Why Do We Use Order of Operations?

Date: 09/18/2008 at 15:50:17
From: Martin
Subject: Why use order of operations?

Why is it necessary to use order of operations?  Why can't you just
write a calculation left to right? 

It seems redundant to follow all those rules, when every equation can
be re-written to work left-to-right.  The only useful part of the
order of operations in my opinion is putting brackets first, but the
rest seems useless.

For example:

5 + 2 ^ 3 x 2
= 5 + 8 x 2
= 5 + 16
= 21

Using the order of operations just makes this confusing as I have to
calculate all over the place.  Left-to-right would be so much easier,
and could be written with the same numbers, just rearranged, like so:

2 ^ 3 x 2 + 5
= 8 x 2 + 5
= 16 + 5
= 21

This also makes it neater because all the calculations would be solved
left-to-right, without any hassle.  Wouldn't that be much easier?

Date: 09/18/2008 at 23:24:15
From: Doctor Peterson
Subject: Re: Why use order of operations?

Hi, Martin.

No, not really.

You're right that any expression can be written regardless of what
rules we follow--but only if you use parentheses pretty heavily.  For 
example, what we write as

  2*3 + 4*5 = 26

would have to be

  2*3 + (4*5)

There is no way to write this using left-to-right evaluation without
parentheses.

One big advantage of the real rules is that they let us write
polynomials, which are very important in algebra, without parentheses:

  2x^2 - 3x + 5

in your form would have to be

  2(x^2) - (3x) + 5

since without the parentheses you would first multiply 2x, then square
that, then subtract 3, then multiply by x, then add 5, like this:

  ((2x)^2 - 3)x + 5

The normal rules were probably invented largely to make it easy and
natural to write polynomials.

[As an aside, to be honest, there actually IS a way to write 
polynomials without parentheses in your form:

  2x-3x+5

This is equivalent to what we would actually write as

  (2x - 3)x + 5 = 2x^2 - 3x + 5

This  is a very efficient way to evaluate polynomials, commonly used
on computers and closely related to synthetic division.  However, this
notation hides such important things as the degree of the polynomial
and the degree of each term, which makes it awkward for actual use,
especially when there are missing terms.]

As a side-effect of this fact, our notation makes it possible for us
to talk about "terms" and "coefficients" in an expression.  In either
of the alternative forms above for the polynomial 2x^2 - 3x + 5, there
is no such thing as terms; they are either obscured by required
parentheses, or not present at all.  The very idea of terms and
coefficients is built on the idea that a polynomial is a SUM of
PRODUCTS, in which multiplication is done before addition.  A term is
a product of coefficients and variables, like 2x^2; these are all
first calculated and then added together.  Our rules are designed to
make this easy to write.

I imagine if the left-to-right rule had been used from the beginning,
we'd have different ideas of what it means to simplify, and of what
kinds of expressions are most interesting to work with; but the
notation that was developed in the 1500's was what they needed in
order to talk about what they already were interested in even before
there were nice symbols (namely polynomials), and it's worked well
ever since.

It's also worth mentioning that our notation is partly designed to
make the rules feel natural.  First, we write multiplication without a
symbol, making "2x" feel like a single quantity and "2x + 3y" look
like you should first multiply and then add.  [In your notation, you
probably would not want to allow this way to indicate multiplication,
but would require use of the symbol, since multiplication would not
play a special role.]  Second, we write exponents as superscripts,
setting them apart and imparting an asymmetry to the notation that
exactly fits the non-commutativity of that operation; as long as you
remember that the exponent is attached only to the item just before
it, it is natural to see "2x^2" as the product of 2 and x^2.  This
takes away some of the force of your objection that the rules are
confusing and hard to follow.

Another very important advantage is that the standard rules fit very
well with the properties of numbers, making algebraic manipulations
feel natural.  For example, addition is commutative, so we can swap
the order of terms, like "2x + 3y" becoming "3y + 2x".  But in your
notation, the former would have to be written as 2x+(3y), and when you
commute the addition you would have to remember to add parentheses
around the term you move to the end: 3y+(2x). It's no longer just a
matter of moving symbols around; you'd have to think more!

One more example: What does the distributive property look like in
your notation?  Here are two examples in standard notation:

  2(3 + 4) = 2*3 + 2*4         (2 + 3)4 = 2*4 + 3*4

In left-to-right notation, the first of these would have to be written as

  2(3 + 4) = 2*3 + (2*4)

You have to remember the parentheses on the right side.  But the
second example is worse.  Here's how you'd write it:

  2 + 3 * 4 = 2*4 + (3*4)

You'd no longer have to use parentheses on the left side, so
distribution would not be thought of as a way to eliminate
parentheses, and would not consistently be applied across parentheses.

Do you see now that the combination of notation and rules that we have
is just about perfect for algebra?  You don't see it when you're just
working with numbers, but as soon as you start using variables and
moving things around, you find that the way we write expressions makes
it really easy to do what algebra requires.

Now imagine going back before the 1500's, when algebra problems were
written out in words (in Latin or Arabic)!  No wonder math and science
made such huge strides in the 1600s, giving us calculus, the orbits of
planets, and worldwide navigation.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/19/2008 at 17:43:17
From: Martin
Subject: Thank you (Why use order of operations?)

Thanks a lot.  I guess that things are smartest the way they are,
especially with millions of people constantly looking over and
criticizing them (cough, cough).  Keep up the good work!

Date: 09/20/2008 at 20:50:03
From: Doctor Peterson
Subject: Re: Thank you (Why use order of operations?)

Hi, Martin.

Just to make sure I haven't overstated my case, not ALL notation in
math is perfect yet!  There are several areas (such as logarithms)
where our notation is still at least a little awkward, and you should
not give up on asking whether they can be improved.  Sometimes nobody
complains just because we're used to it and know that it would be next
to impossible to get everyone to change; yet notation does keep
changing, at least in little ways, and there's always a chance that
some new idea will take over because it is clearly superior.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/