# Chapter 6: Rationals: Multiplication, Division, and Applications

by

Millard Lewis and John Moore

## Day 8: Scientific Notation

In this lesson, you will learn how to convert numbers from standard notation to scientific notation. Scientific notation was invented by scientists in order to abbreviate especially large or small numbers. The conversion process utilizes multiplication and division by rational numbers and, thus, is applicable to this unit.

Converting from standard to scientific notation

Look at the number 4,197; where is the decimal point? Yes, it is at the end of the number.

Step 1: We now need to figure out where you would put the decimal point in order to make a number greater than or equal to 1 and strictly less than 10. In this case, the decimal point would need to be placed between the 4 and the 1, giving us 4.197 which meets the criteria above.

Step 2: Next, figure out how many places and in which direction you need to move the decimal point to make your original number. In this case, you would need to move the decimal 3 places to the right, right?

Now, you have everything you need to write the number in scientific notation. First of all, write down the number that you found in Step 1 (see example below). Then write "x 10". The neat thing here is that you have to multiply the number you found in Step 1 by some power of 10. (Notice: That means you just move the decimal point a number of places equal to the exponent of 10.) Then, use the direction that you found in Step 2 to determine the sign of the exponent of 10, where left is negative and right is positive, and finally, use the number of places as the exponent with the sign you just found.

Example: 4,197 ->

More examples:

1. 0.00438 -> [4.38 is our number from Step 1 multiplied by , or , because we have to move the decimal point 3 places to the left to obtain our original number.]

2. 127,007,658 -> [You may be tempted to use 7.658 as your "Step 1" number, but that would actually be 127,007.658 Your direction is right, or positive, and your "how many places" is 8.]

Converting from scientific to standard notation

Converting numbers the other way requires a much more straightforward process. Given a number in scientific notation, say , just multiply the two numbers as you normally would. (Notice: Most calculators will not display the answers that you obtain in standard notation!!) Therefore, we obtain 0.000002639

Examples:

1. -> 1,295,000,000,000 [Straightforward multiplication!!]

2. -> 0.00000000832 [Remember, is just , or ]

Click here to do a puzzle using conversions from standard to scientific notation and vice versa.