Significant digits, also known as significant figures, is foundational to physics. Scientific notation works hand in hand with significant digits to carry information about the precision and accuracy of measurements. If you did the optional lesson, you had a chance to work with measurements from lab devices. This gives you some experiential understanding related to why significant digits are needed.

Course Competencies and Learning Objectives

A ★ indicates that this page contains content related to that LO.

CC1.1 Solve problems related to scientific units of measurement and significant figures

LO1.1.1 Identify the terms used in scientific communication

★ LO1.1.2 Apply the rules of significant figures

LO1.1.3 Identify SI and imperial units

LO1.1.4 Match the SI prefixes with their corresponding values

LO1.1.5 Apply unit conversion

LO1.1.6 Demonstrate units displayed in the correct style

Reading

Appendix A3 "Significant Figures" of Physical Science, 13th edition by Bill Tillery McGraw Hill Education

Significant Figures

This video goes over the arithmetic rules of significant figures. Be sure to take note of the differences in the rules for addition vs multiplication.

Summary

Explains the purpose of significant figures in scientific measurement and computation.
Emphasizes that measurements have limited precision, and results should not claim more precision than the inputs.
Demonstrates how to add numbers with different precision and why only measured digits are significant.
Shows how to use significant figures in addition, subtraction, multiplication, and division.
Explains the difference between measured and exact quantities.
Provides worked examples and rules for rounding results to the correct number of significant figures.

Time: 11:20

ASL version of Significant Figures

Direct link to non-ASL version

Identifying & Counting Figures

The first step when doing an analysis of significant figures is to identify and count the figures. All non-zero digits are considered significant, but only some types of zero figures are significant depending on their position. The reason for the difficulty with zero is because zero has two functions when writing numbers (i) the value and (ii) a placeholder. Here are the rules for when zero is counted as significant:

for whole numbers, zeros to the right of the last non-zero digit are not significant unless otherwise indicated as such
- 1200 has two significant digits (the 1 and the 2)
- 820 has two significant digits (the 8 and the 2)
- -335,000 has three significant digits (the 3, the 3, and the 5)
- $335 \overset{—}{0} 00$ has four significant figures (the 3, the 3, the 5, and the 0 with a bar over it)
for values with magnitudes less than one, zeros to the left of the first non-zero digit are not significant

0.0035 has two significant digits (the 3 and the 5)
-0.0556 has three significant digits (the 5, the 5, and the 6)
0.000,000,1 has one significant digit (the 1)

for numbers larger than one, all digits after the decimal point are significant

1.0 has two significant digits (the 1 and the 0)
8.20 has three significant digits (the 8, the 2, and the 0)

zeros between any two significant digits are significant

1200.10 has six significant digits (each digit, including the three 0's are significant)
10.04 has four significant digits (each digit, including the two 0's are significant)
$3350 \overset{—}{0} 0$ has five significant figures (the 3, the 3, the 5, the 0 with a bar over it, and the 0 before it)

Exact Numbers

The counting of significant figures is limited to quantities which are measured. Often times, there are values that are used with those measured quantities whose value is exact, and thus has infinite precision and an infinite number of significant figures. Here are some examples:

integers and fractions, e.g.: 1/2 = 0.5
mathematical constants, e.g.: π = 3.1415926....
scientific constants, e.g.: speed of light, c = 299 792 458 m/s
unit conversion factors, e.g.: 1000 to convert kilograms to grams, and 2.54 centimeters per inch

Addition & Subtraction

When adding and/or subtracting values the result has a precision that is limited by the precision of the inputs. The number of digits past the decimal point is limited to the least significant digit of all the inputs. Here is an example:

55.6789 + 3.412 = 59.0909 \to 59.091

In the example above, the first term is precise to four digits past the decimal point, and the second term is precise to only three digits past the decimal point. The resultant sum cannot be more precise than the least precise input, so the answer must be rounded to three digits past the decimal point.

Multiplication & Division

There is a similar, but different rule with respect to multiplication and division. With these operations, the result is limited by the minimum number of significant figures of all the inputs as opposed to the position of the least precise digit. The number of significant figures in the result is the minimum number of significant figures of all the inputs. Here is an example:

22.179 \times 1.3 = 28.8327 \to 29.

