5x5-5÷5+5=?

[LCKurtz]

Anyone who would write the expression in the form [MATH]5 \times 5 - 5\div 5+5[/MATH] deserves any confusion or wrong answers that comes their way. There are several different ways to interpret such an expression. You might want to read about infix, prefix, and postfix (reverse Polish). After doing so, you might come to the conclusion that the proper use of parentheses really does matter, and the order that you press the keys in your calculator is calculator dependent. Even if you insert parentheses in the above expression to remove all ambiguity, you would enter a different sequence of keys on a typical TI calculator versus an HP (reverse-polish) one.

 

[tkhunny]

I go back to the beginning.....How can this be? I find this very worrying. Can someone please explain how this can be and how long ago the second method came about and which method came first and who, on earth, is right etc etc.

Yes, let's...

The straight-forward, succinct answer to your question? Ignorance. Hard to say whose.

The convention is as old as the notation (which isn't that old). As soon as mathematical expressions needed to be communicated and understood, we started to formulate conventions. Sometimes, different conventions developed in different areas at the same time!

The Wiki article is actually quite good: https://en.wikipedia.org/wiki/Order_of_operations It mentions many of the things we have mentioned, here.

There are many such issues in our society. Here's just one example: How did we not know for so long that cigarettes killed people? Short answer? Many did, but some who did lied about it. This creates ignorance and confusion.

How can we get two answer from the same relatively-simple mathematical expression? We shouldn't, but we do. Life isn't perfect. You can talk to your local MP and get some statutes in place - maybe apologize to those of your generation who didn't learn the more accepted standard? On the other hand, maybe your teachers were underpaid and overworked and we should blame those who funded them.

Here in the US, back in the late 19th Century, the State of Indiana notoriously tried to LEGISLATE the value of PI to some convenient value with just a few decimal places. Fortunately, more sensible heads prevailed. Sometimes, like I said, straight up ignorance is the problem.

If you want a forum where you can get volunteers to go storm the palace and demand the confusion be repaired, that will be a harder search. You have been hearing the truth, here. You just don't like it as we don't seem to be outraged. Sadly, with years and years of experience, having addressed this issue literally thousands of times, it seems the problem will not go away anytime soon. Good luck on your crusade. Good work noticing the problem. Definitely kudos for that.

P.S. I once returned home to a group of seven (7) people faced with a much larger expression of just the type we have been discussing. EACH of the seven had gotten a different answer. They were VERY confused. How could there be SEVEN answers? They appealed to me to settle the dispute. I sat down and demonstrated the convention very clearly, one intermediate value at a time, and this resulted in an EIGHTH result. Guess what my seven guests did. They abandoned their consternation at the seven answers and accepted that the more authoritative source had demonstrated the proper procedure and had obtained the proper result. They put their shovels and pitchforks back in their cars and returned to their daily lives, having learned.

 

[tkhunny]

Anyone who would write the expression in the form [MATH]5 \times 5 - 5\div 5+5[/MATH] deserves any confusion or wrong answers that comes their way. There are several different ways to interpret such an expression. You might want to read about infix, prefix, and postfix (reverse Polish). After doing so, you might come to the conclusion that the proper use of parentheses really does matter, and the order that you press the keys in your calculator is calculator dependent. Even if you insert parentheses in the above expression to remove all ambiguity, you would enter a different sequence of keys on a typical TI calculator versus an HP (reverse-polish) one.

Mostly agreed.
1) If you just taught a lesson on the Order of Operations, it may be a legitimate presentation.
2) Please, let us ALL remember that parentheses should be used to clarify intent if there is ANY chance of ambiguity.
3) Didn't know anyone still used RPN. Did you use my HP-12C example, above? I guess it does still live in some circles.

 

[topsquark]

The calculator is not the issue. Please see previous posts..

I did.

Hello. I came across the above sum, the other day, in one of those Facebook puzzle posts. I don't normally look at this sort of thing, but I had a quick glance at this one and am still trying to get to the bottom of things. The way I was taught, the answer to this sum is 9. But there appears to be another school of thought where the answer is 29. At first, I thought it was a wind up, but it seems there are some genuine people, teachers even, who insist the answer is 29. Can someone please explain how there can be two schools of thought to solving this sum. If I do the calculation on one of my personal calculators or on the Windows 10 calculator, the answer is 9. Obviously!! But if I do the calculation on my mobile phone, the answer is in fact 29!! How can this be? I find this very worrying. Can someone please explain how this can be and how long ago the second method came about and which method came first and who, on earth, is right etc etc. Many thanks.

Where have I missed the point? You were given how to solve this and then the conversation turned into the question "Which calculator can do this properly?" Do you or don't you know how to solve this problem without a calculator? That seems to be the essence of your original post: Is the answer 29 or 9? There are no two schools about it..the answer is 29. Which calculator you use doesn't matter at all.

