Scientific notation is a useful way to write large or small numbers in a compact form. It uses powers of 10 to "condense" a lot of digits. Numbers written in scientific notation are of the form

where 1≤a<10 and k∈Z.

Scientific notation is sometimes called "standard form."

Let x=a×10k,y=b×10n.

To find the sum x+y, we need to have k=n. If k=n, choose the number with the smaller power, and rewrite it in terms of the larger one. Calling n the smaller power (i.e. n<k), we rewrite y as y=c×10k where c=b(10n−k). Note c×10k=b(10n−k)(10k)=b×10n, so the two ways of writing y are equivalent, as we wanted.

Once the powers of 10 are equal, we can just add a and b:

To find the difference x−y, we do the same thing, rewriting the number with the smaller power (here we're calling this n) in terms of the larger one (here, k) and then taking the difference of the coefficients,

If the sum or difference of a and b is not between 1 and 10, adjust the final exponent k by adding or subtracting 1 so that the sum or difference can be rewritten with a coefficient between 1 and 10.

We can multiply and divide numbers in scientific form as follows:

Multiplying and dividing numbers in scientific notation relies heavily on exponent rules.