Standard form

In a nutshell

Standard index form (or standard form) is a convenient way of writing very large or very small numbers.

What does standard form look like?

A number is in standard index form if it is written in the form:

$a\times10^b$

Where the number $a$ is always between $1$ and $10$ , and $b$ is a whole number.

Writing a number in standard form

To write a number in standard index form, follow this procedure.

procedure

1.	Move the decimal point until the number is between $1$ and $10$ . Count how many spaces the decimal point was moved.
2.	This number between $1$ and $10$ is the value of $a$ .
3.	The number of spaces the decimal point was moved is the value of $b$ . If the decimal point moved to the left, then $b$ is positive. If the decimal point was moved to the right, then $b$ is negative.

Example 1

What is $76000$ in standard index form?

First, write $76000$ as $76000.00$ and move the decimal point until the number is between $1$  and $10$ :

$7\overset{\curvearrowleft \curvearrowleft \curvearrowleft \curvearrowleft}{.6\, 0\,0\,0\,}0\thinspace 0$ , so $a=7.6$ 

The decimal point moved $4$ spaces to the left, so $b=4$ .

$\underline{76000=7.6\times10^4}$

Example 2

What is $0.0000815$ in standard index form?

First, move the decimal point until the number is between $1$ and $10$ :

$0\thinspace \overset{\curvearrowright \curvearrowright \curvearrowright \curvearrowright \curvearrowright}{0\,0\,0\,0\,8.}15$ , so $a=8.15$

The decimal point moved $5$ spaces to the right, so $b=-5$ .

$\underline{0.0000815=8.15\times10^{-5}}$ 

Multiplying and dividing in standard form

To multiply and divide two numbers in standard form, use your calculator to multiply the numbers. Then, make any adjustments if necessary to make sure the answer is in standard form.

Example 3

What is $(1.5\times10^8)\times(9\times10^{-3})$ ?

Using the calculator gives:

$(1.5\times10^8)\times(9\times10^{-3})=1350000$

Convert $1350000$ into standard form:

$1350000=1.35\times10^6$

$\underline{(1.5\times10^8)\times(9\times10^{-3})=1.35\times10^6}$