# Distances and Sizes in the Universe using Powers-of-Ten

#### Powers of Ten

The distances and sizes of of the objects astronomers study vary from very small, including atoms and atomic nuclei, to very large including galaxies, clusters of galaxies and the size of the universe. To describe such a huge range, astronomers need a way to avoid confusing terms like "a billion trillion" and "a millionth". Astronomers use a system called powers-of-ten notation, which consolidates all of the zeros that you would normally find attached to very large or small numbers such as 1,000,000,000,000 or 0.0000000001. All of the zeros are put in an exponent, which is written as a superscript, and indicates how many zeros you would need to write out the long form of the number. So for example:

100 = 1

101 = 10

102 = 100

103 = 1000

104 = 10,000

and so on.

In powers-of-ten notation, numbers are written as a figure between one and ten multiplied by a power of ten. So for example, the distance to the Moon of 384,000 km can be re-written as 3.84 × 105 km. Notice that 3.84 is between one and ten. The same number could accurately be rewritten as 38.4 × 104 or 0.384 × 106, but the preferred form is to have the first number be between one and ten.

Very small numbers can also be written using powers-of-ten notation. The exponent is negative for numbers less than one and indicates dividing by that number of tens. So for example:

100 = 1

10-1 = 1/10 = 0.1

10-2 = 1/10 × 1/10  = 0.01

10-3 = 1/10 × 1/10 × 1/10  = 0.001

10-4 = 1/10 × 1/10 × 1/10 × 1/10 = 0.0001

and so on.

Once again, numbers are written as a figure between one and ten multiplied by a power of ten. So for example, a number like 0.00000375 would be expressed as 3.75 × 10-6.

Some familiar numbers written as powers-of-ten:
One hundred (100) 102
One thousand (1000) 103
One million (1,000,000) 106
One billion (1,000,000,000) 109
One trillion (1,000,000,000,000) 1012
One one-hundredth (0.01) 10-2
One one-thousandth (0.001) 10-3
One one-millionth (0.000001) 10-6
One one-billionth (0.000000001)10-9
One one-trillionth (0.000000000001) 10-12

A few websites offer demonstrations of powers-of-ten and the scale of the universe.