Now if you are trying to say that you have a more complicated expression to evaluate then a calculator might be easier to work with. But if you are simply relying on the calculator to solve it then you really need to review order of operations. Calculators don't always give you the answer you want, they give you the answer that you programmed in. My advice is to not rely on them at all.

-Dan

 

[giorgiopin]

I did.

Where have I missed the point? You were given how to solve this and then the conversation turned into the question "Which calculator can do this properly?" Do you or don't you know how to solve this problem without a calculator? That seems to be the essence of your original post: Is the answer 29 or 9? There are no two schools about it..the answer is 29. Which calculator you use doesn't matter at all.

Now if you are trying to say that you have a more complicated expression to evaluate then a calculator might be easier to work with. But if you are simply relying on the calculator to solve it then you really need to review order of operations. Calculators don't always give you the answer you want, they give you the answer that you programmed in. My advice is to not rely on them at all.

-Dan

I didn't say anything about, 'Which calculator can do this properly?' We didn't use calculators in my day. I made the point, 'if' I use a calculator (as if the calculator knows all - I do know it doesn't).

I know how to work out the sum, the way I was taught, without any assistance. And I would argue, using common sense, that the way I have been taught is correct. If the 'x' comes first, the '-' second the '/' third and '+' last, that is the order the sum is calculated. I would suspect that was the way it was in the very beginning because it is the most basic, simplest and obvious.

Since the beginning, it seems, people have tried to improve things and ended up messing it up completely. If I wanted a different solution to the sum, I would have to rearrange the numbers at the very least but, in your case, put in brackets, in fact.

'My' system uses simple rules. I am afraid that 'your' system is not so simple. I suspect I would be standing by this viewpoint had I been taught about DEMBAS, or whatever it's called (there appears to be some confusion as to what the exact acronym is - surprise, surprise) in the first place.

And I STILL haven't found anyone, I know, who is aware of 'your' method. And I have been looking.

 

[Harry_the_cat]

When the PEMDAS, BODMAS, or whatever you want to call the convention, came about is an interesting question. It's universal - ie the convention exists wherever you come from, Australia, UK, USA, anywhere. I would venture to say that the convention has been around for a very long time. I was certainly taught it at school and I'm sixty.
There has to be an "order convention", so that arguments like this don't happen.
Following the convention, the correct answer is 29 as others have stated. Many people who don't deal with maths on a daily basis have either forgotten being taught the order convention, or were taught incorrectly in the first place.
It appears that some of your calculators don't follow the convention. That is the worry!

 

[Harry_the_cat]

Giorgiopin, a question for you:
What is \(\displaystyle 3 + 5^2 \)?

 

[giorgiopin]

Yes, let's...

The straight-forward, succinct answer to your question? Ignorance. Hard to say whose.

The convention is as old as the notation (which isn't that old). As soon as mathematical expressions needed to be communicated and understood, we started to formulate conventions. Sometimes, different conventions developed in different areas at the same time!

The Wiki article is actually quite good: https://en.wikipedia.org/wiki/Order_of_operations It mentions many of the things we have mentioned, here.

There are many such issues in our society. Here's just one example: How did we not know for so long that cigarettes killed people? Short answer? Many did, but some who did lied about it. This creates ignorance and confusion.

How can we get two answer from the same relatively-simple mathematical expression? We shouldn't, but we do. Life isn't perfect. You can talk to your local MP and get some statutes in place - maybe apologize to those of your generation who didn't learn the more accepted standard? On the other hand, maybe your teachers were underpaid and overworked and we should blame those who funded them.

Here in the US, back in the late 19th Century, the State of Indiana notoriously tried to LEGISLATE the value of PI to some convenient value with just a few decimal places. Fortunately, more sensible heads prevailed. Sometimes, like I said, straight up ignorance is the problem.

If you want a forum where you can get volunteers to go storm the palace and demand the confusion be repaired, that will be a harder search. You have been hearing the truth, here. You just don't like it as we don't seem to be outraged. Sadly, with years and years of experience, having addressed this issue literally thousands of times, it seems the problem will not go away anytime soon. Good luck on your crusade. Good work noticing the problem. Definitely kudos for that.

P.S. I once returned home to a group of seven (7) people faced with a much larger expression of just the type we have been discussing. EACH of the seven had gotten a different answer. They were VERY confused. How could there be SEVEN answers? They appealed to me to settle the dispute. I sat down and demonstrated the convention very clearly, one intermediate value at a time, and this resulted in an EIGHTH result. Guess what my seven guests did. They abandoned their consternation at the seven answers and accepted that the more authoritative source had demonstrated the proper procedure and had obtained the proper result. They put their shovels and pitchforks back in their cars and returned to their daily lives, having learned.

