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What Are Significant Figures?

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In this tutorial about significant figures, we will cover their definition, relevant guidelines, and historical context. 

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What are Significant Figures?

Significant Figures, often referred to as “Sig Figs,” are specific digits that denote the degrees of precision exemplified by different numbers. We can classify certain digits as significant figures; others, however, we cannot. A given digit’s status as either significant or non-significant stems from a checklist of criteria.

Rules for Determining Significant Figures 

What Constitutes a Significant Figure?

First, let’s review these criteria that define sig figs. We can classify numbers as significant figures if they are: 

  1. Non-zero digits 
  2. Zeros located between two significant digits 
  3. Trailing zeros to the right of the decimal point 
  4. (For digits in scientific notation format, N x 10x)
  • All digits comprising N are significant in accordance with the rules above
  • Neither “10” nor “x” are significant

Specific amounts of precision, designated by significant figures, must appear in our mathematical calculations. These appropriate degrees of precision vary, corresponding to the type of calculation being completed. 

To determine the number of sig figs required in the results of certain calculations, consult the following guidelines. 

Rules for Addition and Subtraction Calculations:

  1. For each number involved in the problem, quantify the amount of digits to the right of the decimal place–these stand as significant figures for the problem.
  2. Add or subtract all of the numbers as you normally would. 
  3. Once arriving at your final answer, round that value so it contains no more significant figures to the right of its decimal than the LEAST number of significant figures to the right of the decimal in any number in the problem.

 Rules for Multiplication and Division Calculations:

  1. For each number involved in the problem, quantify the amount of significant figures using the checklist above. (Look at each whole number, not just the decimal portion).
  2. Multiply or divide all of the numbers as you normally would. 
  3. Once arriving at your final answer, round that value so that it contains no more significant figures than the LEAST number of significant figures in any number in the problem.
Learn amazing facts about significant figures, including their definition, relevant guidelines, and historical context.

Origination of Significant Figures 

We can trace the first usage of significant figures to a few hundred years after Arabic numerals entered Europe, around 1400 BCE. At this time, the term described the nonzero digits positioned to the left of a given value’s rightmost zeros. 

Only in modern times did we implement sig figs in accuracy measurements. The degree of accuracy, or precision, within a number affects our perception of that value. For instance, the number 1200 exhibits accuracy to the nearest 100 digits, while 1200.15 measures to the nearest one hundredth of a digit. These values thus differ in the accuracies that they display. Their amounts of significant figures–2 and 6, respectively–determine these accuracies.

Scientists began exploring the effects of rounding errors on calculations in the 18th century. Specifically, German mathematician Carl Friedrich Gauss studied how limiting significant figures could affect the accuracy of different computation methods. His explorations prompted the creation of our current checklist and associated rules.

Practice Problems

Determine the amount of significant figures in the following values and problems. Use the checklist and rules discussed in this article. 

1. 0.00784 

2. 1.056   

3. 500 

4. 700.  

5. 0.0114 x 104   

6. 8.9568 + 13.75 

7. 33.85 x 806.5988 

Answer Key: 3, 4, 1, 3, 22.71 (round to 2 significant figures), 7. 29672.6 (round to 6 significant figures)

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