**Lesson 3:**

**Rounding and Significant Figures**

**Objective**

In this lesson we will learn the following:

- How to correctly round numbers.
- Why significant figures are important.

**Lesson**

**Rounding Numbers**

When rounding numbers, the following key points need to be considered:

- Numbers are rounded to reduce the number of digits to the right of the decimal point. This is done for convenience, not accuracy.
- Rule: A number is rounded off by dropping one or more numbers from the right and adding zeroes, if necessary, to maintain the decimal point. If the last figure dropped is 5 or more, increase the last retained figure by 1. If the last digit dropped is less than 5, do not increase the last retained figure.

Let's look at a problem showing how to round numbers correctly.

*Example:*

**Round off 10,546 to 4, 3, 2 and 1 significant figures.**

4 significant figures:10,546 = 10,550 (because the 6 after the 4 is greater than 5, so the 4 rounds up and the rest is zeroes, because it only asked for 4 significant figures and the zeroes are placeholders)

3 significant figures:10,546 = 10,500

2 significant figures:10,546 = 11,000 (because of the "5" the number increases by 1)

1 significant figure:10,546 = 10,000 (this is not 11,000 because it only asked for 1 significant figure, which looks to the second digit place to see if it increases or stays the same. The second digit place holds a zero, so it stays the same).

**Significant Figures**

A rule that needs to be followed when determining significant figures is that significant figures are those numbers that are known to be reliable. The position of the decimal point does not determine the number of significant figures.

You can determine the number of significant figures in a number using a few simple rules. In the examples below, I have shown the significant digits in bold red.

- All non-zero numbers are significant. For example:

Numbers |
# of Significant Figures |

251 |
3 |

13.49 |
4 |

8765.1 |
5 |

- Zeroes between significant digits are significant. For example:

Numbers |
# of Significant Figures |

305 |
3 |

42003 |
5 |

70201 |
5 |

- If there is no decimal point, then trailing zeroes are not significant. For example:

Numbers |
# of Significant Figures |

470 |
2 |

10 |
1 |

6,000 |
1 |

- If there is a decimal point, then all trailing zeroes are significant. For example:

Numbers |
# of Significant Figures |

41.00 |
4 |

20.50 |
4 |

10. |
2 |

- If a number is less than one, then the first significant figure is the first non-zero digit after the decimal point.

Numbers |
# of Significant Figures |

0.009 |
1 |

0.156 |
3 |

0.01060 |
4 |

Let's review a video showing you how to determine the appropriate significant figures to use.

**Addition and Subtraction**

When you are adding or subtracting a bunch of numbers and need to be concerned with significant figures, you would first add (or subtract) the numbers given in their entire format, and then round the final answer. When rounding the final answer after adding or subtracting, the answer must be written with the same significant figures as the **least accurate decimal place given**. Let's look at an example:

*Example:*

**Round to the appropriate number of significant digits.**

**13.214 + 234.6 + 7.0350 + 6.38**

So, let's add the numbers together:

13.214 + 234.6 + 7.0350 + 6.38 = 261.2290

Looking at the numbers given, I can see that the second number (234.6) is only accurate to the tenths place making it the **least accurate number**. My answer must be rounded to the same place as the least accurate number:

**261.2290 rounds to 261.2 **(one decimal place)

Let's watch a video showing you how to determine the correct number of significant figures needed when adding and subtracting.

**Multiplication and Division**

When multiplying or dividing multiple numbers you would do these calculations as normal. When the answer must be written in the appropriate significant figure your answer must round to the same number of significant figures as the **least number of significant figures**. Let's look at a couple of examples:

*Example:*

**Simplify, and round, to the appropriate number of significant digits.**

**16.235 x 0.217 x 5**

First off, I see that 5 has only one significant figure, so I will have to round my final answer to one significant digit:

16.235 x 0.217 x 5 = 17.614975

Since I can only claim one accurate significant digit I will need to round 17.614975 to one digit. I'll start with the 1 in the tens place. Immediately to its right is a 7, which is greater than 5, so I'll be rounding the 1 up to 2, and then replacing the 7 with a zero, and dropping the decimal point and everything after it:

17.614975 rounds to 20

*Example:*

**Simplify, and round, to the appropriate number of significant figures.**

**0.00435 x 4.6**

First do the simply multiplication:

0.00435 x 4.6 = 0.02001

We can see that 4.6 has only 2 significant figures, so we well have to round the answer to two significant figures. The 2 is the first significant digit in the final answer, which means the zero following it will have to be the second significant figure to report correctly. That means that 0.02001 rounds to 0.020, which has 2 significant figures (0.0**20**). The answer should not be 0.02, because that indicates only one significant figure when we need 2 significant figures in the final answer.

Let's watch a video showing you how to determine the correct number of significant figures needed when multiplying and dividing.

**Summary**

When rounding numbers, the following key points need to be considered:

- Numbers are rounded to reduce the number of digits to the right of the decimal point. This is done for convenience, not accuracy.
- Rule: A number is rounded off by dropping one or more numbers from the right and adding zeroes, if necessary, to maintain the decimal point. If the last figure dropped is 5 or more, increase the last retained figure by 1. If the last digit dropped is less than 5, do not increase the last retained figure.

When determining the number of significant figures needed in the final answer, follow these rules:

- For adding/subtracting, use the least accurate
**decimal place**given. - For multiplication/division, use the least number of
**significant digits**given.

**Assignment**

Complete the math worksheet for this lesson. You must be logged into Canvas to submit this assignment. Make sure you choose the appropriate semester.