# Scientific Notation

Physicists deal with really big and really small numbers sometimes, and it can get tiresome (and risky) to count all of those zeroes. That's why scientific notation was born! It looks like this$$\underbrace{5.2}_{\text{mantissa}}\times {10^{\!\!\!\overbrace{-3}^\text{exponent}}}$$

The mantissa is typically (but not always! See Formal Scientific Notation) a number between 1 (inclusive) and 10 (exclusive): that is, a single nonzero digit, followed by a decimal point and any number of other digits. The exponent is always an integer: if it's nonnegative then this is a number bigger than (or equal to) 1, and if it is negative then it is less than 1.

To convert a number to scientific notation, count the number of times you would need to move the decimal point so that it is just behind the first non-zero digit. A move to the left means a positive exponent.

$$2_\dotspot\lshift{5}{\mathbf{+3}}\lshift{4}{+2}\lshift{6}{+1}. = 2.546\times 10^3 \qquad 0.\rshift{0}{-1}\rshift{3}{\mathbf{-2}}\!{}_\dotspot 4 = 3.4\times 10^{-2}$$

Sometimes we need to change a number in scientific notation so that it has a different exponent: for example, \(2.3\times 10^3\) is the same as \(23\times 10^2\). Every change of the exponent must also be accompanied by moving the decimal point one place. For example: $$2.\rshift{3}{\color{red}+1}\rshift{0}{\color{red}+2}{}_\dotspot\times 10^{-7\color{red}{-2}} = 230\times 10^{-9}$$ $${}_\dotspot\lshift{2}{\color{red}-1}.3\times 10^{5\color{red}+1}=0.23\times 10^6$$

Or in other words

If you make the mantissa bigger, make the exponent smaller.