## Definition

Significant figures are also called significant digits, as they are established in the form of digits. The number of significant digits can be identified by counting all the values starting from the 1st non-zero digit located on the left. These numbers are reliable and necessary to indicate the quantity of a length, volume, mass, measurement, and so on. Arithmetic operations such as addition, subtraction, multiplication, and division are used while calculating significant figures.^{1}

**Significant figures** (also known as the **significant digits**, *precision* or *resolution*) of a number in positional notation are digits in the number that are reliable and absolutely necessary to indicate the quantity of something.

**If a number expressing the result of a measurement (e.g., length, pressure, volume, or mass) has more digits than the number of digits allowed by the measurement resolution, then only as many digits as allowed by the measurement resolution are reliable, and so only these can be significant figures.**

For example, if a length measurement gives 114.8 mm while the smallest interval between marks on the ruler used in the measurement is 1 mm, then the first three digits (1, 1, and 4, showing 114 mm) are certain and so they are significant figures. Digits which are uncertain but *reliable* are also considered significant figures. In this example, the last digit (8, which adds 0.8 mm) is also considered a significant figure even though there is uncertainty in it.

Another example is a volume measurement of 2.98 L with an uncertainty of ± 0.05 L. The actual volume is somewhere between 2.93 L and 3.03 L. Even when some of the digits are not certain, as long as they are reliable, they are considered significant because they indicate the actual volume within the acceptable degree of uncertainty. In this example the actual volume might be 2.94 L or might instead be 3.02 L. And so all three are significant figures.

The following digits are not significant figures.

- All leading zeros. For example, 013 kg has two significant figures, 1 and 3, and the leading zero is not significant since it is not necessary to indicate the mass; 013 kg = 13 kg so 0 is not necessary. In the case of 0.056 m there are two insignificant leading zeros since 0.056 m = 56 mm and so the leading zeros are not absolutely necessary to indicate the length.
- Trailing zeros when they are merely placeholders. For example, the trailing zeros in 1500 m as a length measurement are not significant if they are just placeholders for ones and tens places as the measurement resolution is 100 m. In this case, 1500 m means the length to measure is close to 1500 m rather than saying that the length is exactly 1500 m.
- Spurious digits, introduced by calculations resulting in a number with a greater precision than the precision of the used data in the calculations, or in a measurement reported to a greater precision than the measurement resolution.

Of the significant figures in a number, the **most significant** is the digit with the highest exponent value (simply the left-most significant figure), and the **least significant** is the digit with the lowest exponent value (simply the right-most significant figure). For example, in the number “123”, the “1” is the most significant figure as it counts hundreds (10^{2}), and “3” is the least significant figure as it counts ones (10^{0}).

Significance arithmetic is a set of approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as propagation of uncertainty.

Numbers are often rounded to avoid reporting insignificant figures. For example, it would create false precision to express a measurement as 12.34525 kg if the scale was only measured to the nearest gram. In this case, the significant figures are the first 5 digits from the left-most digit (1, 2, 3, 4, and 5), and the number needs to be rounded to the significant figures so that it will be 12.345 kg as the reliable value. Numbers can also be rounded merely for simplicity rather than to indicate a precision of measurement, for example, in order to make the numbers faster to pronounce in news broadcasts.^{2}

## Who

The concept of significant digits was not taught when I was in K-12, or in my undergraduate or graduate studies. If you are planning on a career in applied mathematics, this topic is essential for you.

## What

Significant Figures refer to the number of important single digits from 0 to 9 in the coefficient of the expression that conveys the message accurately. These significant figures help engineers or scientists in asserting the quantity of any measurement, length, volume, or mass. For example, 453 has three significant figures.

The two main applications to understand significant figures are – Precision and Accuracy. Let’s learn how these two terms play an important role in real-life situations when it comes to the concept of significant figures.

**Precision** – The closeness between two or more quantities to each other under the same condition is called precision. In precision, the level of measurement when repeated gives the same result and the individual measurement agrees to each other.

