number with the least number of significant figures.

2.10 * 3.607 = 7.5747

Answer: 7.57

Exponentiation and Logarithms
When one takes the logarithm of a number, one gives the answer the same number of decimal places as the significant figures of the input. Conversely, when one takes an exponent, the input is considered to only have the number of significant figures as the number of decimal places, but the answer is written normally using that number of significant figures.

The part of the number that comes after the decimal point is called the mantissa.

Example:

log(32.4) = 1.511 (the input has 3 significant figures, so the answer must go to 3 spaces after the decimal point, i.e. the mantissa must have 3 significant figures)

10^2.1331 = 135.9 (the mantissa has 4 significant figures, so the answer must only have 4 significant figures)

Some Flaws
An example that may be found confusing is:

7.02 * 2

Where 7.02 is a physical number and 2 is a pure number.

At first the answer may appear to be 14.0. However, multiplying a physical number by the pure number 2 is the same thing as adding the physical number to itself, by the definition of multiplication. When you add 7.02 to 7.02, you get an answer of 14.04.

This is an example of one of the flaws in the significant figure system. In this scenario it is probably best to assume the answer 14.0, since multiplication was the original case.

The reverse example is:

7.02 + 7.02

Where both 7.02s are physical numbers.

The answer would seem to be 14.04. However, adding a physical number to itself is the same thing as multiplying it by the pure number 2, by the definition of multiplication. When you multiply 7.02 by the pure number 2, you get 14.0.

In this scenario it is probably best to assume the answer 14.04, since addition was the original case.

Basically, if you run into a scenario in which there are two possible numbers of significant figures for the answer depending on the operation used, you should assume the number of significant figures obtained from the original operation.

Events that involve Significant Figures

 * *This is not a comprehensive list. More events than this may involve Significant Figures.

Simple Machines

Compound Machines

Metric Mastery

Experimental Design (C Division)

Technical Problem Solving