Why people can't math...

[Nikki Nyx]

I came across this on Facebook today. There was much heated argument over the solution to the following simple equation:
6 ÷ 2(2 + 1) = ?

One group insisted that the solution is 9. The second group, which included me, maintained that the solution is 1.

The first group did this:
6 ÷ 2(2 + 1) = ?
6 ÷ 2(3) = ?
3(3) = 9
Their explanation was that once the parenthetical addition is simplified, the 2(3) becomes implied multiplication, therefore the equation is solved from left to right.

My group did this:
6 ÷ 2(2 + 1) = ?
6 ÷ 2(3) = ?
6 ÷ 6 = 1
Our explanation was that the 3 remains parenthetical and, thus, takes precedence over the division in the order of operations, so that 2(3) must be solved before being divided into 6.

I've admitted that my math skills leave a lot to be desired, but I didn't think I'd forgotten them entirely. Would the math experts here weigh in, please?

[Gawdzilla Sama]

I'm mathgnostic.

[Austin Harper]

You have to add a × for the other people's solution to work.
6 ÷ 2 × (2 + 1) = 6 ÷ 2 × (3) = 3 × 3 = 9

Without the × your solution is correct.
6 ÷ 2(2 + 1) = 6 ÷ 2(3) = 6 ÷ 6 = 1

You can try shuffling this around a bit if it helps you to visualize it.
Let x = (2 + 1). Which of the following is correct?

6 ÷ 2x = 9
6 ÷ 2x = 1

Solving the first equation, you get x = ⅓ ≠ (2 + 1).
Solving the second equation, you get x = 3 = (2 + 1).

[gorgeous]

numbers don't exist...that's why...

[Nikki Nyx]

You have to add a × for the other people's solution to work.
6 ÷ 2 × (2 + 1) = 6 ÷ 2 × (3) = 3 × 3 = 9

Without the × your solution is correct.
6 ÷ 2(2 + 1) = 6 ÷ 2(3) = 6 ÷ 6 = 1

I thought so! Thanks, Austin! I didn't think they could have changed math THAT much since I was in school.

[Poodle]

Yeppers - bracketed expressions must be fully evaluated in all cases except in the Gorgeousii V system, where parentheses are always interpreted as a declaration of war.

[scrmbldggs]

[OlegTheBatty]

numbers don't exist...that's why...

Yeah, it's tough counting: one, another one, another one, another one . . . another one.

[scrmbldggs]

[gorgeous]

E. Lee Lady: Do Numbers and Other Mathematical Entities Really Exist?



www.math.hawaii.edu/~lee/exist.html



Certainly numbers do not have a tangible existence in the world. They exist in our collective consciousness --------------------------A friend in the Philosophy Department at the University of Kansas once said to me that numbers do not exist. They are just as fictional, he said, as the character Frodo in Lord of the Rings.

[Gawdzilla Sama]

numbers don't exist...that's why...

Yeah, it's tough counting: one, another one, another one, another one . . . another one.

One is a number, so it would be "uh, another uh, another uh, another uh . . . another uh"

[Poodle]

A friend in the Philosophy Department at the University of Kansas once said to me that numbers do not exist. They are just as fictional, he said, as the character Frodo in Lord of the Rings.

I've heard that an unimaginable number of times.

[Gawdzilla Sama]

A friend in the Philosophy Department at the University of Kansas once said to me that numbers do not exist. They are just as fictional, he said, as the character Frodo in Lord of the Rings.

I've heard that an unimaginable number of times.

I've heard it at least nine times.

[bobbo_the_Pragmatist]

Nikki: I like the way you architected the OP header.

[Gord]

Without the × your solution is correct.
6 ÷ 2(2 + 1) = 6 ÷ 2(3) = 6 ÷ 6 = 1

Yeah, I was always taught to multiply through the brackets first, like this:

6 ÷ 2(2 + 1) = 6 ÷ (4 + 1) = 6 ÷ 6 = 1

But we were also taught to never use the "÷" sign because it always always always led to confuse in new students. Therefore, I have to assume new students were being taught another way to do it. As a result, we would write the equation by putting the "6" in the numerator position and the "2(2 + 1)" in the denominator position:

    6   
2(2+1)

Similar things would come up when using variables, like this:

6 ÷ 2(x + 1) = ?

Again, we were taught to multiply through the bracket first:

6 ÷ 2(x + 1) = 6 ÷ (2x + 2) = 3 ÷ (x + 1)

[Gord]

...and that was still being done in my university in the early to mid 80s.

[bobbo_the_Pragmatist]

Gord: creating that fraction is an excellent way to "visualize" the issue........but the "rule" remains the same.

Makes me think hoomans still think visually. Ooooh......I'm kinda floating away in the universe right now.

[Gord]

Yes bobbo, but the "rule" wasn't the rule back then. Another rule was the rule.

[bobbo_the_Pragmatist]

To my memory, its always been the rule......in Japan and USA. Might be a geographical issue as well as a temporal one.

......and that only again emPHAsizes that making a fraction simply states what rule is being followed.............. until people start violating that one too.

People........are perverse ........ ha, ha.........as we both know.

[scrmbldggs]

(4 + 1) = 6?

[Lance Kennedy]

I am no great mathematician, but I disagree.

6÷2(2+1) is the same as 6/2 x (2+1) which is 9. The parenthesis is the key. It separates the second part of the equation from the first.

To the get answer 1, you would make the equation 6÷[2(2+1)]

But it is very ambiguous, and I think you could make a good case for both outcomes.

