## Representing very large and small numbers

In physics, we often encounter numbers that are very big and very small. For example, in a typical basic physics course, students will have to preform calculations using quantities like the mass of the earth, m_{E}= 60000000000000000000000000kg, and the mass of an electron, m

_{e}= 0.0000000000000000000000000000009kg, sometimes even both at once!

In order to handle numbers like these with many zeros (ie. that are either very large or very small) in a way that is both efficient and reduces the chance of making an error copying from one line to the next – Scientific Notation or “Powers of Ten”. In scientific notation, a number is written as a number between one and ten multiplied by ten raised to some power.

For example, the number 37 800 can be written as 3.78 x 10 x 10 x 10 x 10 = 3.78 x 10^{4}.

The large and small numbers mentioned above, written in scientific notation are:

m_{E}=2 x 10^{24}kg and m_{e}=9 x 10^{-31}kg.

This system also allows significant figures to be easily represented: 3.780 x 10^{4} has four significant figures and 3.78 x 10^{4} has only three. In traditional notation, both these numbers are written as 37800, with no clear way to indicate which digits are significant.

^{ Powers of Ten:}

^{ 100=1}

^{ 101=10}

^{ 102=100}

10^{10}=10000000000

^{ 10-1=1/10}

10^{-2}=1/100

^{ 10-4=1/10000}

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