In the example above, the first factor has five significant figures and the second has two. Because the product cannot be more precise than any of the inputs the number of significant figures in the result is rounded to two. This rule also applies to division, roots, and powers.

Working with Scientific Notation

In this video we go over what scientific notation is, how to use it on a calculator, and arithmetic with it.

Summary

Introduces two methods for computing with scientific notation: using a scientific calculator and manual calculation.
Explains how to enter numbers in scientific notation on calculators and in computer programs.
Demonstrates manual calculation by separating numbers and exponents, performing arithmetic on each, and recombining.
Walks through a detailed example multiplying and dividing numbers in scientific notation.
Provides practice problems for students to try on their own.

Time: 4:20

ASL Version of Working with Scientific Notation

Direct link to non-ASL version

Practice and Apply - Computational

Important: Strict Answer Formatting Required

When entering answers in quizzes and mastery assessments, you must follow strict formatting rules. Answers will only be marked correct if they match the required format exactly.

Significant Figures: Use the correct number of significant digits. Do not add extra digits or zeros unless they are significant.
Scientific Notation: Enter scientific notation as 1.2*10^6 (not 1.20*10^6 unless two decimals are significant).
No Extra Characters: Do not include units, spaces, or unnecessary symbols unless instructed.

Examples:

Correct: 4.42 Incorrect: 4.420 or 4.4
Correct: 1.2*10^6 Incorrect: 1.20*10^6 or 1.2 x 10^6

If your answer is not accepted, double-check your formatting!

PROMPT Compute the following quantity with the correct number of significant figures: 11.2 · 39.18

Answer

The operation here is multiplication, so we can compute:

11.2 · 39.18 = 438.816

Now, we need to round to the correct number of digits. Because this is multiplication, the number of significant figures is the minimum number of significant figures of the inputs. In this problem, the first number has three significant figures (the 1, the 1, and the 2), and the second number has four significant figures (the 3, the 9, the 1, and the 8). Therefore, the number of significant figures in the result is min(3, 4) = 3; don't forget to round:

438.816 → 439.

PROMPT Compute the following quantity with the correct number of significant figures: 1.2×10⁶ + 980

Answer

Both numbers have two significant figures. The digits in the first number that are significant are the 1 and the 2, and the two numbers in the second that are significant are the 9 and the 8; the 0 is not significant. To perform the addition, we need to put both numbers in the same form:

1.2×10⁶ + 0.00098×10⁶ = 1.20098×10⁶

Because the operation is addition, we need to look for the least precise digit. In the first number, the least precise digit is one digit past the decimal point, and in the second it is the fifth. The least precise digit in the result is the minimum of those two, which is 1. Therefore, the result is rounded to the first digit past the decimal point:

1.20098×10⁶ → 1.2×10⁶

Because the first number is so large and has fairly limited precision, the addition of a relatively small number has no change.

To enter this value in Canvas, you would type: 1.2*10^6

PROMPT Compute the following quantity with the correct number of significant figures: 2.47 · 3.1

Answer

This is a multiplication problem. First, multiply the numbers:

2.47 · 3.1 = 7.657

Now, determine the correct number of significant figures. 2.47 has three significant figures, and 3.1 has two significant figures. The result should have the minimum, which is two significant figures.

7.657 → 7.7

So, the answer is 7.7.

PROMPT Compute the following quantity with the correct number of significant figures: 4.0572 · 1.09

Answer

Multiply the numbers:

4.0572 · 1.09 = 4.422348

4.0572 has five significant figures, 1.09 has three. The result should have three significant figures.

4.422348 → 4.42

So, the answer is 4.42.

Note: In Canvas, enter your answer as 4.42 (do not include extra digits or trailing zeros unless they are significant).

PHYS-1315 M1L1 Significant Figures and Scientific Notation

Course Competencies and Learning Objectives

Reading

Significant Figures

Identifying & Counting Figures

Exact Numbers

Addition & Subtraction

Multiplication & Division

Working with Scientific Notation

Practice and Apply - Conceptual

Sort Items

Answer Bank

Two Significant Figures

Three Significant Figures

Practice and Apply - Computational

Important: Strict Answer Formatting Required