# Significant Figures Calculator

The significant figures calculator undertakes calculations with significant figures and works out how many significant figures (sig figs), i.e., digits, a number holds.

Simply input a number or mathematical expression, then click the "Calculate" button for the answer.

## Operators and functions that are supported:

- Arithmetic operators: addition ( + ), subtraction ( - ), division ( / or ÷ ), multiplication ( * or × ) and exponent ( ^ )
- Group symbols: ( )
- Functions: log n, ln n
- Constants: pi, e

## Exact quantities

- In calculating with sig figs one sometimes encounters quantities that are exact as opposed to having limited accuracy. You can enter an exact quantity by appending an # to end of the number; e.g. 3.2#. For instance, if you want to convert 3.00045 g sulfur to moles using sulfur's molecular weight of 32.06 g/mol you would use the following calculation:
- 3.00045 / 32.06# = 0.0935886 mol sulfur
- as opposed to the calculation:
- 3.00045 / 32.06 = 0.09359 mol sulfur
- where 32.06 is taken to only have 4 significant figures and accuracy is thereby lost in this calculation.

## Rules:

This calculator applies a set of rules to determine significant figures. These are outlined below:

- Addition / subtraction rounded to the lowest number of decimal places.
- Multiplication / division rounded to the lowest number of significant figures.
- Logarithms rounded so that a number of significant figures in the input match the number of decimals in the result.
- Exponentiation rounded to the certainty in the base only.
- To enumerate trailing zeros, it places a decimal point after the number (e.g., 100000.) or express it in scientific terms (e.g., 1.00000 × 10^5 or 1.00000e+5).
- Rounds on the last step, following parentheses, when appropriate.

## Describing Significant Figures

When we report values that are derived from a measurement or that were calculated by employing measured values, we need a method by which we can determine the measurement's level of certainty. We can do this by employing significant figures.

Significant figures represent the digits within a value that we have a certain amount of confidence that we know. As the quantity of significant figures rises, the measurement becomes more certain. As the measurement becomes more precise, the number of significant figures increases.

## Rules for significant figures

1) Every digit that is not zero is significant.

- For example:
- 2.437 includes four significant figures
- 327 includes three significant figures

2) When zeros are between digits that are not zeros, they are significant.

- For example:
- 700021 includes six significant figures
- 3049 includes four significant figures

3) When a zero is to the left of the first digit that is not a zero, it is not significant.

- For example:
- 0.003333 includes four significant figures
- 0.00098 includes two significant figures

4) Trailing zeros (zeros which come after the final non-zero digit) are significant if the number contains a decimal point.

- For example:
- 8.000 includes four significant figures
- 800. includes three significant figures
- 0.080 includes two significant figures

5) If the number does not have a decimal point, trailing zeros are not significant.

- For example:
- 500 or 5 × 10^2 only includes one significant figure
- 51000 includes two significant figures

6) In scientific notation, all digits before the multiplication sign are significant.

- For example:
- 1.603 × 10^-4 includes four significant figures

7) The number of significant digits in exact numbers is infinite. This is also true for defined numbers.

- For example:
- 1 meter = 1.0 meters = 1.000 meters = 1.00000000 meters etc.

#### Examples of Significant Figures

Number | # of Sig Figs | Significant Figures |
---|---|---|

100 | 1 | 1 |

100.0 | 4 | 1, 0, 0, 0 |

0.01 | 1 | 1 |

0.05 | 1 | 5 |

71 | 2 | 7, 1 |

125000 | 3 | 1, 2, 5 |

0.10500 | 5 | 1, 0, 5, 0, 0 |

0.0025 | 2 | 2, 5 |

15000.15 | 7 | 1, 5, 0, 0, 0, 1, 5 |

0.0750 | 3 | 7, 5, 0 |

0.10120 | 5 | 1, 0, 1, 2, 0 |

1500. | 4 | 1, 5, 0, 0 |

7.128 × 10^{-3} |
4 | 7, 1, 2, 8 |

## Significant Figures Quiz

Determine the number of significant figures in each of the following measurements.

^{18}atoms

## Significant figures in operations:

**Addition and subtraction**

With addition and subtraction, you should round your final result so its precision (number of decimal places!) matches the precision of the least precise number, no matter how many significant figures any particular term possesses. For example:

87.221 + 1.2 = 88.421 but you should round this value down to 88.4 (so that it matches the precision of the least precise number in the sum, 1.2)

**Multiplication, division, and roots**

In multiplication, division or when taking roots, your results should be rounded so that the final result has the same number of significant figures as the number with the least number of significant figures. For example:

3.14 × 2.2048 = 6.923072 but you should round this value down to 6.92 (the measurement with the least significant figures is 3.14, which has 3 significant figures, rounding to 3 sig figs gives 6.92)

**Logarithms**

If you are calculating the logarithm of a number, you should make sure that the mantissa (the figure to the right of the decimal point in the answer) contains an identical number of significant figures as the number of significant figures of the number of which the logarithm is being calculated. For example:

log (2×10^5) = 5.301029995663981 - you should round this figure to 5.3

**Multiple Mathematical Operations**

Should a calculation require a number of mathematical operations to be combined, do it with more figures than the number that will be significant to get your value. Then review the calculation and, by applying the rules above, calculate the number of significant figures needed in the final result.

You may also be interested in our Scientific Notation Calculator