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Scientific Notation

Objective Learn how to write very large and small numbers in scientific notation and to compare and order numbers written in scientific notation.

Scientific notation is a useful topic for many applications. It also provides a great opportunity to reinforce your understanding of place value in terms of powers of ten.


Very Large and Very Small Numbers

Let's see some very large and very small numbers. A few examples are listed.

• The mass of the planet Pluto is roughly 12,900,000,000,000,000,000,000 kilograms.

• The number of molecules in a cubic centimeter of oxygen at standard temperature and pressure is about 602,000,000,000,000,000,000,000.

• A typical cell membrane is about 0.00000001 meter thick.

• A large virus has a diameter of roughly 0.0000001 meter.

Each of the numbers shown above contains many zeros. Thus, it may be difficult to read and compare them. However, there is a notation that can be used to express all of the zeros in a simple, easy to read form. Consider the following examples.


Example 1

7,000,000,000 (seven billion) = 7 × (one billion)
  = 7 × 10 9 One billion is equal to 10 9.

Notice how few symbols are required to write 7 × 10 9 compared to 7,000,000.


Example 2

13,000,000 (thirteen million) = 13 × (one million)
  = 13 × 10 6 One billion is equal to 10 6.
  = 1.3 10 7 1.3 is one tenth of 13.


Example 3

0.0000056 (fifty-six ten-millionths)

= 5 × (one millionth) + 6 × (one ten-millionth)  
1,000,000 = 10 6 and 10,000,000 = 10 7
= 5 × 10 -6 + 6 ×10 -7 Laws of Negative Exponents
= (5 + 6 ×10 -1 ) × 10 -6 Factor 10 -6 .
= 5.6 × 10 -6  


Key Idea

Every nonzero number can be written as a number greater than or equal to one and less than ten times a power of ten.


Definition of Scientific Notation

A number that is written as a number between 1 and 10 times a power of ten is said to be written in scientific notation.


All of the following numbers are written in scientific notation.

1.2 ×10 6

2.4 ×10 -5

5.8 ×10 18

The following numbers are not written in scientific notation because the number in front of the multiplication sign is not between 1 and 10.

12.5 ×10 12

0.15 ×10 11

235 ×10 -13


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