Using Scientific Notation to Approximate

Instructor: David Karsner

David holds a Master of Arts in Education

When you have really large numbers or really small numbers, the exact number is not always that important. In this lesson, you'll learn how scientific notation allows you to approximate these numbers and make them easier to use.

Exact Numbers and Approximation

There are times when having an exact number is exactly what you need. For example, when taking medicine an exact amount is your best option. At other times, exact numbers can be very cumbersome to manipulate when an approximation will work just as well.

An approximation is a value or amount that is close but not exact. For example, the distance to the sun is 149,597,887.5 kilometers. This large number is difficult to remember, and using it is overly complicated. An approximation would make everything much easier. Scientific notation is one method of approximating very large or very small numbers when an exact number is not a necessity.

Scientific Notation

Scientific notation is an alternate way to express numbers. It is typically used on very large numbers or very small numbers; but it can be used on any rational number. Scientific notation begins with one digit (1 to 9, including negatives) in front of the decimal point. It then has a few non-zero digits after the decimal (the number of digits here depends on the question). Finally it will be multiplied to 10 raised to some power.

Example of Scientific Notation

  • 3 x 105
  • 4.56 x 1013
  • 2.75 x 10-3

When Numbers Are Large

When numbers are really large, scientific notation is very convenient. It saves you from having to write all those numbers. To convert a large number to scientific notation, you are basically dividing the decimal number by a power of 10. The powers of ten are: 10; 100; 1,000; 10,000; 100,000...).

You need to divide your decimal number by a large enough power of ten so that you only have one digit in front of your decimal point. For example, 6,430 divided by 1,000 is 6.43. Since 1,000 has three zeros, 3 will be the power of ten. In scientific notation 6,430 is 6.43 x 103.

Another way to look at it is to count how many times the decimal needs to be moved so that there is only one digit in front of it. With 6,430, the decimal point is after the zero (keep in mind that if no decimal point is present, it is understood to be at the end). Moving it one place to the left you have 643.0, two places 64.30, three places gives 6.430. There is only one digit in front of the decimal and we had to move it three times. We end up with 6.43 x 103.

Large number conversion

Example of an Approximation

The number 6,430 is a fairly small number so scientific notation doesn't save a lot of time. Also 6,430 and 6.43 x 103 are both exact numbers. Let's look at an example where you would use an approximation.

The sun is 149,597,887.5 kilometers away from the Earth. Let's convert that to scientific notation. Notice that the decimal point is between the 7 and the 5. If we move the decimal point 8 spots to the left (you simply count the spots) you will have 1.495978875. Let's approximate this number to be 1.5 (a much easier number to work with). In scientific notation the distance from the earth to the sun is 1.5 x 108 kilometers.

If you were ever asked how far it would be to go the sun and back three times, scientific notation makes this computation easy. You would multiply the distance to the sun by 6.

6(1.5 x 108) would be 9 x 108 kilometers

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