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Order of Operations

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Date: 05/19/99 at 13:54:24
From: Stephanie Wu  and Meghan Heil
Subject: Algebraic expressions and order of operation

The problem was presented like this:

a = 1.56
b = 1.2
x = 7.2
y = 0.2

ax/by = ?

Here are two ways that I solved it:

1) I first rewrote the problem as [1.56(7.2)/ 1.2](0.2). Second, a was
multiplied by x. The product was 11.232. Then, since no parentheses
were present, I followed the order of operations and divided 11.232 by
b, which was 1.2. The quotient was 9.36. Then I multiplied 9.36 by y,
which was 0.2. The final answer was 1.872.

2) The other way, the first thing I did was multiply a by x. The
product, which was 11.232, was set aside for the time being. Then b
was multiplied by y, which gave the product of 0.24. The problem was
now solved by dividing 11.232 (or ax) by 0.24 (or by) to reach a final

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Date: 05/19/99 at 17:03:49
From: Doctor Peterson
Subject: Re: Algebraic expressions and order of operation

Hi, Stephanie and Meghan.

questions about expressions similar to yours, from confused teachers
and students who have found that different books or teachers have
different answers, and even calculators disagree.

ax/by

should be evaluated left to right: a times x, divided by b, times y.
The multiplication is not done before the division, but both are done
in the order they appear. Your first solution is right.

Some texts make a rule, as in your second solution, that
multiplication without a symbol ("implied multiplication") should be
done before any other operations in an expression, including "explicit
multiplication" using a symbol. Following this rule, you would
multiply a by x, then multiply b and y, then divide one  by the other.
Some (probably most) texts don't mention such a rule - but some of
those may use it without saying so, which is far worse.

I don't know of a general rule among mathematicians that implied
multiplication should be done before explicit multiplication. As far
as I'm concerned, all multiplications fit in the same place in the
order of operations. It's not an unreasonable rule, though, since it
does seem that implied multiplication ties the operands together more
tightly, at least visually; but the idea of Order of Operations (or
precedence, as it is called in the computer world) is supposed to be
to ensure that everyone will interpret an otherwise ambiguous
expression the same way - so if some texts change the rules, or if
people do what feels natural, the purpose has been lost.

The problem here is that the expression looks as if it were meant to
be

ax
----
by

In the Dr. Math FAQ about writing math in e-mail, one of our
recommendations is to use parentheses wherever possible to avoid
ambiguity, even where the rules should make it clear, because it can
be easy to forget them in some situations:

http://mathforum.org/dr.math/faq/faq.typing.math.html

So in e-mail we would write it like this:

ax/(by)  or  (ax/b)*y

depending on what is intended.

In my research for another Dr. Math "patient," I found that some
calculators have experimented with this rule. Calculators have
somewhat different needs than mathematicians, since they have to take
input linearly, one character after another, so they are forced to
make a decision about it. On the TI Web site I learned that they
deliberately put this "feature" into the TI 82, and then took it out
of the TI 83, probably because they decided it was not a standard rule
and would confuse people. Take a look at their explanation:

http://www.ti.com/calc/docs/faq/83faq039.htm

They also talk about a similar issue for exponentiation of the form
a^b^c, and give the same conclusion we give: always use parentheses
where a statement is ambiguous without special rules:

http://www.ti.com/calc/docs/faq/83faq058.htm

right - which means, of course, that the question itself is wrong. I
unless their own text gives the "implicit multiplication first" rule;
but in practice if I came across that expression, I would probably
first check where it came from to see if I could tell what was
intended. The main lesson to learn is not which rule to follow, but
how to avoid ambiguity in what you write yourself. Don't give other
people this kind of trouble.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra
High School Calculators, Computers
Middle School Algebra

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