A way of expressing numbers in the form $a \cdot 10^b$, where $a$ is a real number and $1 < a < 10$, and $b$ is an integer, otherwise known as standard form. This is a concise way to express very large or very small numbers.
Scientific notation, otherwise known as standard form or standard index form, is a way of expressing numbers that have very large or very small orders of magnitude. This shortens the representation of the number and thus simplifies arithmetic with these numbers.
In scientific notation, all numbers are written in the form:
$$a \times 10^b$$
where $a$ is a real number and $1 < a < 10$, and $b$ is an integer. $b$ is specifically chosen so that $a$ is restricted to $1 < a < 10$.
Given two numbers in scientific notation, say $a_0 \times 10^{b_0}$ and $a \times 10^{b_1}$, then the four operations can be easily performed. By the laws of indices: $$(a_0 \times 10^{b_0})(a_1 \times 10^{b_1}) = a_0 a_1 \times 10^{b_0 + b_1}$$
and $$\frac{a_0 \times 10^{b_0}}{a_1 \times 10^{b_1}} = \frac{a_0}{a_1} \times 10^{b_0 - b_1}$$
Addition and subtraction can only be performed if $b_ 0 = b_1$. Then they can be added using the distributive law, which gives: $a_0 \times 10^{b_0} ± a_1 \times 10^{b_1} = (a_0 ± a_1) \times 10^{b_0}$.
One advantage of scientific notation is its ability to clearly indicate precision. If a number is known to $6$ significant figures, then it can be written with all $6$ digits clearly displayed, such as $1.230 \ 40 \times 10^6$. Rounding is made much easier as well: if this number is rounded to $3$ significant figures, $1.23 \times 10^6$, it is easy to verify that it has been rounded to the specified number of significant figures.
On calculators and many computer programs, scientific notation is commonly displayed in the form $a \ \text{E} \ b$ or $a \ \text{EXP} \ b$, which has the same meaning as $a \times 10^b$.
