Scientific Notation

Courtesy of C. Keeton

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.

- Ö(900)
= (9*10
^{2})^{1/2}= (9)^{1/2}* (10^{2})^{1/2}= 3 * 10 = 30