**Accuracy** – The closeness between the measurement and the accurate number is called accuracy. Accuracy helps in providing consistent results with no error along with accuracy in the result.^{1}

### Rounding Significant Figures Rules^{3}

- Non-zero digits are always significant
- Zeros between non-zero digits are always significant
- Leading zeros are never significant
- Trailing zeros are only significant if the number contains a decimal point

## Why

See Theoretical Knowledge Vs Practical Application.

## How

Many of the **References** and **Additional Reading** websites and **Videos** will assist you with determining significant digits.

As some professors say: “It is intuitively obvious to even the most casual observer.”

## References

^{1} “Significant Figures – Definition, Rules, Rounding Significant Figures, Solved Examples, Practice Questions, Faqs”. 2022. *Cuemath*. https://www.cuemath.com/numbers/significant-figures/.

^{2} “Significant Figures – Wikipedia”. 2013. *en.wikipedia.org*. https://en.wikipedia.org/wiki/Significant_figures.

^{3} “Rounding Significant Figures Calculator”. *CalculatorSoup, LLC*. 2022. https://www.calculatorsoup.com/calculators/math/significant-figures-rounding.php.

## Additional Reading

Cole, T J. 2015. “Too Many Digits: The Presentation Of Numerical Data”. *Archives Of Disease In Childhood* 100 (7): 608-609. doi:10.1136/archdischild-2014-307149. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4483789/.

“Lesson 3: Rounding And Significant Figures”. 2022. *water.mecc.edu*. https://water.mecc.edu/courses/env148/lesson3b.htm.

“Significant Digits Definition (Illustrated Mathematics Dictionary)”. 2022. *mathsisfun.com*. https://www.mathsisfun.com/definitions/significant-digits.html.

“Intro To Significant Figures (Video) | Khan Academy”. 2022. *Khan Academy*. https://www.khanacademy.org/math/arithmetic-home/arith-review-decimals/arithmetic-significant-figures-tutorial/v/significant-figures.

“Following are Summaries from Two Chemistry Education Web Sites Concerning Significant Figure Rules”. 2022. *ruf.rice.edu*. http://www.ruf.rice.edu/~kekule/SignificantFigureRules1.pdf.

“Rules for Significant Figures (sig figs, s.f.)” 2022. *physics.smu.edu*. http://www.physics.smu.edu/cooley/phy3305/sigfigs.pdf.

“Significant Digits — From Wolfram Mathworld”. 2022. *mathworld.wolfram.com*. https://mathworld.wolfram.com/SignificantDigits.html.

“Significant Figures”. 2022. *cxp.cengage.com*. https://cxp.cengage.com/contentservice/assets/owms01h/references/significantfigures/index.html.

“What Are The Rules For Significant Figures – Precision, Accuracy & Examples”. 2022. *BYJUS*. https://byjus.com/chemistry/significant-figures/.

## Videos

Craig Beals answers the question, “How many places after the decimal should my answer have?”. He shows the importance of significant figures, how to count significant figures, and how to accurately show a mathematical answer in the correct number of significant figures.

Significant figures are the number of digits in a value, often a measurement, that contribute to the degree of accuracy of the value. We start counting significant figures at the first non-zero digit.

This video tutorial provides a fast review on significant figures. It explains how to count the number of significant figures by identifying nonzero digits, trailing zeros, in between zeros, and leading zeros. It also explains how to round a number to the appropriate number of sig figs when adding or subtracting and when multiplying or dividing.

What’s the point of significant figures?!? Really, significant figures (or significant digits) can be such a pain in the neck. What is the point? **Significant figures tell us how to round, but they also make sure that the answer we get from a math problem is not more precise than the numbers that we started with.** Accuracy and precision are both important with significant figures and digits.

Craig Beals answers the question, “How many places after the decimal should my answer have?”. He shows the importance of significant figures, how to count significant figures, and how to accurately show a mathematical answer in the correct number of significant figures.