### Maths Significant Figure

##### Significant Figures

Significant figures are the digits in a measurement that are known with certainty, plus one uncertain digit. The number of significant figures in a measurement depends on the precision of the measuring instrument used.

##### Rules for Determining Significant Figures

The following rules can be used to determine the number of significant figures in a measurement:

- All nonzero digits are significant.
- Zeros between nonzero digits are significant.
- Zeros at the end of a number are significant if the number has a decimal point.
- Zeros at the end of a number are not significant if the number does not have a decimal point.

##### Examples

The following are examples of measurements with different numbers of significant figures:

- 123.45 has five significant figures.
- 123.4 has four significant figures.
- 123 has three significant figures.
- 120 has two significant figures.
- 100 has one significant figure.

##### Importance of Significant Figures

Significant figures are important because they allow us to communicate the precision of our measurements. When we report a measurement, we should always include the number of significant figures so that others know how precise our measurement is.

Significant figures are a fundamental concept in science and engineering. By understanding the rules for determining significant figures, we can communicate the precision of our measurements accurately.

Significant figures are the digits in a measurement that are known with certainty, plus one uncertain digit. The number of significant figures in a measurement is determined by the precision of the measuring instrument used.

##### Rules for Determining Significant Figures

The following rules are used to determine the number of significant figures in a measurement:

**All nonzero digits are significant.**For example, the number 123 has three significant figures.**Zeros between nonzero digits are significant.**For example, the number 1002 has four significant figures.**Zeros at the end of a number are significant if they are to the right of the decimal point.**For example, the number 1.00 has three significant figures, while the number 100.0 has four significant figures.**Zeros at the end of a number are not significant if they are to the left of the decimal point.**For example, the number 100 has two significant figures.**In scientific notation, the exponent is not a significant figure.**For example, the number 1.23 x 10$^2$ has three significant figures.

##### Examples

The following table shows some examples of measurements and the number of significant figures in each:

Measurement | Number of Significant Figures |
---|---|

123 | 3 |

1002 | 4 |

1.00 | 3 |

100.0 | 4 |

100 | 2 |

1.23 x 10$^2$ | 3 |

Significant figures are important because they allow us to communicate the precision of our measurements. By following the rules for determining significant figures, we can ensure that our measurements are accurate and reliable.

##### Operations on Significant Figures

Significant figures are the digits in a measurement that are known with certainty, plus one uncertain digit. They are used to express the precision of a measurement.

##### Rules for Operations on Significant Figures

When performing operations on measurements with different numbers of significant figures, the following rules apply:

**Addition and subtraction:**The result should have the same number of decimal places as the measurement with the fewest decimal places.**Multiplication and division:**The result should have the same number of significant figures as the measurement with the fewest significant figures.**Exponentiation:**The result should have the same number of significant figures as the base.**Logarithms:**The result should have the same number of significant figures as the argument.

##### Examples

**Addition and subtraction:**

- 12.34 + 5.678 = 18.02 (the result has the same number of decimal places as the measurement with the fewest decimal places, which is 5.678)
- 100 - 56.78 = 43.22 (the result has the same number of decimal places as the measurement with the fewest decimal places, which is 56.78)

**Multiplication and division:**

- 12.34 x 5.678 = 70.39 (the result has the same number of significant figures as the measurement with the fewest significant figures, which is 5.678)
- 100 / 56.78 = 1.761 (the result has the same number of significant figures as the measurement with the fewest significant figures, which is 56.78)

**Exponentiation:**

- 10$^2$ = 100 (the result has the same number of significant figures as the base, which is 10)

**Logarithms:**

- log(100) = 2 (the result has the same number of significant figures as the argument, which is 100)

Significant figures are an important tool for expressing the precision of measurements. By following the rules for operations on significant figures, you can ensure that your results are accurate and meaningful.

##### Rounding Significant Figures

Significant figures are the digits in a measurement that are considered reliable and meaningful. They include all the digits that are known with certainty, plus one estimated digit. The estimated digit is the last digit in the measurement that is not known with certainty.

##### Why are Significant Figures Important?

Significant figures are important because they allow us to communicate the precision of our measurements. When we report a measurement, we should include all of the significant figures so that others know how accurate our measurement is.

