## Chemistry for Dummies

**Appendix B. How to Handle Really Big or Really Small Numbers**

Those who work in chemistry become quite comfortable working with very large and very small numbers. For example, when chemists talk about the number of sucrose molecules in a gram of table sugar, they’re talking about a very large number. But when they talk about how much a single sucrose molecule weighs in grams, they’re talking about a very small number. Chemists can use regular longhand expressions, but they become very bulky. It’s far easier and quicker to use exponential or scientific notation.

*Exponential Notation*

In exponential notation, a number is represented as a value raised to a power of ten. The decimal point can be located anywhere within the number as long as the power of ten is correct. In scientific notation, the decimal point is always located between the first and second digit — and the first digit must be a number other them zero.

Suppose, for example, that you have an object that’s 0.00125 meters in length. You can express that number in a variety of exponential forms:

0.00125 m = 0.0125 x 10^{-1} m, or 0.125 x 10^{-2} m, or 1.25 x 10^{-3} m, or 12.5 x 10^{-4} m, and so on.

All these forms are mathematically correct as numbers expressed in exponential notation. In scientific notation, the decimal point is placed so that there’s one digit other than zero to the left of the decimal point. In the preceding example, the number expressed in scientific notation is 1.25 x 10^{-3} m. Most scientists automatically express numbers in scientific notation.

Here are some positive and negative powers of ten and the numbers they represent:

*Addition and Subtraction*

To add or subtract numbers in exponential or scientific notation, both numbers must have the same power of ten. If they don’t, you must convert them to the same power. Here’s an addition example:

(1.5 x 10^{3 }g) + (2.3 x 10^{2 }g) = (15 x 10^{2 }g) + (2.3 x 10^{2 }g) = 17.3 x 10^{2 }g (exponential notation) = 1.73 x 10^{3 }g (scientific notation)

Subtraction is done exactly the same way.

*Multiplication and Division*

To multiply numbers expressed in exponential notation, multiply the coefficients (the numbers) and add the exponents (powers of ten):

(9.25 x 10^{-2} m) x (1.37 x 10^{-5} m) = (9.25 x 1.37) x 10^{(-2 + -5)} = 12.7 x 10^{-7} = 1.27 x 10^{-6}

To divide numbers expressed in exponential notation, divide the coefficients and subtract the exponent of the denominator from the exponent of the numerator:

(8.27 x 10^{5 }g) ÷ (3.25 x 10^{3 }mL) = (8.27 ÷ 3.25) x 10^{53} g/mL = 2.54 x 10^{2} g/mL

*Raising a Number to a Power*

To raise a number in exponential notation to a certain power, raise the coefficient to the power and then multiply the exponent by the power:

(4.33 x 10^{-5 }cm)^{3} = (4.33)^{3} x 10^{-5x3} cm^{3} = 81.2 x 10^{-15} cm^{3} = 8.12 x 10^{-14} cm^{3}

*Using a Calculator*

Scientific calculators take a lot of drudgery out of doing calculations. They enable you to spend more time thinking about the problem itself.

You can use a calculator to add and subtract numbers in exponential notation without first converting them to the same power of ten. The only thing you need to be careful about is entering the exponential number correctly. I’m going to show you how to do that right now:

I assume that your calculator has a key labeled EXP. The EXP stands for x 10. After you press the EXP key, you enter the power. For example, to enter the number 6.25 x 10^{3}, you type 6.25, press the EXP key, and then type 3.

What about a negative exponent? If you want to enter the number 6.05 x 10^{-12}, you type 6.05, press the EXP key, type 12, and then press the +/- key.

REMEMBER. When using a scientific calculator, don’t enter the x 10 part of your exponential number. Press the EXP key to enter this part of the number.