### Powers of Ten

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For numbers larger than 10, the power of 10 is a positive value and negative for numbers less than 1. For numbers between 0 and 10, the power is a positive fraction. In the examples that follow, notice what happens to the decimal point:

 100 = 1. = 1. with the decimal point moved 0 places 101 = 10. = 1. with the decimal point moved 1 place to the right 102 = 100. = 1. with the decimal point moved 2 places to the right 106 = 1000000. = 1. with the decimal point moved 6 places to the right and 10-1 = 0.1 = 1. with the decimal point moved 1 place to the left 10-2 = 0.01 = 1. with the decimal point moved 2 places to the left 10-6 = 0.000001 = 1. with the decimal point moved 6 places to the left.

The exponent of 10 tells you how many places to move the decimal point to the right for positive exponents or left for negative exponents. These rules come in especially handy for writing very large or very small numbers.

### Scientific Notation

Since you will be working with very large and very small numbers, use scientific notation to cut down on all of the zeroes you need to write. Proper scientific notation specifies a value as a number between 1 and 10 (called the mantissa below) multiplied by some power of ten, as in

mantissa×10exponent.
The power of ten tells you which way to move the decimal point and by how many places. As a quick review:

10 = 1 × 101, 253 = 2.53 × 100 = 2.53 × 102 and 15,000,000,000 = 1.5 × 1010 which you will sometimes see written as 15 × 109 even though this is not proper scientific notation. For small numbers we have: = 1 × 10-1, × 10-2 or about 0.395 × 10-2 = 3.95 × 10-3.

#### Multiplying and Dividing with Scientific Notation

When you multiply two values given in scientific notation, multiply the mantissa numbers and add the exponents in the power of ten. Then adjust the mantissa and exponent so that the mantissa is between 1 and 10 with the appropriate exponent in the power of ten. For example: (3×1010) × (6× 1023), you'd have 3×6 × 10(10+23) = 18. × 1033 = 1.8×1034.

When you divide two values given in scientific notation, divide the mantissa numbers and subtract the exponents in the power of ten. Then adjust the mantissa and exponent so that the mantissa is between 1 and 10 with the appropriate exponent in the power of ten. For example: × 1010-23 = 0.5 × 10-13 = 5 × 10-14.

Notice what happened to the decimal point and exponent in the examples. You subtract one from the exponent for every space you move the decimal to the right. You add one to the exponent for every space you move the decimal to the left.

#### Entering scientific notation on your scientific calculator

Please read this if you have not used a scientific calculator for a while!

Most scientific calculators work with scientific notation. Your calculator will have either an ``EE'' key or an ``EXP'' key. That is for entering scientific notation. To enter 253 (2.53 × 102), you would punch 2 . 5 3 EE or EXP 2. To enter 3.95 × 10-3, you would punch 3 . 9 5 EE or EXP 3 [ key]. Note that if the calculator displays ``3.53 -14'' (a space between the 3.53 and -14), it means 3.53 × 10-14 NOT 3.53-14! The value of 3.53-14 = 0.00000002144 = 2.144×10-8 which is vastly different than the number 3.53×10-14. Also if you have the number 4 × 103 and you enter 4 × 1 0 EE or EXP 3, the calculator will interpret that as 4 × 10 × 103 = 4 × 104 or ten times greater than the number you really want!

One other word of warning: the EE or EXP key is used only for scientific notation and NOT for raising some number to a power. To raise a number to some exponent use the ``yx'' or ``xy'' key depending on the calculator. For example, to raise 3 to the 4th power as in 34 enter 3 yx or xy 4. If you instead entered it using the EE or EXP key as in 3 EE or EXP 4, the calculator would interpret that as 3×104 which is much different than 34 = 81.

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last updated: 27 May 2001

##### Is this page a copy of Strobel's Astronomy Notes?

Author of original content: Nick Strobel