Why Does Order of Operations Work the Way It Does?

Date: 05/04/2004 at 19:48:01
From: Aviv
Subject: Logic for the Order of Operations

In order of operations, it is understandable that brackets are 
focused on initially since they represent individual terms.  However, 
why do exponents then come before the other possible operations?  And
most importantly, why are multiplication and division done before
addition and subtraction?  What would happen otherwise?

What are the reasons and logic for this, if any?  I imagine that 
mathematicians a long time ago didn't just decide on it for the sake 
of agreement.

Date: 05/04/2004 at 22:43:14
From: Doctor Peterson
Subject: Re: Logic for the Order of Operations

Hi, Aviv.

I don't know that anyone ever said just why they did what they did, 
but I give my impressions in these pages, each of which points out a 
slightly different aspect:

  History of the Order of Operations
    http://mathforum.org/library/drmath/view/52582.html 

  Explaining Order of Operations
    http://mathforum.org/library/drmath/view/57199.html 

  Why Rules?
    http://mathforum.org/library/drmath/view/57031.html 

  Negative Squared, or Squared Negative?
    http://mathforum.org/library/drmath/view/61633.html 

The basic idea, as I see it, is that exponentiation is in a sense 
more powerful than multiplication, in that it distributes over 
multiplication; and likewise multiplication is more powerful than 
addition.  Another aspect is that the order we use makes the most 
interesting expressions, such as polynomials, as easy as possible to 
write.  Finally, there is a good chance that the order just arose 
naturally out of the way human language tends to express arithmetic, 
as the symbolism developed gradually from abbreviations of Latin or 
other languages; for example, "two cats and three dogs" naturally 
means (2c) + (3d) rather than 2(c + 3)d!

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

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