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# Introduction to standard form

Small or big numbers are generally written in standard form or scientific notation. This makes the numbers easier to write, shows the scale of the number, and makes it easy to compare quantities.

A positive number is in standard form if it is written as $$\Tblue{A} \times 10^\Tred{n}.$$ The number $\Tblue{A}$ must be between 1 (included) and 10 (excluded) in absolute value. It is sometimes called the mantissa. The number $\Tred{n}$ is an integer, called the exponent.

$7600$ is written in standard form $\Tblue{7.6}\cdot 10^\Tred{3}$. The mantissa is $\Tblue{7.6}$ and the exponent is $\Tred{3}$. The scientific notation for $-0.02$ is $\Tblue{-2}\cdot 10^{\Tred{-2}}$

Some examples:

\begin{align*} 123.4 & = \Tblue{1.234}\times 10^\Tred{2},\quad-32,\!124 = \Tblue{-3.2124}\times 10^{\Tred{5}},\\ &0.32 = \Tblue{3.2}\times 10^{\Tred{-1}},\quad 0.032 = \Tblue{3.2}\times 10^{\Tred{-2}}. \end{align*}

The exponent gives the position of the first significant figure.

$100 = 10^\Tred{2}$ has $\Tred{2}+1 = 3$ digits left from the decimal point.

$0.01 = 10^{\Tred{-2}}$ has $\Tred{2}-1 = 1$ zero right from the decimal point.

Saturn is 116 000 000 m in diameter. E. coli cells are 0.000 002 m long. These numbers are easier to handle using standard form.