Idiot\'s Guides Basic Math and Pre-Algebra

(Marvins-Underground-K-12) #1
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


  1. Write 0.00001 as a power of ten.

  2. 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.

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