I read the Wikipedia article. Now we're getting somewhere. It seems this alternative method (29) originated around 15th/16th century. As I figured - not that old. Good point about the cigarettes - I always figured some people must have deduced the earth was round a long time before Pythagoras worked it out.
Also, I would suspect that, one day, sense will prevail and anyone advocating PEMBAS is going to have to start using brackets

 

[Harry_the_cat]

Still interested to know how would you answer \(\displaystyle 3 + 5^2\)? Is it 64 or 28?

 

[giorgiopin]

Giorgiopin, a question for you:
What is \(\displaystyle 3 + 5^2 \)?

The way I was taught - 28. If you want 64, that would be (3+5)squared.

 

[Harry_the_cat]

But if you performed the operations from left to right as you did with the expression including all the 5s, wouldn't you do the addition before the power?

 

[giorgiopin]

But if you performed the operations from left to right as you did with the expression including all the 5s, wouldn't you do the addition before the power?

That's a good point. I would argue that the square is applying to the 5 and not the whole equation. I mean, it's sitting on the 5 - it is completely separated from the 3 by the plus symbol. I don't know for a fact if that is what my teacher would have said, but it makes sense. Also, I guess I'm starting to figure how this confusion has come about - as maths has evolved, it has got more complicated with more rules and more systems etc. Some have tried to simplify things but ended up complicating things. I suspect one or two have just thrown a spanner in the works just for the fun of it.
PS. How about if the sum was 3+5 squared x 3 ÷ 2?

 

[Harry_the_cat]

\(\displaystyle 3+5^2 * 3 ÷ 2 = 3 + 25 * 3 ÷ 2 = 3 + 75 ÷ 2 = 3 + 37.5 = 40.5 \)

Interesting to note that \(\displaystyle 3 + 5^2\) is now often typed as 3+5^2 which puts a symbol between the 5 and the 2, which weakens your argument above. The order convention, which says that exponents are applied before addition, makes it clear in both cases.

 

[giorgiopin]

\(\displaystyle 3+5^2 * 3 ÷ 2 = 3 + 25 * 3 ÷ 2 = 3 + 75 ÷ 2 = 3 + 37.5 = 40.5 \)

Interesting to note that \(\displaystyle 3 + 5^2\) is now often typed as 3+5^2 which puts a symbol between the 5 and the 2, which weakens your argument above. The order convention, which says that exponents are applied before addition, makes it clear in both cases.

It might strengthen your argument, but it doesn't weaken mine. How can it when I've never even seen that symbol before?

 

[tkhunny]

Also, I would suspect that, one day, sense will prevail ...

As for the study of mathematics, it already does. We've been through enough pain exchanging carelessness for rigor since at least the early 19th century. I've rather casually counted as a cutoff somewhere between Euler and Gauss. I doubt every historian would agree. We do have a pretty good grip on where we still need additional rigorous basis. Again, not everyone would agree. There was a nice shock in 1976 concerning computer-assisted exhaustion, but we got over that. Then there is Gödel's Incompleteness. Lots for reading for you. Work up to it.

For physics, there are lots of holes. Physicists are pretty honest about it. Take a look at the Flux-Luminosity equation (or maybe proportion). We can talk about all sorts of things with sincere folks trying honestly to plug holes. No need to panic when we trip over one. The real scientists are likely already working on it.

As far as much of the rest of the world, You Dreamer!!

 

[giorgiopin]

As for the study of mathematics, it already does.
For the rest of the world, You Dreamer!!

No it doesn't - students of mathematics are on a quest to MAKE sense and, no doubt, they will come up with a whole bunch of other systems etc. long before they ever do. Hence more confusion.

I don't know what you mean by your other comment. The rest of the world is dreaming? I would argue otherwise.

 

[Harry_the_cat]

It might strengthen your argument, but it doesn't weaken mine. How can it when I've never even seen that symbol before?

What I'm saying is that, written that way, the 2 can no longer be assumed to apply to the 5 only. There are no assumptions involved. It is the order convention which still says to apply the exponent first.

 

[Harry_the_cat]

I'm not even sure what we are arguing about. In mathematics there are axioms, rules and conventions. If they are not followed, we become unstuck. Simple as that.

 

[giorgiopin]

What I'm saying is that, written that way, the 2 can no longer be assumed to apply to the 5 only. There are no assumptions involved. It is the order convention which still says to apply the exponent first.

If they are teaching that now that is all very well. You keep mentioning the order convention. I don't know why. It is not recognised or used by many people. It does appear to be widely accepted in the States, however.

 

[giorgiopin]

I'm not even sure what we are arguing about. In mathematics there are axioms, rules and conventions. If they are not followed, we become unstuck. Simple as that.

I agree totally. It just appears that there are conflicting, 'axioms, rules and conventions'.