Rounding Numbers

"To five places" means "to five decimal places". First, I count out the five decimal places, and then I look at the sixth place:

I've drawn a little line separating the fifth place from the sixth place. This can be a handy way of "keeping your place", especially if you are dealing with lots of digits.

The fifth place has a 9 in it. Looking at the sixth place, I see that it has a 2 in it. Since 2 is less than five, I won't round the 9 up; that is, I'll leave the 9 as it is. In addition, I will delete the digits after the 9. Then π, rounded to five places, is:

First, I go back to the original number (not the one I just rounded in the previous example). I count off four places, and look at the number in the fifth place:

The number in the fifth place is a 9, which is greater than 5, so I'll round up in the fourth place, truncating the expansion at four decimal places. That is, the 5 becomes a 6, the 9265... part disappears, and π, rounded to four decimal places, is:

First, I ignore my result from the previous exercise and instead go back to the original number. I count off three decimal places, and look at the digit in the fourth place:

The number in the fourth place is a 5, which is the cut-off for rounding: if the number in the next place (after the one you're rounding to) is 5 or greater, you round up. In this case, the 1 becomes a 2, the 59265... part disappears, and π, rounded to three decimal places, is:

...(ten-thousands) (thousands) (hundreds) (tens) (ones) (decimal point)

(decimal point) (tenths) (hundredths) (thousandths) (ten-thousandths)...

"The nearest thousandth" means that I need to count off three decimal places (tenths, hundredths, thousandths), and then round:

Then π, rounded to the nearest thousandth, is:

The hundredths place is two decimal places (just like "hundred" has two zeroes), so I'll count off two decimal places, and round according to the third decimal place:

Since the third decimal place contains a 6, which is greater than 5, I have to round up. But rounding up a 9 gives a 10. In this case, I round the 79 up to an 80: