Significant Figures

Rules for Identifying Significant Figures

Any non-zero digits and zeros in between the non-zeros are significant

1. ex: 61.48126 - all the digits are significant because they are all non-zeros.
2. ex: 12.0075 - all digits are significant because the zeros are in between non-zero digits.

Leading zeros are not significant

Leading zeros come before all of the non-zero digits.

1. ex: 00.713326 - only the digits after the decimal are significant because the zeros leading up to them are not considered significant. That's why they are called the leading zeros.
2. ex: 0.03813 - even though there is a zero after the decimal, it is still leading up to the non-zero digits so it is a leading zero and is not significant.

Trailing zeros

Trailing zeros are the zeros that come after all of the non-zero digits.

• When there is a decimal, count the trailing zeros as significant.

1. ex: 0.69383500 - leading zeros are still not significant but since there is a decimal the trailing zeros are considered significant along with the non-zero digits.

• When there isn't a decimal, don't count the trailing zeros.

1. ex: 1500 - only count the non-zero digits as significant because writing it like this assumes there was rounding to the nearest hundred so the trailing zeros are not significant.

Multiplication and Division

• When multiplying or dividing, the product or quotient can only have the same number of significant figures as the least amount of significant figures of either numbers you are multiplying or dividing. 

1. ex: 3.45 x 1.3 - the least amount of significant figures out of the numbers is two because the 1.3 has two and the 3.45 has three. Therefore, instead of having a product of 4.58, it can only have two significant figures so you round up to 4.9.

Addition and Subtraction

• The actual number of significant digits are not as important with adding and subtracting. Now it has to do with how many numbers are after the decimal point. Or rather how precise your least precise number is.

1. ex: 1.26 + 2.3 - the least precise number only goes one digit after the decimal or to the tenth. This means that even though the sum of the two numbers is 3.56, you round to the nearest tenth making it 3.6.
2. ex: 2.4 + 103.7 - with these numbers it may look like your answer should only have two digits since the 2.4 only has two significant digits. However, with adding and subtracting, only look at the number of digits after the decimal of your least precise number. Therefore, the answer would be 106.1 with a total of four digits because there is only one number after the decimal.

• Even if there is no decimal, the rule of rounding from the least precise number still stands.

1. ex: 560 + 5 - now your least precise number can be assumed to have only been measured to the nearest ten while the other was measured to the nearest one. So, the sum would be 565 however since the least precise was the nearest ten, your sum has to be rounded to the nearest ten making it 570.

Problems

Determine the number of significant figures.

1. 6.002 - 4 sig. figures
2. 10.0500 - 4 sig. figures

Carry out the following calculations.

1. 52.13 g + 1.7502 = 53.8802 = 53.88
2. 12 m x 6.41 m = 76.92 = 77

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