Scientific Notation

Scientific notation is a way of expressing numbers that are either very large or very small in a short and convenient form. The scientific notation takes the following form:

\[s \cdot 10^n \]

where \(s\) is the so-called significand or mantissa and is multiplied with 10 to the power of \(n\), with \(n\) being the so-called exponent. If the number is negative then a minus sign precedes the significand \(s\).

Example \[\begin{align*} 2 &= 2 \cdot 10^0 \\ 300 &= 300 \cdot 10^0 = 30 \cdot 10^1 = 3 \cdot 10^2 \\ -10.5 &= -1.5 \cdot 10^1 \end{align*} \]

The exponent of 10 indicates how many places the decimal point is moved. A positive exponent indicates that the decimal point is moved to the right, while a negative exponent indicates that the decimal point is moved to the left.

Example \[\begin{align*} 0\textcolor{red}{.}00004 =& 0\textcolor{red}{.}00004 \cdot 10^0 \\ =& 00\textcolor{red}{.}0004 \cdot 10^{-1} \\ =& 000\textcolor{red}{.}004 \cdot 10^{-2} \\ =& 0000\textcolor{red}{.}04 \cdot 10^{-3} \\ =& 00000\textcolor{red}{.}4 \cdot 10^{-4} \\ =& 000004\textcolor{red}{.} \cdot 10^{-5} \\ =& 4 \cdot 10^{-5} \end{align*} \]

Normalized Notation

As you can see in the second example there can be ambigous representations, which is why the normalized notation was introduced. In the normalized form the exponent is chosen so that the significand is at least 1 but less than 10, so \(s \in [1,10)\).

Example \[\begin{align*} 2 &= 2 \cdot 10^0 \\ 300 &= 3 \cdot 10^2 \\ -53'000 &= -5.3 \cdot 10^4 \\ 0.2 &= 2 \cdot 10^{-1} \end{align*} \]

The “E” Notation

Because displaying exponents like \(10^-5\) can be inconvenient to display or type on a computer or calculator, the letter “E” or “e”, for “exponent”, is often used to represent “s times ten raised to the power of n”.

Example \[\begin{align*} 0.00004 &= 4 \cdot 10^{-5} = 4E-5 = 4e-5 \\ 300 &= 3 \cdot 10^2 = 3E2 = 3e2 \end{align*} \]