# Scientific Notation

## Key Questions

• A number in scientific notation is in the form

$a \cdot {10}^{b}$

To convert to a real number, write out $a$ then move the decimal point depending on $b$'s sign.

If $b$ is positive, move the decimal point to the right
If $b$ is negative, move the decimal point to the left

For example,

$1.23 \cdot {10}^{5}$

Moving the decimal point 5 places to the right, we have

$123000$

$4.56 \cdot {10}^{-} 5$

Moving the decimal point 5 places to the left, we have

$0.0000456$

• First, observe what happens when a particular number is multiplied or divided by multiples of 10.

$123.45 \cdot 10 = 1234.5$ Decimal place moved by 1 place to the right

$123.45 \cdot 100 = 12345$ Decimal place moved by 2 places to the right

$123.45 \cdot 10000 = 1234500$ Decimal place moved by 4 places to the right

$67.89 \cdot \frac{1}{10} = 6.789$ Decimal place moved by 1 place to the left

$67.89 \cdot \frac{1}{100} = 0.6789$ Decimal place moved by 2 places to the left

$67.89 \cdot \frac{1}{10000} = 0.6789$ Decimal place moved by 4 places to the left

Remember that a number's multiples can also be expressed exponential form

$1 = {10}^{0}$
$10 = {10}^{1}$
$100 = {10}^{2}$
$10000 = {10}^{4}$
$\frac{1}{10} = {10}^{-} 1$
$\frac{1}{100} = {10}^{-} 2$
$\frac{1}{10000} = {10}^{-} 4$

A number in scientific notation form is in the form

$A \cdot {10}^{b}$

where $A$ is a rational number in decimal form.

To convert to a number in scientific notation form,
move the decimal place by $b$ places. If $b$ is negative, move to the left. If $b$ is positive, move to the right