# Scientific Notation

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
100 = 1.
101 = 10.
102 = 100.
103 = 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*108
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*102) + (5*100) = (1.21*102) + (0.05*102) = (1.21+0.05)*102 = 1.26*102

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

121*50 = (1.21*102) * (5*101) = (1.21*5) * 102+1 = 6.05*103
250/50 = (2.5*102) / (5*101) = (2.5/5) * 102-1 = 0.5*101 = 5
300,000,000*31,000,000 = (3*108) * (3.1*107) = (3*3.1) * 108+7 = 9.3*1015

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

(102)4 =
102*4 = 108

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

(2*102)4 = (2)4 * (102)4
16 * 108
= 1.6 * 109

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

√(900)
= (9*102)
1/2 = (9)1/2 * (102)1/2 =
3 * 10 = 30