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Significant Figures and Scientific Notation

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  • 0:55 Measurements &…
  • 2:56 Identifying…
  • 4:33 Calculations &…
  • 7:27 Scientific Notation
  • 9:46 Lesson Summary
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Lesson Transcript
Instructor: Chelsea Schuyler
Are 7.5 grams and 7.50 grams the same? How do scientists represent very large and very small quantities? Find out the answers to these questions in this video.

Introduction

When scientists do experiments, they're always recording data and making measurements. Sometimes the information they record is based on observation. This is called qualitative, meaning that it is based on an observation, but it's not directly measured and recorded numerically. For example, 'The water in the beaker is warm' would be a qualitative observation. I didn't actually go and measure the temperature of the water in the beaker. The other type of information that scientists record is quantitative, meaning that it is based on a measurement, and it's reported numerically. An example would be 'The water in the beaker is 87 degrees.' Notice how there's a number in the quantitative observation and not in the qualitative observation.

Measurements & Significant Figures

If you have ever measured something more than once, you may have noticed that each time you may get a slightly different result. Any time you make a measurement there is some degree of uncertainty related to that measurement. This is because no measuring device is perfect. Usually the more high-quality the measuring instrument is the more precise your measurement will be. The precision of an instrument refers to the smallest repeatable digit that the instrument can measure to. For example, if you are measuring the mass of a pen and one balance reads 7.5 grams while another - more precise - balance measures 7.50 grams, the second balance will give you a more precise measurement.

When reporting these measurements, it's extremely important to report all the digits that are given. In math class, you may have learned that 7.50 is equivalent to 7.5, but when it comes to making and recording a measurement, the zero at the end is just as important as the seven and the five. This is because the zero tells the person reading the number that the balance measured out to the nearest hundredth place, which just happened to be a zero. It pretty much tells us that the second balance we used is a little more 'high-tech' than the first one because it measures out farther. This zero is so important that it is called a significant figure. A significant figure is a number that plays a role in the precision of a measurement. Don't confuse the word 'significant' with 'important' or 'certain.' If a number is significant, it's just kept track of when reporting measured results and making calculations. It's very important to be able work with significant figures correctly so both the measurement and the precision of the instrument used are communicated.

So, if the last zero in 7.50 is significant, what numbers are not significant? First of all, all non-zero numbers are considered significant, as in the number 524, which has three significant figures. Also, zeros between two non-zero numbers - I like to call them 'sandwiched zeros' - are significant, as in the number 9,201, which has four significant figures. Leading zeros are not significant, as in the number 0.003, which has one significant figure. These zeros just serve as placeholders. The same number could have easily been written as 3-3 without those leading zeros. Trailing zeros are only significant in numbers with a decimal point, as in the number 7.50, which has three significant figures. Trailing zeros in numbers that do not contain a decimal point are not significant, as in the number 25,000, which only has two significant figures. These zeros also just serve as placeholders. For example, when finding the population of a town, the number 25,000 implies that the actual value is around 25,000 rounded to the nearest thousand, whereas the number 25,000. - with a decimal at the end - implies that the actual value is 25,000. When you see a number, keep an eye out for significant figures and decimal points.

Calculations & Significant Figures

You will also need to pay close attention to numbers when you're doing calculations. Calculators do not identify significant figures, so you'll need to make sure you report the results of calculations with the correct number of significant figures. Any time you're adding or subtracting, you will always report your answer with the same number of decimal places as the number with the least number of decimal places. For example, if you are adding 5.113 and 2.0, your answer should be rounded to the nearest tenth place, or 7.1. You can't assume that the second measurement was 2.000, so your answer should never be more precise than the measurements used in the calculations.

Have you ever wanted to divide two numbers (like 54 divided by 7), and you end up getting a number with LOTS of numbers past the decimal? When you take 54 divided by 7 on a calculator, you'll get an answer of 7.715284714… and so on. Hopefully you know that it would be a little absurd to report the entire answer given on the calculator, but how do you know where to round? When multiplying and dividing, your answer must have the same number of significant figures as the number with the least number of significant figures. For example, if you are multiplying 40 (with one significant figure) and 9.2 (with two significant figures), your answer should contain only one significant figure, so it should be reported as 400 instead of the 368 your calculator would give you.

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