[bobbo_the_Pragmatist]

But it is very ambiguous, and I think you could make a good case for both outcomes.

Well, you were succeeding with an even "one"...but rapidly derailed. There is no "good case." You either follow "the rule" or you don't/follow some other rule.

An excellent demonstration of what "It's definitional" means in practice.

[Gord]

But it is very ambiguous, and I think you could make a good case for both outcomes.

Exactly what I've been trying to say. I just added "it used to be this way, now it's that way" 'cause I'm o-o-o-o-o-o-l-l-l-l-l-l-d.

[Austin Harper]

I agree that adding the second set of brackets does make it more clear. The real issue I think is that ÷ makes things difficult to read all the time. Were I writing this down myself, I would write everything as an unambiguous form like

____6____
2(2 + 1)

[Gord]

I agree that adding the second set of brackets does make it more clear. The real issue I think is that ÷ makes things difficult to read all the time. Were I writing this down myself, I would write everything as an unambiguous form like

____6____
2(2 + 1)

You have me on ignore, right?

[Austin Harper]

I agree that adding the second set of brackets does make it more clear. The real issue I think is that ÷ makes things difficult to read all the time. Were I writing this down myself, I would write everything as an unambiguous form like

____6____
2(2 + 1)

You have me on ignore, right?

Obviously.

[Nikki Nyx]

Without the × your solution is correct.
6 ÷ 2(2 + 1) = 6 ÷ 2(3) = 6 ÷ 6 = 1

Yeah, I was always taught to multiply through the brackets first, like this:

6 ÷ 2(2 + 1) = 6 ÷ (4 + 1) = 6 ÷ 6 = 1

But we were also taught to never use the "÷" sign because it always always always led to confuse in new students. Therefore, I have to assume new students were being taught another way to do it. As a result, we would write the equation by putting the "6" in the numerator position and the "2(2 + 1)" in the denominator position:

    6   
2(2+1)

I actually rewrote the equation as a fraction. It didn't help them. They claimed I had changed the equation. When I explained to them that the fraction one-half was the exact same thing as 1 ÷ 2, they became angry with me. Several of them posted this idiot's musings as proof that 9 was the correct answer...and I quote:

If you type 6÷2(3) into a calculator, Google or WolframAlpha, the input has to be parsed and then computed. All of these will first convert the parentheses into an implied multiplication. The expression becomes the following.

6÷2(3)
= 6÷2×3

According to the order of operations, division and multiplication have the same precedence, so the correct order is to evaluate from left to right. First take 6 and divide it by 2, and then multiply by 3.

6÷2×3
= 3×3
= 9

This gets to the correct answer of 9.

This is without argument the correct answer of how to evaluate this expression according to current usage.

Some people have a different interpretation. And while it’s not the correct answer today, it would have been regarded as the correct answer 100 years ago.

First, what moron requires a calculator to solve this equation? Second, when did evaluating expressions change? My daughter is 27, and she solved this equation correctly.

[Austin Harper]

Isn't the ÷ symbol meant to represent a fraction?
It's the horizontal bar with dots in place of the values that are written to the left and right instead?

[Nikki Nyx]

Isn't the ÷ symbol meant to represent a fraction?
It's the horizontal bar with dots in place of the values that are written to the left and right instead?

I would agree. I mean, one-half (or 1/2) and 1 ÷ 2 are the same expression, are they not?

[Gord]

I agree that adding the second set of brackets does make it more clear. The real issue I think is that ÷ makes things difficult to read all the time. Were I writing this down myself, I would write everything as an unambiguous form like

____6____
2(2 + 1)

You have me on ignore, right?

Obviously.

Who said that?

[Gord]

...Second, when did evaluating expressions change? My daughter is 27, and she solved this equation correctly.

I kind of thought it was during the early to mid 80s, from my experiences. But that's just anecdotal evidence. I'm no expert.

[Poodle]

Gorgeous, here are the opening lines of the example you presented as supportive evidence ...

"A friend in the Philosophy Department at the University of Kansas once said to me that numbers do not exist. They are just as fictional, he said, as the character Frodo in Lord of the Rings.
Certainly my own knowledge of philosophy is at best that of a dilettante. But I know enough to know for certain that on this matter he was wrong."

Is there a deep-seated reason why you would do this when it begins by stating that the author believes YOUR claim to be wrong? Or - (could it be?) - did you simply not read it at all?

[Nikki Nyx]

What?! Gorgeous quoted out-of-context, deliberately omitting the relevant portion? I'm shocked. Shocked, I say!

[Poodle]

I know. I'm still struggling with it myself

[bobbo_the_Pragmatist]

Its worthy of opprobrium.... which I don't often say.

[Nikki Nyx]

Its worthy of opprobrium.... which I don't often say.

Hey...a word that's new to me...awesome! I love it when that happens. I shall add it to my collection.

"I've got gadgets and gizmos aplenty.
I've got whosits and whatsits galore.
You want thingamabobs? I've got twenty."

[bobbo_the_Pragmatist]

Yes, I often think it, but rarely say it, and just about zero ever write it. Quite salubrious.

[Nikki Nyx]

I find it salubrious to learn new vocabulary. It makes me smile, which automatically lifts my mood.

[Austin Harper]

For the record, I know a lot of people at the University of Kansas who also agree that numbers do exist.

[Nikki Nyx]

For the record, I know a lot of people at the University of Kansas who also agree that numbers do exist.

Numbers lead such interesting lives.