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Here's a common question among math teachers and students (and math dabblers who just like to raise people's hackles!):
"If you see 3/2x, how do you interpret it? Is it 3 divided by 2x? Or 3 divided by 2, times x? Order of operations says we do division and multiplication left-to-right, which leads to the second answer. However, if you look at the slash as a division symbol, it appears to be the other way: 3 is the numerator and 2x is the denominator."
The correct answer to this question is: it's neither.
That's right. It's neither 3/(2x) nor 3/2 times x.
So no matter which way you were arguing, you're wrong. Let me explain.
Whenever you come across something like this: 3/2, the standard reading is not "three divided by two." You read it as "three over two," (this is considered to be the proper designation for the slash symbol when used in this context). This lends credence to the notion that the slash is being used as a fraction bar, and therefore, our example should be read as a fraction: 3/(2x).
But did you know that there are specific rules for how you write fractions using standard typographic practices? First, you are expected to use a specific slash symbol, which is not your standard "forward slash" on your keyboard - it's a unicode symbol called "fraction slash." The fraction slash is designed with minimal kerning (space between characters), and there's a very good reason for this.
There's another typographical practice we must follow: we superscript the numerator and subscript the denominator. The superscripts and subscripts, combined with the minimal kerning, result in the numerator being above the slash, and the denominator below.
Thus, we would either write: 3/2x or 3/2x, and now you can see that proper typographic practices makes it clear which way we intended it to be interpreted.
In other words, 3/2x is actually just a typographical error, and not a real mathematical expression. It's the result of someone being lazy. (Don't worry, I've done it too!). With the sophisticated word processors we have these days, with powerful equation editors, there's no longer any excuse for any mathematician to type 3/2x. In fact, with equation editors, you can get expressions that appear much nicer than the ones that you create with superscripts and subscripts.
Of course, there is one place where this typographical error still shows up: calculators.
Many calculators are not designed for proper typographical display of fractions. So what do we do? We do one of the following:
- Figure out which way your particular calculator handles this expression, and always do it that way.
- The safer approach: When dealing with a calculator, always clarify your meaning by including parentheses.
Once you've settled on one of these practices, it's time to accept the fact that you haven't been arguing about some standard of mathematics, but about typography. It's now time to do some real math, and leave behind the arguments about typographical quirks!
Sixth grader Elise asks, "I don't get BODMAS. Can you help me?"
Well, Elise, this is one of my favorite questions, and I get asked this a lot. But I've never written it up on the "Ask Professor Puzzler" blog, so here we go!
Before I get started explaining BODMAS, I need to mention that in different parts of the world, this "rule" is known by different names. You call it BODMAS, but some people call it PEMDAS. So when I explain what it means, in parentheses I'll explain what it means to people who call it something different.
The six letters each stand for something you can do to combine numbers in mathematics:
B = Brackets (P = Parentheses)
O = Order (E = Exponents)
D = Division
M = Multiplication
A = Addition
S = Subtraction
These six letters indicate the order in which you do the operations in a mathematical expression. For example, if you see the following:
2 - (3 + 2),
You notice that "3 + 2" is in brackets (parentheses), so you do that FIRST: 3 + 2 = 5
Now you have
2 - 5 = -3
Why does it matter which order you do things? It matters because you would get a different answer if you did the subtraction first:
2 - 3 = -1
-1 + 2 = 1
Uh oh! One way we get -3, and the other way we get 1!
Here's another example. Suppose you have 1 + 2^3 (1 + 2 cubed). If you did that from left to right, you would add 1 + 2 and get 3. Then you would cube that and get 27.
But what you're SUPPOSED to do, is evaluate the exponent first: 2 cubed is 8. Then you add 1 + 8, and get 9. It all depends on what order you do things, so you have to get the order right!
You see, we have to have an order of operations, or nobody would ever calculate expressions the same way. Order of Operations is a rule that helps to make sure EVERYONE evaluates the same expression in exactly the same way. If we didn't have order of operations, people would get different answers for the same problem, and that would be horrible - nothing would ever get done, and none of our technology would work right because teams of engineers would always be fighting over how to evaluate the equations and formulas they work with, and if they didn't use proper order of operations, not only would things not work right, you could end up with some pretty horrible catastrophes (imagine engineers using heat formulas in nuclear reactors or power plants, and not calculating correctly how much cooling they need!).
Now here's the tricky part (and I've even had emails from MATH TEACHERS who don't understand this part!): you DON'T do all your multiplication before all your division, and you DON'T do all your addition before all your subtraction. Multiplication and division are on the same order of priority, and addition and subtraction are on the same order of priority. If a problem has both multiplication and division operations, you do them from left to right. If a problem has both addition and subtraction, you do them from left to right as well.
So really, it should be written BO (DM) (AS)* to remind you that division and multiplication go together, and addition and subtraction go together.
* Or PE (MD) (AS)
If you want some practice using BODMAS (PEMDAS), you can try our One To Ten Game, which challenges you to put toghether expressions that add up to all the numbers from one to ten.
We have a new math 6th grade series where Order of Operations has changed and PEMDAS is no longer the standard. Have you encountered this yet? We have Glenco, 6th grade. Multiplication and division still come before addition and substranction, however, now it works left to right whereas before it was multiplication before division.
In some areas of the world they use a different acronym (such as BODMAS or BEDMAS), but these are still the same thing as PEMDAS (Please Excuse My Dear Aunt Sally).
Believe it or not, the PEMDAS order of operations is not only still correct, but it's always been what you just described.
The acronym PEMDAS can be deceptive, if it’s not taught correctly. How it should be taught is:
P: Parenthesis first
E: Exponents next
MD: Multiplication and Division next
AS: Addition and Subtraction last
Notice that the M and D are grouped together, as are the A and S. This is because Multiplication and Division are at the same priority level, and should be done in left to right order. Likewise, Addition and Subtraction are at the same priority level, and should be done in left to right order.
Unfortunately, many teachers don’t realize this, and teach that all multiplication is done before all division, and all addition is done before all subtraction. I was taught that way all through elementary school, and it wasn’t until I hit Jr/Sr high that I found out that Multiplication and Division are at the same priority level, as are Addition and Subtraction.
If you are looking for a game that forces students to think through Order of Operations, here's a game I created several years ago: One To Ten.