APPROXIMATIONS

We frequently don ’ t need to know an answer to many decimal places. When we give someone directions to drive to our house, we usually say something like, “Get off the highway at exit 14, go right and follow the road for about 12mi until you see an old church on the right.”

We could have said “Follow the road for 11.87 mi” but “about 12” gives enough information to tell someone when to start looking for the church. I don’t need to delve into the theory of approximations. Instead, I’ll use some commonsense rules, such as “about 14 mi” means that the number is closer to 14 than it is to 13 or 15.

The mathematical expression
means that “x is approximately equal to 14.” Other ways of writing this are x ~ 14 and x = 14.

The number 2,123,774 has seven significant figures. It is approximately 2 million. I have to be careful, however, not to say that the number is approximately 2,000,000 because when I do this I am implying that I precisely know all the digits I put down: that is, that the last digit is indeed 0 and not 4.

All of the above issues are resolved by using a notation sometimes called “sci­entific” notation, which specifies exactly how many digits of a number are known and how many are just place keepers (zeros to the left of the decimal place). For the purposes of this book, I’ll be careful just to use commonsense approximations and not worry about the mathematical implications.

Another name for approximating is rounding. When I know a currency amount to the nearest penny and I round it to the nearest dollar, I’ m approximating the amount to two fewer significant figures than I started with. The rules are simple. If the amount after the decimal place is 50 or less, just drop this amount. If the amount after the decimal place is 51 or more, add 1 to the number of dollars and drop the amount after the decimal place. A few examples:

$151.23 rounded to the nearest dollar is $151.

$26.76 rounded to the nearest dollar is $27.

$315.00 rounded to the nearest dollar is $315.

You can also round to the nearest ten dollars, to the nearest million dollars, and so on. For example,

$514,676.26 rounded to the nearest hundred thousand dollars is $515 hundred

thousand dollars—not $515,000.00 or $515,000.

Rounding to the nearest billion dollars is usually restricted to members of Congress.

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