How does scientific notation work?

Answer 1

Scientific notation is a convenient way of writing and working with large (or small) numbers. Such numbers are common in physics and science in general.

It works by writing down how many trailing zeros there are instead of writing down all zeros.

For example one might encounter the number 63000. One can write this down somewhat shorter by 63 * 10^3. Also often used is an even shorter notation:

63000 = 63e3

So you have an 'e' (or sometimes 'E') followed by a number. This is the amount of zeros that follows the number in front of 'e'. If this amount is negative you instead move the decimal point to the left. So:

63e-3 = 0.063

For 63000 it is still viable to write down the number itself but for a number like 1e100 (a googol) it is already too much work.

As a somewhat side note, one might wonder how often one does encounter a number with a large number of trailing zeros. This is actually very often because measurements done in science are never perfect, they contains uncertainties so only the first few digits are significant; the rest can easily be replaced by zeros.

There are methods for writing down even larger numbers. A googolplex is a 1 with a googel (=1e100) zeros following. This is cumbersome to work with even using the scientific notation (although one could perhaps do something like 1e(1e100)), but there are methods in which you can easily write down and work with this number. Scientific notation is a way of writing numbers that are too big or too small to be conveniently written in decimal form.

In normalized scientific notation all numbers are written in the form a x 10^b (a times ten raised to the power of b) where a is a nonzero single-digit integer and b is an integer.