Order of Operations

So if you’re on social media, you will sooner or later get exposed to one of those math questions tagged with “What’s the answer:  9 out of 10 people get this one wrong!!!!!”

Many of these deal with mathematical problems:  8 \div 4 \times 2 - 1, for example.  Before talking about this, here’s an important insight:

Order of operations is like driving on the left side of the road:  it’s a convention, decided by society, and there’s no right or wrong as long as everyone agrees on it.  

In fact, the existence of a standard order of operations is because there was no general agreement, so you couldn’t be sure how to answer a question like 8 \div 4 \times 2 - 1.  It was only during the twentieth century that order of operations became standardized.

So what’s the standard?  The basic rule of order of operations is this:  All operations are to be performed from left to right unless

It’s the unless that makes it complicated:

  • Expressions inside grouping symbols (parentheses usually, but there are implied grouping symbols; I’ll return to this topic later) go first.
  • Multiplication and division are equiprecedent:  they’re dealt with, left to right, at the same time.
  • Addition and subtraction are equiprecedent:  they’re also dealt with, left to right, at the same time.

Those who remember PEMDAS may think I’ve omitted something.  But read on…

Let’s take a look at that 8 \div 4 \times 2 - 1.  We have a division and a multiplication, which are equiprecedent, so we proceed from left to right:

  • We find 8 \div 4 = 2,
  • Next we find 2 \times 2 = 4,
  • Finally we find 4 - 1 = 3.

Now many of us learned PEMDAS:  parentheses, exponents, multiplication and division, addition and subtraction.  One problem with this nifty little mnemonic is that it makes it appear that multiplication should be done before division, and addition before subtraction.  But again, multiplication and division are equiprecedent:  they’re done at the same time, left to right; likewise addition and subtraction.

What about exponents?  This one’s a little tricky, but it comes down to this:  Exponentiation isn’t an operation:  it’s shorthand.  In particular, when you write 2^{3}, you are not actually performing an arithmetic operation; you’re using mathematical shorthand:  2^{3} = 2 \times 2 \times 2.

So consider:  3 \times 2^{3} \div 4.  You don’t do exponents first.  You do recognize that 2^{3} = 2 \times 2 \times 2, so this problem is really 3 \times 2 \times 2 \times 2 \div 4.  Now you can do the multiplication and division from left to right to get the final answer:  6.

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