Enter numb. or expr. (e.g. "5.13 * 3.78")
Number or expression
Round to sig fig (optional)

# Significant Figures Calculator - Sig Fig

By Mateusz Mucha, Daniel Trojanowski and Bogna Haponiuk

The significant figures calculator converts any number into a new number with the desired amount of sig figs AND solves expressions with sig figs (try doing `3.14/7.58-3.15`). What are significant figures rules? Those concepts will be explained throughout this page as well as how to use a sig fig calculator.

## What are significant figures?

Significant figures are all numbers that add to the meaning of the overall value of the number. To prevent repeating figures that aren't significant, numbers are often rounded. One must be careful not to lose precision when rounding. Many times rounding numbers are for the purpose of simplicity only. Use the rounding calculator to assist in such problems.

## What are the significant figures rules?

To determine what numbers are significant and which aren't, use the following rules:

1. The zero to the left of the decimal value less than `1` is not significant.
2. All trailing zeros that are placeholders are not significant.
3. Zeros between non zero numbers are significant.
4. All non zero numbers are significant.
5. If a number has more numbers than the desired numbers of significant digits, the number is rounded. For example, `432,500` is `433,000` to `3` significant digits.
6. Zeros at the end of numbers which are not significant but are not removed, as removing would affect the value of the number. In the above example, cannot remove `000` in `433,000` unless changing the number into scientific notation.

## How to use the sig fig calculator

Our significant figures calculator works in two modes - it performs arithmetic operations on multiple numbers (for example `4.18 / 2.33`) or simply rounds a number to a desired number or sig figs.

Following the rules noted above, we can calculate sig figs by hand or by using the significant figures counter. Suppose we have the number `0.004562` and want `2` significant figures. The trailing zeros are placeholders, so we do not count them. Next we round `4562` to `2` digits, leaving us with `0.0046`.

Now we'll consider an example that is not a decimal. Suppose we want `3,453,528` to `4` significant figures. We simply round the entire number to the nearest thousand, giving us `3,454,000`.

What if a number is in scientific notation? In such cases the same rules apply. Note that the significant figures calculator does not convert numbers into scientific notation. The scientific notation calculator is an excellent tool to accomplish this.

When dealing with estimation, the number of significant digits should not be more taking the log base `10` of the sample size and rounding to the nearest integer. For example, if the sample size is `150`, log of `150` is approximately `2.18`, so we use `2` significant figures.

## Significant figures in operations

There are additional rules regarding the operations - addition, subtraction, multiplication and division.

• The result of an operation cannot have more significant figures that the value with the least number of significant figures. For example, when performing the operation `12.13 + 1.72 + 0.45`, the value with the least number of sig figs (2) is 0.45. Hence, the result must have two significant figures as well: `12.13 + 1.72 + 0.45 = 14.30 = 14`.
• If performing addition and subtraction only, it is sufficient to do all calculations at once and apply the significant figures rules to the end result.
• If performing multiplication and division only, it is sufficient to do all calculations at once and apply the significant figures rules to the end result.
• If, however, you do mixed calculations - addition/subtraction and multiplication/division - you need to round the value for each step of calculations to the correct number of significant figures. For example, for the calculation `12.13 + 1.72 * 3.4`, after first step you will obtain the following result: `12.13 + 5.848`. Then, you have to round the result of multiplication to 2 significant figures, obtaining `12.13 + 5.8`. Now, simply add the numbers and leave two significant figures, obtaining the result of `12.13 + 5.8 = 17.93 = 18`.
• Exact values, including defined numbers such as conversion factors and 'pure' numbers, don't affect the accuracy of calculation. They can be treated as if they had an infinite number of significant figures. For example, when using the speed conversion, you need to multiply the value in m/s by 3.6 if you want to obtain the value in km/h. The number of significant figures is still determined by the accuracy of the initial speed value in m/s - for example, `15.23 * 3.6 = 54.83`.

Mateusz Mucha, Daniel Trojanowski and Bogna Haponiuk

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