Chapter 1

Noting Numbers Scientifically

In This Chapter

Crunching numbers in scientific and exponential notation

Telling the difference between accuracy and precision

Doing math with significant figures

Like any other kind of scientist, a chemist tests hypotheses by doing experiments. Better tests require more reliable measurements, and better measurements are those that have more accuracy and precision. This explains why chemists get so giggly and twitchy about high-tech instruments: Those instruments take better measurements!

How do chemists report their precious measurements? What’s the difference between accuracy and precision? And how do chemists do math with measurements? These questions may not keep you awake at night, but knowing the answers to them will keep you from making rookie mistakes in chemistry.

Using Exponential and Scientific Notation to Report Measurements

Because chemistry concerns itself with ridiculously tiny things like atoms and molecules, chemists often find themselves dealing with extraordinarily small or extraordinarily large numbers. Numbers describing the distance between two atoms joined by a bond, for example, run in the ten-billionths of a meter. Numbers describing how many water molecules populate a drop of water run into the trillions of trillions.

To make working with such extreme numbers easier, chemists turn to scientific notation, which is a special kind of exponential notation. Exponential notation simply means writing a number in a way that includes exponents. In scientific notation, every number is written as the product of two numbers, a coefficient and a power of 10. In plain old exponential notation, a coefficient can be any value of a number multiplied by a power with a base of 10 (such as 104). But scientists have rules for coefficients in scientific notation. In scientific notation, the coefficient is always at least 1 and always less than 10. For example, the coefficient could be 7, 3.48, or 6.0001.

To convert a very large or very small number to scientific notation, move the decimal point so it falls between the first and second digits. Count how many places you moved the decimal point to the right or left, and that’s the power of 10. If you moved the decimal point to the left, the exponent on the 10 is positive; to the right, it’s negative. (Here’s another easy way to remember the sign on the exponent: If the initial number value is greater than 1, the exponent will be positive; if the initial number value is between 0 and 1, the exponent will be negative.)

To convert a number written in scientific notation back into decimal form, just multiply the coefficient by the accompanying power of 10.

Q. Convert 47,000 to scientific notation.

A. . First, imagine the number as a decimal:

Next, move the decimal point so it comes between the first two digits:

Then count how many places to the left you moved the decimal (four, in this case) and write that as a power of 10: .

Q. Convert 0.007345 to scientific notation.

A. . First, put the decimal point between the first two nonzero digits:

Then count how many places to the right you moved the decimal (three, in this case) and write that as a power of 10: .

1. Convert 200,000 into scientific notation.

2. Convert 80,736 into scientific notation.

3. Convert 0.00002 into scientific notation.

4. Convert from scientific notation into decimal form.

Multiplying and Dividing in Scientific Notation

A major benefit of presenting numbers in scientific notation is that it simplifies common arithmetic operations. The simplifying abilities of scientific notation are most evident in multiplication and division. (As we note in the next section, addition and subtraction benefit from exponential notation but not necessarily from strict scientific notation.)

To multiply two numbers written in scientific notation, multiply the coefficients and then add the exponents. To divide two numbers, simply divide the coefficients and then subtract the exponent of the denominator (the bottom number) from the exponent of the numerator (the top number).

Q. Multiply using the shortcuts of scientific notation: .

A. . First, multiply the coefficients:

Next, add the exponents of the powers of 10:

Finally, join your new coefficient to your new power of 10:

Q. Divide using the shortcuts of scientific notation: .

A. . First, divide the coefficients:

Next, subtract the exponent in the denominator from the exponent in the numerator:

Then join your new coefficient to your new power of 10:

5. Multiply .

6. Divide .

7. Using scientific notation, multiply .

8. Using scientific notation, divide .

Using Exponential Notation to Add and Subtract

Addition or subtraction gets easier when you express your numbers as coefficients of identical powers of 10. To wrestle your numbers into this form, you may need to use coefficients less than 1 or greater than 10. So scientific notation is a bit too strict for addition and subtraction, but exponential notation still serves you well.

To add two numbers easily by using exponential notation, first express each number as a coefficient and a power of 10, making sure that 10 is raised to the same exponent in each number. Then add the coefficients. To subtract numbers in exponential notation, follow the same steps but subtract the coefficients.

Q. Use exponential notation to add these numbers: .

A. . First, convert both numbers to the same power of 10:

Next, add the coefficients:

Finally, join your new coefficient to the shared power of 10:

Q. Use exponential notation to subtract: .

A. . First, convert both numbers to the same power of 10:

Next, subtract the coefficients:

Then join your new coefficient to the shared power of 10:

9. Add .

10. Subtract .

11. Use exponential notation to add .

12. Use exponential notation to subtract .

Distinguishing between Accuracy and Precision

Accuracy and precision, precision and accuracy … same thing, right? Chemists everywhere gasp in horror, reflexively clutching their pocket protectors — accuracy and precision are different!

• Accuracy: Accuracy describes how closely a measurement approaches an actual, true value.
• Precision: Precision, which we discuss more in the next section, describes how close repeated measurements are to one another, regardless of how close those measurements are to the actual value. The bigger the difference between the largest and smallest values of a repeated measurement, the less precision you have.

The two most common measurements related to accuracy are error and percent error:

• Error: Error measures accuracy, the difference between a measured value and the actual value:
• Percent error: Percent error compares error to the size of the thing being measured:

  Being off by 1 meter isn’t such a big deal when measuring the altitude of a mountain, but it’s a shameful amount of error when measuring the height of an individual mountain climber.

Q. A police officer uses a radar gun to clock a passing Ferrari at 131 miles per hour (mph). The Ferrari was really speeding at 127 mph. Calculate the error in the officer’s measurement.

A. –4 mph. First, determine which value is the actual value and which is the measured value:

• Actual value = 127 mph
• Measured value = 131 mph

Then calculate the error by subtracting the measured value from the actual value:

Q. Calculate the percent error in the officer’s measurement of the Ferrari’s speed.

A. 3.15%. First, divide the error’s absolute value (the size, as a positive number) by the actual value:

Next, multiply the result by 100 to obtain the percent error:

13. Two people, Reginald and Dagmar, measure their weight in the morning by using typical bathroom scales, instruments that are famously unreliable. The scale reports that Reginald weighs 237 pounds, though he actually weighs 256 pounds. Dagmar’s scale reports her weight as 117 pounds, though she really weighs 129 pounds. Whose measurement incurred the greater error? Who incurred a greater percent error?

14. Two jewelers were asked to measure the mass of a gold nugget. The true mass of the nugget is 0.856 grams (g). Each jeweler took three measurements. The average of the three measurements was reported as the “official” measurement with the following results:

• Jeweler A: 0.863 g, 0.869 g, 0.859 g
• Jeweler B: 0.875 g, 0.834 g, 0.858 g

Which jeweler’s official measurement was more accurate? Which jeweler’s measurements were more precise? In each case, what was the error and percent error in the official...