Scientific notation, also known as "standard form," is a way to more conveniently write numbers that are very large or very small. This method is commonly used by mathematicians, scientists, and engineers.

For example, the numbers   $43,000,000,000,000,000,000$ (the number of different possible configurations of a Rubik's cube) and $0.000000000000000000000340$ (the mass of the amino acid tryptophan) are extremely inconvenient to write and read. Therefore, they can be rewritten as a power of 10 using scientific notation.

Scientific notation is written as follows:

$m \cdot 10^n$

This is read "$m$ times 10 raised to the power of $n$."

How to Use Scientific Notation

To write a number in scientific notation:

• Move the decimal point so that there is one nonzero digit to its left.
• Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

For example, let's write the number 43,500 in scientific notation. There are four digits in this number, so the decimal should be moved 4 places to the left to leave one nonzero digit left of the decimal point:

$43,500 = 4.35 \cdot 10^{4}$

The exponent is -4 because the decimal point was moved to the left (the exponent would be positive had the decimal been moved to the right) by exactly 4 places.A number written in scientific notation can also be converted to standard form by reversing the process described above. For example, let's write the number $2.15 \cdot 10^{-3}$ in standard form:

$2.15 \cdot 10^{-3}= 0.00215$

To reverse the process, we move the decimal point three places to the left, adding leading zeroes where necessary.

Normalized Scientific Notation

Any given number can be written in the form of $m \cdot 10^{n}$ in many ways; for example, 350 can be written as $3.5 \cdot 10^{2}$ or $35 \cdot 10^{1}$or $350 \cdot 10^{0}$, etc.

In normalized scientific notation, also called exponential notation, the exponent $n$ is chosen so that the absolute value of $m$ remains at least 1 but less than 10. In other words, $ 1 \leq | m | < 10 $. This form allows easy comparison of two numbers of the same sign with $m$ as a base, as the exponent $n$ gives the number's order of magnitude.

Following these rules, 350 would always be written as $3.5 \cdot 10^{2}$ and $0.015$ would always be written as $1.5 \cdot 10^{-2}$.  Note that $0$ cannot be written in normalized scientific notation since it cannot be expressed as $m \cdot 10^n$ or any non-zero $m$. 

Normalized scientific form is the typical form of expression for large numbers in many fields, except during intermediate calculations or when an unnormalized form, such as engineering notation, is desired.

Calculations involving Scientific Notation

When numbers written in scientific notation are multiplied or divided, the standard rules for operations with exponentiation apply. For example:

$\begin{aligned} \displaystyle (3.12 \cdot 10^2) \cdot (4.06 \cdot 10^5) &= 3.12 \cdot 4.06 \cdot 10^{\left(2+5\right)} \\&=12.67 \cdot 10^7 \\&=1.267 \cdot 10^8 \end{aligned} $

$\begin{aligned} \displaystyle \frac{1.85 \cdot 10^3}{4.25 \cdot 10^{-2}} &= \frac{1.85}{4.25} \cdot 10^{3-(-2)} \\&=.435 \cdot 10^5 \\&= 4.35 \cdot 10^4 \end{aligned}$

When numbers written in scientific notation are added to or subtracted from each other, the terms first must be rewritten so the exponents are the same. Then, the constant value, or $m$, can simply be added or subtracted. For example:

$\begin{aligned} \displaystyle (3.12 \cdot 10^6)+(1.24 \cdot 10^7)&= (3.12 \cdot 10^6)+(12.4 \cdot 10^6) \\&= 15.52 \cdot 10^6 \end{aligned}$

E Notation

Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like $10^7$ cannot always be conveniently displayed, the letter E or e is often used to represent the phrase "times ten raised to the power of" (which would be written as "$\cdot 10^n$") and is followed by the value of the exponent. Note that in this usage, the character e is not related to the mathematical constant $\mathbf{e}$ or the exponential function $e^x$ (a confusion that is less likely if a capital E is used), and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation. The use of this notation is not encouraged by publications, however.