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# Significant Figures Calculator - Sig Fig

By Daniel Trojanowski and Steven Wooding

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 the 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 the goal of rounding numbers is just to simplify them. Use the rounding calculator to assist with 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 number of significant digits, the number is rounded. For example, `432,500` is `433,000` to `3` significant digits (using half up (regular) rounding).
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, we 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 your desired number of 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. To enter scientific notation into the sig fig calculator, use E notation, which replaces `x 10` with either a lower or upper case letter 'e'. For example, the number `5.033 x 1023` is equivalent to `5.033E23` (or `5.033e23`). For a very small number such as `6.674 x 10-11` the E notation representation is `6.674E-11` (or `6.674e-11`).

When dealing with estimation, the number of significant digits should be no more than the log base `10` of the sample size and rounding to the nearest integer. For example, if the sample size is `150`, the 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.

• For addition and subtraction operations, the result should have no more decimal places than the number in the operation with the least precision. For example, when performing the operation `128.1 + 1.72 + 0.457`, the value with the least number of decimal places (1) is `128.1`. Hence, the result must have one decimal place as well: `128.̲1 + 1.7̲2 + 0.45̲7 = 130.̲277 = 130.̲3`. The position of the last significant number is indicated by underlining it.

• For multiplication and division operations, the result should have no more significant figures than the number in the operation with the least number of significant figures. For example, when performing the operation `4.321 * 3.14`, the value with the least significant figures (3) is `3.14`. So the result must also be given to three significant figures: `4.32̲1 * 3.1̲4 = 13.̲56974 = 13.̲6`.

• If performing addition and subtraction only, it is sufficient to do all calculations at once and apply the significant figures rules to the final result.

• If performing multiplication and division only, it is sufficient to do all calculations at once and apply the significant figures rules to the final result.

• If, however, you do mixed calculations - addition/subtraction and multiplication/division - you need to note the number of significant figures for each step of the calculation. For example, for the calculation `12.1̲3 + 1.7̲2 * 3.̲4`, after the first step, you will obtain the following result: `12.1̲3 + 5.̲848`. Now, note that the result of the multiplication operation is accurate to 2 significant figures, and more importantly, one decimal place. You shouldn't round the intermediate result and only apply the significant digit rules to the final result. So for this example, the final steps of the calculation are `12.1̲3 + 5.̲848 = 17.̲978 = 18.̲0`.

• Exact values, including defined numbers such as conversion factors and 'pure' numbers, don't affect the accuracy of the 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`.

To use an exact value in the calculator, give the value to the greatest number of significant figures in the calculation. So for this example, you would enter `15.23 * 3.600` into the calculator.

Daniel Trojanowski and Steven Wooding