﻿ Scientific Notation - Measurements and Calculations

# Measurements and CalculationsScientific Notation Leelu/Shutterstock.com

A variety of chemical glassware.

As we pointed out in Chapter 1, making observations is a key part of the scientific process. Sometimes observations are qualitative (“the substance is a yellow solid”), and sometimes they are quantitative (“the substance weighs grams”). A quantitative observation is called a measurement. Measurements are very important in our daily lives. For example, we pay for gasoline by the gallon, so the gas pump must accurately measure the gas delivered to the fuel tank. The efficiency of the modern automobile engine depends on various measurements, including the amount of oxygen in the exhaust gases, the temperature of the coolant, and the pressure of the lubricating oil. In addition, cars with traction control systems have devices to measure and compare the rates of rotation of all four wheels. As we will see in the “Chemistry in Focus” discussion in this chapter, measuring devices have become very sophisticated in dealing with our fast-moving and complicated society.

As we will discuss in this chapter, a measurement always consists of two parts: a number and a unit. Both parts are necessary to make the measurement meaningful. For example, suppose a friend tells you that she saw a bug long. This statement is meaningless as it stands. Five what? If it’s millimeters, the bug is quite small. If it’s centimeters, the bug is quite large. If it’s meters, run for cover!

The point is that for a measurement to be meaningful, it must consist of both a number and a unit that tells us the scale being used.

In this chapter we will consider the characteristics of measurements and the calculations that involve measurements. IPGGutenbergUKLtd/iStockphoto.com

A gas pump measures the amount of gasoline delivered.

Scientific Notation

Objective

· To show how very large or very small numbers can be expressed as the product of a number between and and a power of .

The numbers associated with scientific measurements are often very large or very small. For example, the distance from the earth to the sun is approximately ( million) miles. Written out, this number is rather bulky. Scientific notation is a method for making very large or very small numbers more compact and easier to write.

To see how this is done, consider the number , which can be written as the product

Because , we can write

Similarly, the number can be written

and because , we can write

Scientific notation simply expresses a number as a product of a number between and and the appropriate power of . For example, the number can be expressed as

The easiest way to determine the appropriate power of for scientific notation is to start with the number being represented and count the number of places the decimal point must be moved to obtain a number between and . For example, for the number we must move the decimal point seven places to the left to get (a number between and ). To compensate for every move of the decimal point to the left, we must multiply by . That is, each time we move the decimal point to the left, we make the number smaller by one power of . So for each move of the decimal point to the left, we must multiply by to restore the number to its original magnitude. Thus moving the decimal point seven places to the left means we must multiply by seven times, which equals :

Remember: whenever the decimal point is moved to the left, the exponent of is positive.

We can represent numbers smaller than by using the same convention, but in this case the power of is negative. For example, for the number we must move the decimal point two places to the right to obtain a number between and : This requires an exponent of , so . Remember: whenever the decimal point is moved to the right, the exponent of is negative.

Next consider the number . In this case we must move the decimal point four places to the right to obtain (a number between and ): Moving the decimal point four places to the right requires an exponent of . Therefore,

We summarize these procedures below.

Using Scientific Notation

· Any number can be represented as the product of a number between and and a power of (either positive or negative).

· The power of depends on the number of places the decimal point is moved and in which direction. The number of places the decimal point is moved determines the power of . The direction of the move determines whether the power of is positive or negative. If the decimal point is moved to the left, the power of is positive; if the decimal point is moved to the right, the power of is negative.

Interactive Example 2.1. Scientific Notation: Powers of (Positive)

Represent the following numbers in scientific notation.

a.

b.

Solution

a. First we move the decimal point until we have a number between and , in this case . Because we moved the decimal point five places to the left, the power of is positive . Thus .

b. Thus .

Interactive Example 2.2. Scientific Notation: Powers of (Negative)

Represent the following numbers in scientific notation.

a.

b.

Solution

a. First we move the decimal point until we have a number between and , in this case . Because we moved the decimal point four places to the right, the power of is negative . Thus .

b. Thus .

Self-Check: Exercise 2.1

· Write the numbers and in scientific notation. If you are having difficulty with scientific notation at this point, reread the Appendix.

See Problems 2.5, 2.6, 2.7, 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, and 2.14.

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