Use this tool in **significant figures calculator** mode to perform algebraic operations with numbers (adding, subtracting, multiplying and dividing) with the appropriate significant digit rounding. In **significant figures counter** mode it will count the number of significant digits in a number. Use the calculator in **rounding** mode to round a number to a given number of significant figures.

Quick navigation:

- What are significant figures?
- Significant figures rules

- Significant digits examples

- Significant figures addition rule
- Significant figures multiplication rule

## What are significant figures?

Significant figures (a.k.a. significant digits or precision) of a number written in positional notation are all digits that carry meaningful contributions to its measurement resolution. Of the significant figures in a number the one in the position with the highest exponent value (the left-most) is the most significant, while the one in the position with the lowest exponent value (the right-most) is the least significant.

Significant digits are important in different areas where measurements apply and are usually used to express the precision of measurements. Numbers can be rounded to a given number of significant figures, for example when the measurement device cannot produce accurate results to a given resolution. Counting how many digits are significant is done by following several simple rules. Our sig fig calculator can help with all of these operations.

## Significant figures rules

The rules for which digits in a positional notation are significant are simple. All figures are significant except the following:

- Leading zeroes, e.g. "00123" has three significant figures: 1, 2, and 3. 0.0012 has just two significant figures: 1 and 2.
- Trailing zeros when they are merely placeholders to indicate the scale of the number. This means that zeroes to the right of the decimal point and zeroes between significant figures are themselves significant.
- Digits beyond the required or supported precision. E.g. figures introduced by division or multiplication or measurements reported to a greater precision than the measurement equipment supports.

Note that the above rules mean that all non-zero digits (1-9) are significant, regardless of their position. Use our significant digits calculator in "counter" mode to count and examine the significant figures in any number.

See below for the rules for rounding when performing arithmetic operations with numbers with a given precision.

### Significant digits examples

The following table contains examples of applying the significant digits rules above in a variety of cases that cover everything you should see in practice.

Original number | Significant figures | Count of significant figures | Rules applied |
---|---|---|---|

0.005 | 5 | 1 | #1 |

038 | 3,8 | 2 | #1 |

470 | 4,7 | 2 | #2 |

470. | 4,7,0 | 3 | #2 |

470.0 | 4,7,0,0 | 4 | #2 |

0205.60 | 2,0,5,6,0 | 5 | #1,#2 |

1001.05 | 1,0,0,1,0,5 | 6 | #2 |

12100 | 1,2,1 | 3 | #2 |

121.090 | 1,2,1,0,9,0 | 6 | #2 |

Of the above examples, the most tricky to understand are:

- 0205.60: here the leading zero is dropped via rule #1, the zero between 2 and 5 is preserved as it is between two significant digits and the trailing zero is preserved as it is to the right of the decimal point (both following rule #2)
- 470.0: here the trailing zero is significant as it is to the right of the decimal point, while the other zero is also significant since it sits between two significant figures: the seven to the left and the zero beyond the decimal point
- 1001.05: the first two zeroes are between significant digits greater than zero, the third zero is also significant since it is both to the right of the decimal point and is between two significant digits

## Counting significant figures

Counting the number of significant digits is done simply by identifying them using the rules, and then performing a simple count. For examples, see the table above. To count significant figures using this calculator, simply put the tool in "counter" mode and enter the number you want to count the significant digits of.

## Rounding significant figures

Numbers are often rounded to a specified number of significant figures for practicality, e.g. to present in a news broadcast or to put down in a table neatly. Rounding a number to *n* significant figures happens in a similar way to rounding to *n* decimal places, with an important difference. We start by counting from the first non-zero digit for *n* significant digits and then round the last digit. However, we do not fill in the remaining places to the right of the decimal point with zeroes.

In more detail, the process of rounding to *n* significant digits is as follows:

- Identify the first
*n*significant figures in the number, left to right. - If the digit immediately to the right of the
*n*-th digit is greater than or equal to 5, and to its right there are non-zero digits, add 1 to the*n*-th digit. - If the digit immediately to the right of the
*n*-th digit is 5, and to its right there are only zeroes or nothing, there is a tie. To break the tie using the half away from zero rule, add 1 to the*n*-th digit. If using the half to even method, preferred in scientific settings as it does not produce upwardly skewed numbers, round down to the nearest even number. - Replace non-significant figures in front of the decimal point by zeroes.
- Drop all the digits after the decimal point to the right of the
*n*significant figures.

An example of the rounding rule application, consider the number 1.55 and rounding it to 2 significant figures. Using both methods would result in rounding it to 1.6 since this is also the nearest even number. However, if the original number was 1.45, rounded to two significant figures it would become 1.5 under the half away from zero method, but 1.4 under the half to even method.

### Examples of rounding to *n* significant figures

Rounding with a given precision based on decimal places differs from rounding to the same precision of significant figures. Examples of rounding of the number **12.345** are presented in the table below.

Rounding precision (n) | To n significant figures | To n decimal places |
---|---|---|

7 | 12.34500 | 12.3450000 |

6 | 12.3450 | 12.345000 |

5 | 12.345 | 12.34500 |

4 | 12.35* | 12.3450 |

3 | 12.3 | 12.345 |

2 | 12 | 12.35* |

1 | 10 | 12.3 |

0 | N/A | 12 |

* using the half to even rule it would round to 12.34.

In another example, take the number **0.012345**. The rounding calculations are presented in the table below.

Rounding precision (n) | To n significant figures | To n decimal places |
---|---|---|

7 | 0.01234500 | 0.0123450 |

6 | 0.0123450 | 0.012345 |

5 | 0.012345 | 0.01235* |

4 | 0.01235* | 0.0123 |

3 | 0.0123 | 0.012 |

2 | 0.012 | 0.01 |

1 | 0.01 | 0.0 |

0 | N/A | 0 |

* using the half to even rule it would round to 0.01234.

You can check the accuracy of by using our rounding significant figures calculator.

## Algebraic operations with rounding

Using our tool in significant figures calculator mode you can perform addition, subtraction, multiplication and division of numbers expressed in a scientific notation to a given degree of precision. Our tool will automatically apply the appropriate rounding rule depending on the selected mathematical operation, as explained below.

### Significant figures addition rule

The rule for adding significant figures is to round the result to the least accurate place. The sum or difference is to be rounded to the same number of decimal places as that of the measurement with the fewest decimal places which reflects the fact that the answer is just as precise as the least-precise measurement used to compute it. For example, 2.24 + 4.1 = 5.34 which has to be rounded to one place after the decimal dot, since 4.1 is only precise to that level, giving a result of 5.3. If we were adding 2.24 and 4.10 though, the result would be 5.34.

Our significant figures calculator uses this rule automatically. You can choose if the rounding is done using the half away from zero rule or by the half to even rule. The rule for adding is also used for subtraction of numbers with a given number of significant digits.

### Significant figures multiplication rule

Multiplication rounding and division rounding is performed based on the number of significant figures in the measurement with the lowest count of significant digits. For example, multiplying 20.0 by 10 will result in 200. Since only a single digit ("1") is significant in the second number rounding to the first significant digit gives us 200 of which only the "2" is significant. In another example, let us say we multiply 2.5 by 10.05 and get 25.125. Since 2.5 has only two significant digits, we must round the result to two significant digits as well, giving us an answer of 25. This rounding rule is applied automatically in our tool.

The least number of significant digits rule is used both for multiplication and for division of numbers in our calculator.