##### How to Round Significant Figures

When rounding significant figures, we follow these rules:

- If the digit to be dropped is less than 5, the last digit retained is unchanged.
- If the digit to be dropped is greater than 5, the last digit retained is increased by 1.
- If the digit to be dropped is 5, the last digit retained is increased by 1 if it is odd, and unchanged if it is even.

##### Examples of Rounding Significant Figures

Here are some examples of how to round significant figures:

- 12.345 rounds to 12.35
- 12.355 rounds to 12.36
- 12.346 rounds to 12.35
- 12.347 rounds to 12.35

Significant figures are an important part of scientific communication. By understanding how to round significant figures, we can ensure that we are communicating the precision of our measurements accurately.

##### Solved Examples of Significant Figures

##### Example 1: Counting Significant Figures

Count the number of significant figures in each of the following measurements:

- 123.45 g
- 0.0045 m
- 3.14159
- 1000000000 m
- 2.00 x 10$^{23}$ molecules

**Solution:**

- 123.45 g: 5 significant figures
- 0.0045 m: 2 significant figures
- 3.14159: 5 significant figures
- 1000000000 m: 1 significant figure
- 2.00 x 10$^{23}$ molecules: 3 significant figures

##### Example 2: Rounding Numbers with Significant Figures

Round the following numbers to the specified number of significant figures:

- 123.4567 to 3 significant figures
- 0.00456789 to 4 significant figures
- 3.14159265 to 6 significant figures

**Solution:**

- 123.4567 to 3 significant figures: 123
- 0.00456789 to 4 significant figures: 0.00457
- 3.14159265 to 6 significant figures: 3.141593

##### Example 3: Adding and Subtracting Numbers with Significant Figures

Add the following numbers and express the result with the correct number of significant figures:

- 12.34 g + 5.67 g + 0.0045 g
- 0.0045 m - 0.0023 m

**Solution:**

- 12.34 g + 5.67 g + 0.0045 g = 18.0 g (3 significant figures)
- 0.0045 m - 0.0023 m = 0.0022 m (3 significant figures)

##### Example 4: Multiplying and Dividing Numbers with Significant Figures

Multiply the following numbers and express the result with the correct number of significant figures:

- 12.34 g x 5.67 g
- 0.0045 m / 0.0023 m

**Solution:**

- 12.34 g x 5.67 g = 70.1 g (3 significant figures)
- 0.0045 m / 0.0023 m = 1.96 (3 significant figures)

##### Example 5: Using Significant Figures in Calculations

Calculate the density of a substance with a mass of 12.34 g and a volume of 5.67 mL. Express the result with the correct number of significant figures.

**Solution:**

Density = mass / volume = 12.34 g / 5.67 mL = 2.18 g/mL (3 significant figures)

##### Significant Figure FAQs

**What are significant figures?**

Significant figures are the digits in a measurement that are considered reliable and meaningful. They include all the digits that are known with certainty, plus one estimated digit.

**How do I determine the number of significant figures in a measurement?**

There are a few rules for determining the number of significant figures in a measurement:

- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Zeros at the end of a number are significant if there is a decimal point.
- Zeros at the end of a number are not significant if there is no decimal point.

**What are some examples of significant figures?**

- The measurement 12.34 has four significant figures.
- The measurement 12.00 has three significant figures.
- The measurement 1200 has two significant figures.
- The measurement 12000 has one significant figure.

**Why are significant figures important?**

Significant figures are important because they allow us to communicate the precision of our measurements. When we know the number of significant figures in a measurement, we know how much confidence we can have in the accuracy of that measurement.

**How do I use significant figures in calculations?**

When performing calculations with measurements, it is important to keep track of the number of significant figures in each measurement. The final answer to a calculation should have the same number of significant figures as the measurement with the fewest significant figures.

**What are some common mistakes people make with significant figures?**

Some common mistakes people make with significant figures include:

- Not counting all of the significant figures in a measurement.
- Counting zeros at the end of a number as significant when there is no decimal point.
- Not rounding the final answer to a calculation to the correct number of significant figures.

**Conclusion**

Significant figures are an important part of scientific communication. By understanding the rules for determining the number of significant figures in a measurement, we can communicate the precision of our measurements and perform accurate calculations.