Basic Math and Pre-Algebra

PART 1. The World of Numbers

CHAPTER 6. Decimals

 

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 100, 101, 102, 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 ÷ 100, or 1 ÷ 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.

Scientific Notation for Small Numbers

Writing small numbers in scientific notation is almost the same process as you used for large numbers. A small number like 0.00045 can be written in scientific notation by first writing the digits of the number and placing a decimal point after the first nonzero digit. This gives you the number between 1 and 10. (Later you can drop any leading zeros.)

Count from where you just placed the decimal point to where it actually should be. Notice that you’re counting in the opposite direction from what you did with large numbers. You’ll show that by making the exponent negative. The number of places tells you the exponent to put on the ten, but it will have a minus sign on it.

Suppose you want to write 0.00045 in scientific notation.

Copy the number and put a decimal point after the first nonzero digit.

Count to where the decimal point ought to be.

Make the exponent negative.

needs an exponent of -4.

Write as a number between 1 and 10 times a (negative) power of ten.

00004.5 x 10-4

Drop any leading zeros.

4.5 x 10-4

Now you can see that 0.00045 = 4.5 x 10-4.

MATH TRAP

It's easy to get confused with so many zeros in the number. Count carefully, and place a mark over or under each digit as you count your way to the decimal point. Have a system so you don't get confused with all those zeros.

To change a small number that is written in scientific notation to standard form, copy the digits of the number between 1 and 10 and move the decimal point to the left as many places as the exponent on the 10. This is the same process you used with large numbers, but you move to the left because, for small numbers, the exponent is negative. You can add zeros if you run out of digits. The number 4.193 x 10-4 becomes or .0004193. It’s customary to put a zero in the ones place, so this could be written 0.0004193.

CHECK POINT

11. Write 0.492 in scientific notation.

12. Write 0.0000051 in scientific notation.

13. Write 2.7 x 10-5 in standard notation.

14. Write 8.19 x 10-7 in standard notation.

15. Write 5.302 x 10-4 in standard notation.