By C. Keeton and A. Zabludoff

Astronomy deals with everything from atoms (0.0000000001 m) to the universe (100,000,000,000,000,000,000,000,000 m). To make it easier to write such small and large numbers we use *scientific* (or *power of ten*) notation.

**Simple powers of ten:** Write 1.0, and then move the decimal the number of spaces given by the exponent.

- 10
^{-3}= 0.001 - 10
^{-2}= 0.01 - 10
^{-1}= 0.1 - 10
^{0}= 1. - 10
^{1}= 10. - 10
^{2}= 100. - 10
^{3}= 1000.

**More complicated numbers:** Pull out the largest power of ten you can. The number left out front is called the *mantissa*.

- 390,100,000 = 3.901*100,000,000 = 3.901*10
^{8} - 0.00000558 = 5.58*0.000001 = 5.58*10
^{-6}

**Addition and subtraction:** Make sure the exponents are the same, then just add or subtract the mantissas.

- 121+5 = (1.21*10
^{2}) + (5*10^{0}) = (1.21*10^{2}) + (0.05*10^{2}) = (1.21+0.05)*10^{2}= 1.26*10^{2}

**Multiplication and division:** Simply multiply or divide the mantissas, and add or subtract the exponents.

- 121*50 = (1.21*10
^{2}) * (5*10^{1}) = (1.21*5) * 10^{2+1}= 6.05*10^{3} - 250/50 = (2.5*10
^{2}) / (5*10^{1}) = (2.5/5) * 10^{2-1}= 0.5*10^{1}= 5 - 300,000,000*31,000,000 = (3*10
^{8}) * (3.1*10^{7}) = (3*3.1) * 10^{8+7}= 9.3*10^{15}

**Powers and square roots:** To raise a power of 10 to a power, multiply the two powers.

- (10
^{2})^{4}=

10^{2*4}= 10^{8}

If you have a mantissa and a power of 10, raise both the mantissa and the power of 10.

- (2*10
^{2})^{4}= (2)^{4}* (10^{2})^{4}

= 16 * 10^{8}

= 1.6 * 10^{9}

Taking the square root is equivalent to raising a number to the power 1/2.

- √(900)

= (9*10^{2})^{1/2}= (9)^{1/2}* (10^{2})^{1/2}=

3 * 10 = 30