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) * 10^2-1 = 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 =
10^2*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