Updated: Nov 7, 2022

How Many Significant Figures? A question that caused all sorts of problems for me in my younger days. It seems that others have similar problems. Here, then, is a short speedy summary to get raise your confidence...

There are 3 rules to guide you when you are determining the number of significant figures in your calculations..

1. Non-zero digits are significant.

2. Zeros between digits are significant.

3. Zeros to the right of a decimal point and right of a non-zero digit are significant.

Scan these rules a couple of times then try the following quiz (answers at the end).

**Quiz: Work out how many significant figures are in these numbers**

0.40

0.00403

4.30

430

4.3000

430.00

4.3

4 321 000

4 321.123

432

4.0003

40.000030

## Adding and Subtracting Numbers

In the number 192.67, the ‘7’ is the digit whose value holds the least confidence.

Confidence level *decreases* as you move through numbers from left to right. It follows that an answer cannot be quoted to a greater precision than your most confidently estimated number. You should quote your answer to the left-most place containing an estimated figure.

Look at the following column addition...

I have coloured the least confident/estimated figures red. The answer shows that all the decimal places are estimates since the sum was found by using estimated figures in the tenths, hundredths and thousandths positions of the numbers.

The left-most estimate is ‘9’ which represents the most confident estimate - the answer should be quoted to this decimal place. Rounding from the ‘6’, we arrive at 119.0 to 4 significant figures.

Let’s look at a subtraction (estimated digits in red).

**98.63 - 7.6 = 91.03**

In the answer as shown, the last digits are considered estimates. This number should be rounded to the *left-most estimated digit* to give 91.0 (to 3 significant figures).

## Multiplying and Dividing Numbers

With multiplying and dividing, the answer we arrive at needs to have the same number of significant digits as the value having the least number of significant digits.

Suppose I want to multiply 87.6 by 1.5

87.6 has 3 significant figures and 1.5 has 2. Our answer, 131.40 with 5 significant figures cannot be quoted as is. Looking at the two numbers used, we need to quote our answer to two significant figures, since this is the smallest number of significant figures in the calculation.

In our answer, the 1,4 and 0 are considered estimates.

Our answer becomes 130 to 2 significant figures (131 would be 3 significant figures).

Worth repeating…

When multiplying or dividing, your answer should be quoted to the same precision as the number with the smallest significant figures in your calculation.

Answers to the Quiz

0.40 2 significant figures

0.00403 3 significant figures

4.30 3 significant figures

430 2 significant figures

4.3000 5 significant figures

430.00 5 significant figures

4.3 2 significant figures

4 321 000 4 significant figures

4 321.123 7 significant figures

432 3 significant figures

4.0003 5 significant figures

40.000030 8 significant figures

Like/comment as you see fit. Any mistakes are all my own.

Adrian