Hi, Filippo.

That’s what I do in a case like 8^(2/3), where I’d rather do the cube root first when it gives a nice answer (here 2^2 = 4), though both orders give the same real result; and as we saw here, it’s also good when you want to find all values of something like (-4)^(2/4). But perhaps the more important lesson is simply that we need to consider *all* roots, not just the *principal* root, when we are dealing with complex numbers.

Thanks in advance.

Filippo,

a math entusiast

Dave,

The last example you wrote is pretty much what I use to argue to miscommunication of the expressions. I would notice if you changed (2-3(4-5))÷(7-2)(2+3) to (2-3(4-5))÷(7-2)·(2+3) . And that’s often how those arguing for strict PEDMAS will “rewrite” the expressions by saying “just think of it THIS way”. I would then say, “but that’s not how it is written and why there is ambiguity in how to solve the equation.” I was taught the IMF method beginning with Algebra. And yes I first learned, in Elementary school, strict PEDMAS except they also used strict notations without implied multiplication.

]]>Please explain how this answers the question.

Can you justify the formula?

And is there a reason it doesn’t use “f¹”? And did you mean L+[f²/(f⁰-f²)]i?

]]>Apply

Mode or Z= L+[f²/f⁰-f²]i

L: lower limit of modal class

f¹: frequency of modal class

f⁰: f of pre-Z class

f²: f of post-Z class

i: size of the class

Hi, Don.

Yes, this is redundant, in the sense of summarizing what I’ve said in this series. I’m okay with that!

It happens that I had two conversations in my tutoring center today that touched on this issue. First, while things were quiet (first week of the semester), one tutor gave another an example of this sort. We all agreed that it’s best not to write it, though one focused on the “multiplication and division left to right” idea and PEMDAS; I pointed out that 8**÷**2(2+2) is rather naturally read as 8÷[2(2+2)] for visual reasons, but 8**/**2(2+2) can be easily seen as a fraction times a sum, so it *feels* more ambiguous. Both ambiguity and disagreement are (related but somewhat different) reasons not to write it.

Later I was working with a pre-algebra student, making up order of operations problems for her, and wrote down something like (2-3(4-5))÷(7-2)(2+3), in order to see if she had absorbed the left-to-right concept. But then I realized that this would be taken differently by IMF followers if they pay close attention, so I quietly added a dot: (2-3(4-5))÷(7-2)·(2+3). Why? Just to avoid accidentally doing something I recommend avoiding, though she’d never notice it. I wonder how many of the people who argue about this would notice?

]]>I think people are making things way more complicated than necessary.

When evaluating 8/2(2+2), or any variant of the same form, performing the addition is undeniably a valid first step. That leaves us with 8/2(4). Two operations remain: multiplication and division.

In order to complete the evaluation, there is one, and only one, decision to be made: which of the two operations should be done first?

We make a choice, do the calculation, and we’re done, end of story.

But there are endless discussions are about which one to choose and why; discussions full of flawed reasoning and various misunderstandings.

There is no decisive and authoritative ruling that I know of, nor any sign of a consensus as far as I can tell. That being the case, I think this form has to be considered badly notated. There is some support for this view in the ISO 80000 documents. (side opinion: “ambiguous” is the word most often used, but I don’t think disagreement and ambiguity are the same thing).

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