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:

$$\,\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$$

DEFINITION scientific notation

A number is in scientific notation if and only if it has the form

$$d\times 10^n$$

where $\,d\,$ is a number satisfying $\,1 \le d \lt 10\,$ and $\,n\,$ is an integer. The number $\,d\,$ should be expressed as a decimal (as needed).

Examples

$1.027 \times 10^{-2}\,$ is in scientific notation. Here, $\,d = 1.027\,$ is a number between $\,1\,$ and $\,10\,,$ and $\,n = -2\,$ is an integer.

$7 \times 10^{1024}\,$ is in scientific notation. Here, $\,d = 7\,$ is a number between $\,1\,$ and $\,10\,,$ and $\,n = 1024\,$ is an integer. Notice that no decimal point is needed in the number $\,7\,.$

$10 \times 10^5\,$ is not in scientific notation, because the first part must be strictly less than 10.

$2 \times 10^{0.7}\,$ is 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:

Examples

$1.027 \times 10^{-2} = 0.01027$ (Move decimal point two places to the left; it's good style to put the extra zero to the left of the decimal point.)

$7.1 \times 10^4 = 71,000$ (Move decimal point four places to the right; insert appropriate commas for easy readability.)

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:

$0.000037$

$= 0.000037 \times 1$

(Now, what name for $\,1\,$ shall we use?)

$= 0.000037 \times (10^5 \times 10^{-5})$

(Why this name for $\,1\,$? To turn $\,0.000037\,$ into $\,3.7\,,$ move the decimal point $\,5\,$ places!)

$= (0.000037 \times 10^5) \times 10^{-5}$

(re-group)

$= 3.7 \times 10^{-5}\,$

(the idea: multiplying by $10^5$ 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 $\,10^n\,.$

Examples

$0.000037 = 3.7 \times 10^{-5}$ (Move decimal point $\,5\,$ places to the right.)

$590.27 = 5.9027 \times 10^2$ (Move decimal point $\,2\,$ places to the left.)

$39{,}000{,}000 = 3.9 \times 10^7$ (Insert decimal point to right of last zero, move $\,7\,$ places to the left)

Scientific Notation

Examples

Concept Practice