When a number is so large that it would be unwieldy to list all of its digits on a page, we often use a power of ten to represent some of the digits.

For example, one million is written [latex]1,000,000[/latex]: a one followed by six zeros: [latex]10^{6}[/latex].

A billion is a thousand million. It’s written [latex]1,000,000,000[/latex]: a one followed by nine zeros; [latex]10^{9}[/latex].

Or, [latex]1,000[/latex] followed by six zeros, since a thousand million is [latex]1,000[/latex] times [latex]1,000,000[/latex].

[latex]10^{3}\times10^{6}=10^{3+6}=10^{9}[/latex]

Recall, when we multiply by a million, we move the decimal point six places to the right in the number we are multiplying. That is, we multiply by [latex]10^{6}[/latex]

We can express multiples of millions or billions using a combination of digits and words.

Ex. Write [latex]35[/latex] million as a number.

[latex]35\times1,000,000=35,000,000[/latex]

Ex. Write [latex]350[/latex] million as a number.

[latex]350\times1,000,000=350,000,000[/latex]

Ex. Write [latex]3500[/latex] million as a number.

[latex]3500\times1,000,000=3,500,000,000[/latex], which is 3 billion, 500 million.

Note that the final row of the table above gives [latex]11,227[/latex] millions of dollars in hurricane damage. How much is that in billions?

Ex. Write [latex]11,227[/latex] million as a number.

[latex]11,227\times1,000,000=11,227,000,000[/latex], which is [latex]11[/latex] billion, [latex]227[/latex] million.

We can also write this as [latex]11.227[/latex] billion.

Ex. [latex]11.227\times1,000,000,000=11.227\times10^{9}=11,227,000,000[/latex] by moving the decimal [latex]9[/latex] places to the right.

Say the mean of a sample is given as [latex]\bar{x}=12[/latex] and the observations [latex]7[/latex] and [latex]15[/latex] are contained in the sample. Which value is closer to the mean?

For the value of [latex]7[/latex], [latex]x-\bar{x} = 7-12=-5[/latex]

For the value of [latex]15[/latex], [latex]x-\bar{x} = 15-12=3[/latex]

We might be tempted to conclude that [latex]7[/latex] is closer since [latex]-5[/latex] is a smaller number than [latex]3[/latex]. But distance is calculated using absolute value. The value of [latex]7[/latex] is [latex]5[/latex] units away from the mean (to the left) while the value of [latex]15[/latex] is only [latex]3[/latex] units away from the mean (to the right). To calculate which is closer, use absolute value.

For the value of [latex]7[/latex], [latex]|7-12|=|-5|=5[/latex]

For the value of [latex]15[/latex], [latex]|15-12|=|3|=3[/latex]