# Scientific Notation

Recall that *large* (or *big*)
numbers are far away from zero, and
*small* numbers are close to zero.

Large and small numbers can be positive (to the right of zero) or negative (to the left of zero).

In this section, you'll do lots of work with powers of ten.

Recall that $\,10^3 = 1000\,$ and $\displaystyle\,10^{-3} = \frac{1}{10^3} = \frac{1}{1000}\,$.

More generally, $\,10^n\,$ is a ‘$\,1\,$’ followed by $\,n\,$ zeros, and $\,10^{-n}\,$ is $\,1\,$ over $\,10^n\,$.

Multiplying by
$\,10^3\,$ is the same as multiplying by
$\,1000\,$.
Multiplying by ten to a *positive* power
makes a number bigger.

Multiplying by
$\,10^{-3}\,$ is the same as dividing by
$\,1000\,$.
Multiplying by ten to a *negative* power
makes a number smaller.

Numbers that are very large,

like

$\,B = 23{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000{,}000\,$

and very small,

like

$\,S = 0{\bf.}00000000000000000000000000000000000000000000000006789\,$

have long, inconvenient representations
using standard notation.
*Scientific notation* gives a
representation for large and small numbers that is much more compact
and easier to work with.

Here are the scientific notation names for the big number
$\,B\,$ and the small number
$\,S\,$ given above:

$B = 2.3 \times 10^{43}$

$S = 6.789 \times 10^{-50}$

Notice that there are two ‘parts’ to each number, separated by the ‘times’ symbol, ‘$\times$’:

- the first part is a number between $\,1\,$ and $\,10\,$;
- the second part is a power of ten.

Notice that big numbers have ten raised to a positive power, and small numbers have ten raised to a negative power.

Here are some more examples:

$3.1 \times 10^{-5}\,$ | is a small positive number (close to zero, to right of zero) |

$-3.1 \times 10^{-5}\,$ | is a small negative number (close to zero, to left of zero) |

$3.1 \times 10^{5}\,$ | is a big positive number (far from zero, to right of zero) |

$-3.1 \times 10^{5}\,$ | is a big negative number (far from zero, to left of zero) |

Usually, in algebra and beyond, the
‘$\,\times\,$’ symbol is
*not* used for multiplication,
because
it can too easily be confused with the variable $\,x\,$.
Scientific notation is the exception to this rule!

The precise definition of scientific notation follows.

Recall that the * integers*
are the numbers:

## Examples

*not*in scientific notation, because the first part must be strictly less than 10.

*not*in scientific notation, because $\,0.7\,$ is not an integer.

*
Changing a number from scientific notation back
to standard notation
*
is a repeated application of the following
simple rules:

- When you multiply by ten, you move the decimal point one place to the right. Thus, multiplying by (say) $\,10^5\,$ moves the decimal point five places to the right.
- When you divide by ten, you move the decimal point one place to the left. Thus, multiplying by (say) $\,10^{-5}\,$, which is the same as dividing by $\,10^5\,$, moves the decimal point five places to the left.

In both cases, you fill in spaces with zeroes as needed, as the following examples illustrate:

Numbers have lots of different names, and one of the most common ways to rename a number is to multiply by one in an appropriate form. This idea is used to convert a number in standard notation to scientific notation: you first ‘stretch or shrink’ to a number between $\,1\,$ and $\,10\,$, and then restore to the original size, as illustrated next:

*stretches*to get between $\,1\,$ and $\,10\,$; multiplying by $\,10^{-5}\,$

*shrinks*back!)

People don't typically write out all these steps (phew)! Instead, most people use the following ‘shortcut’ :

*Changing a number from standard notation to scientific notation:*

- Move the decimal point to obtain a number between $\,1\,$ and $\,10\,$, counting the number of places it is moved.
- If you moved it $\,n\,$ places to the right, then multiply by $\,10^{-n}\,$.
- If you moved it $\,n\,$ places to the left, then multiply by $\,10^n\,$.