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General Studies Science: Help & Review24 chapters | 338 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Richard Cardenas*

Richard Cardenas has taught Physics for 15 years. He has a Ph.D. in Physics with a focus on Biological Physics.

In this lesson, you'll learn about the concept of significant figures and apply it to addition, subtraction, multiplication, and division. You'll learn what numbers are significant, why they are significant, and how to deal with zeroes in a number.

A **significant figure** is a figure or a digit that contributes to how accurately something can be measured. Measuring anything is limited by the measuring device you use. For example, a standard meter stick with centimeters as the smallest division can be used to accurately measure to a tenth of the smallest unit, which is a millimeter. Anything smaller than that will be meaningless, since your measuring device only measures to the nearest millimeter. Therefore, if you use this meter stick then measurements like 0.567 meters, or 0.600 meters, are good measurements since they both indicate that you can measure to the nearest millimeter. A measurement like 0.6 meters does not accurately depict that the meter stick can measure to the nearest millimeter. Similarly, a measurement like 0.6123 meters goes beyond the accuracy of the meter stick.

The digits up to the thousandths place then are all significant figures, or sig figs, because together they maximize the accuracy of the measurement being indicated. Given a measuring stick like this, which measures only as small as centimeters, let's consider the process of trying to measure the length of an object. You can only estimate the length of the object to the nearest tenth of a centimeter, or a millimeter; any further isn't possible with the information the ruler is giving us. Let's say the object looks to be about 4.8 centimeters (cm) long, with only two significant figures. The exact length may be different, but with our two significant figures, that's as close as we can get.

But what if the ruler had millimeter markings in addition to centimeters? You could now estimate the length of the object to the nearest tenth of a millimeter. That could give us a measurement like 4.84 cm, with three significant figures. The more significant figures are available to us when measuring, the more exact the measurement will be.

It can be tricky to decide which digits in a complex number are significant figures. Because significant figures are used to increase the accuracy of measurements, numbers which use them will often include long strings of extra zeroes, making it even more difficult to identify the sig figs. Luckily, we have some tools to help.

The most useful one is the Atlantic-Pacific rule, a mnemonic device for identifying significant figures. The rule states that if a decimal point is absent, for example in the case of whole numbers, zeroes on the right side of the number (the Atlantic) are not significant. If a decimal point is present and there are non-integer digits involved in the measurement, then the zeroes on the Pacific side (the left side) are not significant. In both cases, the opposite side contains the significant zeroes.

For example, let's take an imaginary measurement of:

0015.00120

The nonzero digits are all significant figures, but are all of the four zeroes? Notice that a decimal point is present, meaning that there are significant digits on the right side of the decimal. According to the Atlantic-Pacific rule, that means that zeroes on the decimal side are significant, and we can ignore the extra zeroes on the left side of the number. So this number has seven significant figures.

When performing mathematical actions, such as addition and multiplication on numbers with the same number of significant digits, the solution will always have the same number of significant figures as the original terms. But what about when the numbers being added, subtracted, divided, or multiplied have different amounts of significant figures?

For example, 1.001 has four significant digits with three digits after the decimal point. Similarly, 1.01 has three significant digits and two digits after the decimal point. So what happens when we add these two numbers? The number of significant figures in the final answer is limited by the number with the smallest number of digits after the decimal point. In this example, 1.01 has two digits after the decimal point, so our final sum can only be accurate to two digits after the decimal point. After doing some arithmetic, that will leave our final solution as 2.01.

What about subtracting? The rule is the same. The final difference will have the same number of significant digits as the term with the lowest number of post-decimal digits. Our solution is -0.01 which has two digits after the decimal.

Something similar is true when multiplying or dividing decimals. the final answers must have the same number of significant figures as the term with the least amount of significant figures. For example, consider the following numbers: 4.420 and 2.2. The first term has four significant figures, while the second has two. Multiplying the numbers will yield a product of 9.724 which is a total of four sig figs. But, our final answer must have as many significant figures as the number with the fewest, that's 2.2, which has two total significant figures. Therefore, we need to drop two significant figures from our final answer, simplifying the result to 9.7. If we divide the same numbers, the calculator would give an answer of 2.009, which has four significant figures. Remember though, that we can only have two significant figures, so the answer simplifies to 2.0.

**Significant figures** are numbers or digits of a measurement that provide information about the accuracy of the measurement. As a device or instrument gets more accurate, the number of significant figures, or sig figs, in its measurement increases. One tool for determining a measurement's significant digits is the Atlantic-Pacific rule, which states that if a decimal point is present, the significant figures start at the first nonzero digit from the left (the Pacific).

When adding, subtracting, multiplying, or dividing numbers that have significant figures, remember that the solution must have the same number of significant figures as the term with the lowest number. That means if two numbers are multiplied, one with three sig figs and one with seven, the answer will only have three significant digits.

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General Studies Science: Help & Review24 chapters | 338 lessons | 1 flashcard set

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