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© Copyright 1999, Jim Loy
When the arithmetic process of rounding is mentioned, someone often asks, "Do you mean rounding up or down?" "Rounding down" is ambiguous, and is not an expression used by mathematicians, nor is it found in arithmetic text books in grade school. It shows a small misunderstanding of the term "rounding."
Rounding is a way of dealing with fractions. Sometimes fractions are not needed. They may just get in the way and complicate a simple situation. It may be handy to just eliminate the fraction. Rounding does this by producing the nearest integer:
3.00 --> 3 3.14 --> 3 3.5 --> ? 3.76 --> 4
As you can see 3.14 gets rounded to 3, and 3.76 gets rounded to 4. The case of 3.5 is a slight problem, as it is equally distant from 3 and 4. Traditionally, 3.5 is rounded to 4. 1/2 becomes 1. People often call that "rounding up" or "we round .5 to the next higher integer." We will examine the problems involved in the expressions "rounding up" and "rounding down," later.
Why should we ever want to get rid of a fraction at all? Consider this number:
$3,728,567.56
Let's say that that is how much money someone has in the bank, according to his/her last statement. But ask this person how much he/she has in the bank, and he/she may say "about $4 million," or maybe "over $3 million," or maybe "none of your business." For most purposes, "about $4 million" is much more useful than $3,728,567.56. It is easier to remember, and easier to do arithmetic with. And the super accuracy of $3,728,567.56 may be deceptive. This person may have made deposits and withdrawals since the statement, rendering $3,728,567.56 inaccurate. $4 million may be just as accurate. If he/she made a deposit of $100, then $3,728,567.56 is incorrect (the accuracy is misleading), and $4 million is still relatively accurate.
In accounting, interest often comes out to fractions of a penny. So rounding is done there. It has to be rounded, as you are never paid a fraction of a penny. Interest may produce this situation: 2/3=.6666... This infinitely repeating decimal is unhandy. So we usually round: 2/3=.667, for example. Here we didn't round off the entire fraction (2/3=1). We rounded the third decimal place. We might have stopped at other decimal places (.67 if we were dealing with money). In essence, we multiplied by 1000, rounded, and then divided by 1000:
1000 x 2/3 = 666.666...
= 667 (rounded)
667/1000 = .667
In the above paragraph, I used an equal sign for 2/3=.667. It is not equal, is it? I could have used approximately equal (an equal sign with two wavy lines, similar to a ~). But, for many practical purposes, the equal sign is fine, especially since I do not have an approximately equal sign available to me. An equal sign is often used when it technically should not be used: "Billy is 7 years old and Mary is 6..." Well, we know that today is probably not their birthdays (or that they were born at the same second of different years). But to solve the puzzle, we say B=7 and M=6.
Another useful way to get rid of a fraction is to just drop the fraction:
3.00 --> 3 3.14 --> 3 3.5 --> 3 3.76 --> 3
This is called "truncation" or "dropping the fraction." People often call this "rounding down." Computers often truncate. It is easier than rounding. You often have to explicitly tell the computer (by programming it) to round rather than truncate. People sometimes (rarely) choose to truncate rather than round. Bowling averages are truncated. If I bowled 180+180+182=542, then my average was 180 (not 180.666...). That is the law.
So, people say "rounding down" when they mean "drop the fraction." What is wrong with that? Consider a person who is supposed to round 3.5. He or she asks, "Do I round that up or down?" There is rampant confusion here. If we are dropping fractions, this is 3. We are probably not dropping fractions, however. We are probably rounding. The person's question is still valid. He/she probably cannot remember the rule. 3.5 is midway between 3 and 4. Which way should he/she round this? Above, I said that "traditionally" 3.5 is rounded to 4. We could have rounded it to 3 (rounding down, as the expression should be used). But the rule is that we round .5 to the next larger integer. We never round down.
"Rounding down" has two conflicting meanings. Technically, we never round down. For most purposes, we round (and .5 goes to the next higher integer, rounding up). For a few purposes (bowling) we just drop the fraction (which is not rounding down). Am I being picky? Well, when I hear "rounding down" I am confused, as I do not know which meaning the person meant.
There is a story that a smart programmer, for some bank, got rich by truncating instead of rounding. Supposedly, he made his accounting program deposit all of the fractions of pennies (when figuring interest) into his own account. And he got rich. Is this urban legend (i.e. false) or the truth? Apparently, the story is true. But, he should have gotten caught right away. The books balance, but in an odd way that violates accounting practices. Few of the customers received the right amount of interest (over a period of time). The customers didn't care (it was only a penny every other month or so). But the bank cares. And a simple hand calculation shows that the interest is wrong.
What if I write this short program, in BASIC:
10 IF 1/3+1/3+1/3=1 THEN PRINT "EUREKA!"
Will this program print "EUREKA!"? Probably not, because in a computer there is no exact representation for 1/3. We have three numbers which don't quite equal 1/3. There sum does not quite equal 1. This is called rounding error, or truncation error (depending on how BASIC deals with fractions like 1/3). This kind of error has many devious effects.
Addendum:
In computer programming (and sometimes in mathematics) truncation (dropping the fraction) is also called the floor() function. This is abbreviated with a little L shaped bracket, like the "[" square bracket without the top horizontal bar.
Also, in computer programming, there is another operation, similar to truncation (floor()) and rounding, in which any fraction (other than .0000...) goes to the next higher integer. This is called the ceiling() function, and is abbreviated with a sign like the "[" square bracket without the bottom horizontal bar. Grocery stores actually use this function. When something is on sale in certain quantities, like three for a dollar, most stores (not all stores and not every time) give you the benefit of the sale for fewer items. Three for a dollar becomes one for 34 cents. They are not being greedy here (much), they are just preventing the ridiculous situation of customers buying one item (at 33 cents) three times to save a penny.