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 10⁴.

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

m_E=2 x 10²⁴kg and m_e=9 x 10^-31kg.

This system also allows significant figures to be easily represented: 3.780 x 10⁴ has four significant figures and 3.78 x 10⁴ 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

10¹⁰=10000000000

^10-1=1/10

10^-2=1/100

^10-4=1/10000