Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 6: Decimals 71

Powers and Scientific Notation Revisited ...................................................................

Earlier you learned about using exponents to write powers of ten and how to write large numbers

as the product of a number between 1 and 10 and a power of 10. What about the other side of the

decimal point? How can you write the values of the places that hold decimal fractions as powers

of ten? And how do you use scientific notation to express tiny decimal fractions?

Powers of Ten for Numbers Less than 1

Our place values to the left of the decimal point are 10^0 , 10^1 , 10^2 , and so on, with the exponent

getting larger as you move to the left. Each decimal fraction was created by breaking a whole

into ten pieces or a hundred pieces or a thousand pieces and so on. You’re still using powers of 10,

but you’re dividing by 10 each time you move to the right. You’re looking at place values that are

getting smaller instead of bigger.

To write powers of 10 for these fractional values, you’re going to use a very similar system of

exponents, but you have to add a signal that you’re going to the other side of the decimal point,

going in the opposite direction. The signal that you’re moving to the right of the decimal point

is a negative exponent. The negative signals an opposite, in this case, a move in the opposite

direction.

The tenths place is worth a tenth of a whole, or the fraction you get by dividing 1 by 10. You

signal that value as 10-1. The hundredths place is worth 1 z 100, or 1 z 102 , which you can write

as 10-2. The thousandths place is worth 10-3. To express the value of a place to the right of the

decimal point as a power of ten, count the number of digits to the right of the decimal point, put

a minus sign in front of that number, and use that number as an exponent on 10.

Numbers Less than 1 as Powers of 10

Tenths Hundredths Thousandths Ten-thousandths Hundred-thousands

0.1 0.01 0.001 0.0001 0.00001

1 digit after

decimal point

2 digits after

decimal point

3 digits after

decimal point

4 digits after

decimal point

5 digits after

decimal point

10 -1 10 -2 10 -3 10 -4 10 -5

CHECK POINT

6. Write 0.00001 as a power of ten.


7. Write 0.000000001 as a power of
   ten.
   8. Write 0.000000000000001 as a
   power of ten.
   9. Write 10-6 in standard form.
   10. Write 10-10 in standard form.