You’ve all seen it. British TV, British movies, or you may have visited the UK.
They drive on the wrong side of the road there.
Of course, they’d say that we do. We drive on the right side of the road and we very sensibly put the driver’s steering wheel where that is the natural thing. But in the UK, it is the other way around.
So who’s right?
The obvious answer is, they both are. It’s just that we all have to agree on which side of the road to drive, or there would be chaos.
It’s the same in mathematics. There is a debate among philosophers of mathematics as to whether numerical relationships are “real,” or simply a matter of convention. That is, whether “2 + 2 =4” is true because, as a matter of fact, 2 + 2 =4, or do we just define ‘2’ so that if we add it to itself, it equals 4.
That debate will be done, eventually, by intelligent cockroaches, long after human beings are gone. But there is this much truth to the position that concepts, in mathematics, are defined to be true, and that is, there are a lot of conventions in math just like in English, or, for that matter, just like in driving.
One of them is the Order of Operations.
There are several basic operations we do when we do math. We use parentheses, exponents, we add and subtract, we multiply and divide. We also square-root and some of what I’m talking about also applies to roots. When we do these operations one way, we get one result, and if we do them another way, we get a quite different result. Which is the right way?
There is no right way, if you mean which one corresponds to the reality of operations out there somewhere in Plato’s Heaven. But we all have to agree on them, just like we agree to the rules of the road.
So which is the right answer for the following:
4 + 2×3 = (4 + 2) ×3 = 6×3 = 18
Or I could multiply first:
4 + 2×3 = 4 + (2×3) = 4 + 6 = 10
Obviously, balancing a checkbook or sending a human to Mars will depend on which is right. JPL once crashed a Mars explorer when it switched from Imperial to metric in its calculations. (Yes, there is no evidence for the alternative theory that an ancient automated Martian defense system was activated. Sorry.) That’s a similar mistake.
So we agree. We do parentheses first, then exponents, then multiplication/division, then addition/subtraction.
If the operations are equal, we do them left to right.
For instance, 15 ÷ 3 × 4 is not 15 ÷ (3 × 4) = 15 ÷ 12 but is rather (15 ÷ 3) × 4 = 5 × 4, because, going from left to right, you get to the division sign first. Check it out on your calculator, you’ll see I’m right. (Or just accept that in math, I’m always right. Which would be foolish, as I make mistakes.)
What if the operations involve exponents?
4 x 3^3.
We do the exponent first, so this becomes:
4 x 9.
Which is 36.
What about 4 x (3+1) ^2?
Parentheses come first. So it is 3 +1 or 4. Then Exponents. So it is 4 x 4 (or four squared, but I don’t say it that way because when I do, I get an irresistible impulse to recite the Gettysburg Address.). 4 x 4 is 16. So this is 4 x 16, which is 64 or 4^3.
Easy, peasy. Just like driving.
But what if you have nested parentheses, that is, parentheses within parentheses?
4 + [–1(–2 – 1)] ^2.
You do the () first, which makes for -3
Then you do the  next. Which makes it 3.
Then the exponent. Which makes it 3 x 3 or 3 squared. (Doesn’t make me recite the Gettysburg Address. Be weird if it did. Now, 4-squared? Perfectly normal to think of Lincoln.)
4 + 9 =13.
For the triskaidekaphobes out there, change the example. Make the 4 into a 5. That would make it 14. If you’re afraid of the number 14, what can I tell you? I’m a math teacher, not a psychologist.
What if it involves fractions? Same thing. You hear that a lot in math. Same thing. That’s because a lot of things you do in math look different but are—the same thing.
4(2/3 +1/4) ^2
Do what’s in the parentheses first:
2/3 + 1/4
You find a common denominator.
Everyone get 12? Good.
So, we have 8/12 + 3/12
That makes an improper fraction, 11/12
We then square that.
Then we multiply all of that by 4
When we multiply a fraction by a whole number, we give the whole number a denominator (that’s the number on the bottom) of 1, and we get:
4/1 X 121/144 or just 4 x 121/144
Oh, hey, the order of operations again. (I knew we’d get back to it.) 4 x 121 divided by 144, as we go left to right with equal operations.
That gives us 484/144
Simplified, that’s 121/36 (We divided by 4 to simplify. You always divide by the greatest common multiple. Why? To get the smallest number as the answer. Why? Because smaller numbers are easier to work with than larger numbers.)
We do that because that’s a heck of a lot easier to divide to get the mixed number:
So, there you are.
I hope you enjoyed today’s lesson.