How to Accurately Calculate Significant Figures

When you discover the world of measurements and calculations, Significant figures become your trusty guide. Particularly, they help you express the precision of your measurements. Every experimental value is reported according to some rules which are given by German mathematician, Carl Friedrich Gauss. Imagine you’re measuring the length of a pencil. You might say it’s 7.25 inches long. Here, the digits 7, 2, and 5 are your significant digits.

Rules to Find Significant Figures

Understanding the rules is crucial in scientific calculations. Additionally, these rules help you maintain precision and accuracy in your measurements.

Identifying Significant Digits

When you look at a number, how do you know which digits are significant? Here’s a simple guide:

Non-zero digits

All non-zero digits are always significant. They represent the actual measurement and contribute to its precision. For example, in the number 123.45, each digit is significant.

Leading zeros

Leading zeros are those that appear before the first non-zero digit. Besides this, they are not counted. They merely serve as placeholders. For instance, in 0.0025, only the digits 2 and 5 are significant.

Captive zeros

Captive zeros, or zeros between non-zero digits, are always significant. Consequently, they are part of the measurement. Therefore, in the number 1002, all four digits are significant.

Trailing zeros

Trailing zeros can be tricky. In a whole number without a decimal point, they are not significant. However, if there’s a decimal point, they become significant. For example, in 1500, only the digits 1 and 5 are significant. But in 1500.0, all five digits are significant.

Determining the Least Number of Significant Figures

When performing calculations, it’s essential to determine the least number of these figures in your data. This ensures your results are as precise as the least precise measurement.

Identifying the least number in given data

To find the smallest number of significant digits, examine each measurement in your data set. Identify the one with the fewest significant digits. For example, if you have measurements with 3, 4, and 5 significant digits, your result should be reported with 3 significant digits.

Try it Yourself

Take a number like 0.004560 and identify the significant DIgits. Remember the rules: non-zero digits, captive zeros, and trailing zeros with a decimal point are significant. How many significant figures does it have? Comment your answer below!

Addition and Subtraction

When you’re adding or subtracting numbers, understanding how to report correct answer is crucial. This ensures that your results maintain the right level of precision.

Rule for Addition and Subtraction

In addition and subtraction, the focus shifts from the total number of significant digits to the position of the last significant digit.

Aligning decimal points

First, line up the decimal points of the numbers you’re working with. Also, this alignment helps you see which digits are in the same place value. For example, if you’re adding 12.345 and 7.8, you should write them as:

  12.345
+  7.800

Limiting to the least precise decimal place

After aligning, identify the number with the fewest decimal places. Your final answer should have the same number of decimal places as this number. In our example, 7.8 has one decimal place, so your result should also have one decimal place. Therefore, the sum becomes 20.1.

“Precision is the soul of science.” – Henri Poincaré

Examples and Calculations

Physical Chemistry Fundamentals for Young MindsRead more

Let’s look at some examples to see these rules in action.

Example 1

Consider adding 123.456 and 78.9. Align the decimal points:

  123.456
+  78.900

The number 78.9 has the fewest decimal places (one decimal place). Therefore, your result should also have one decimal place. The resulting sum is 202.4.

Example 2

Now, let’s subtract 45.678 from 123.4. Also, align the decimal points:

  123.400
-  45.678

SI Units: The Secret Language of Chemistry!Read more

Here, 123.4 has the fewest decimal places (one decimal place). Therefore, the result should be rounded to one decimal place, giving you 77.7 as the precise answer.

Try it yourself

Try adding 56.789 and 12.3. Remember to align the decimal points and limit your answer to the fewest decimal places. How many decimal places does your final answer have?

Significant Figures in Multiplication and Division

When you multiply or divide numbers, handling significant digits correctly ensures your results are precise and reliable. Let’s explore the rules and see how they apply in practice.

Rule for Multiplication and Division

In multiplication and division, the focus is on the number of significant digits in each measurement. Further, this ensures a precise answer of any calculation. Here’s how you can manage it:

Limiting to the least number of significant figures

1. Identify the Measurement with the Fewest Figures: Look at each number involved in your calculation. Determine which one has the least number of significant digits. This number dictates the precision of your final result.

2. Round Your Answer Accordingly: Once you’ve completed the calculation, round your answer to match the number of significant figures in the least precise measurement. This ensures your result reflects the precision of your data.

“Precision is the mark of a true scientist.” – Marie Curie

Examples and Calculations

Let’s look at some examples to see these rules in action.

Example 1

Consider multiplying 4.56 by 1.4. First, identify the number of significant figures in each number:

• 4.56 has three figures.

• 1.4 has two figures.

Perform the multiplication:

Since 1.4 has the fewest figures (two), round 6.384 to two significant digits. Therefore, the result is 6.4.

Example 2

Now, let’s divide 123.45 by 6.7. Identify the number of significant figures:

• 123.45 has five figures.

• 6.7 has two figures.

Perform the division:

Specifically, round the result to two figures, as 6.7 dictates. The final answer is 18.

Try it yourself

Try multiplying 7.89 by 0.34. Also, remember to limit your answer to the fewest significant figures. Further, calculate how many significant digits does your final answer have? Comment your answer below!

To stay updated with the latest developments in STEM research, visit ENTECH Online. This is our digital magazine for science, technology, engineering, and mathematics.

At ENTECH Online, you’ll find a wealth of information. Besides this, we offer insights and resources to fuel your curiosity. Specifically, our goal is to inspire your passion for new scientific discoveries.

Everything a teen wants to know for career planning.

FAQ

How do you determine the number of significant figures?

To find the number, start counting from the first non-zero digit. Particularly, include all non-zero digits, captive zeros, and trailing zeros if there’s a decimal point. For example, in 0.004560, the significant digits are 4, 5, 6, and 0, totaling four.

How is answer reported in addition and subtraction?

In addition and subtraction, align the decimal points of the numbers involved. Furthermore, limit your answer to the least precise decimal place. For instance, when adding 1.2 and 4.71, the result should be limited to the tenths column, giving you 5.9.

How are significant figures handled in multiplication and division?

For multiplication and division, the result should have the same number of figures as the factor with the fewest significant digits. Consequently, if you multiply 4.56 by 1.4, the result should be rounded to two significant digits, resulting in 6.4 as the answer.