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        riddles >> hard >> Hard: 24
        
(Message started by: ootte on Jul 24^th, 2002, 3:55pm)

Title: Hard: 24
Post by ootte on Jul 24^th, 2002, 3:55pm

24 = 7 * (3 + 3/7)

--
Oliver

Title: Re: Hard: 24
Post by icon on Jul 31^st, 2002, 11:59am

i have this answer am not sure if its valid.... but

(sqrt(7*7)*3)+3

Title: Re: Hard: 24
Post by Gamer555 on Jul 31^st, 2002, 6:19pm

I don't think that's too legal....

(Powers and parentheses, and the 4 operations)

For more fun, make 24 with

3 3 3

3 3 3 3

3 7 7 7 (Duh!)

3 3 7 7 7 (The one I got is a bit sneaky)

3 3 5 5 5

Also, with 3 3 7, get 24 1/3

With 3 3 3 7 7, get 24 1/2

(3^3)-3

(3*3*3)-3

3+7+7+7

(73-(7/7))/3

((3^5)-3)/(5+5)

(73/3)=24 1/3

(7*7)/(3-(3/3))=24 1/2

Title: Re: Hard: 24
Post by icon on Jul 31^st, 2002, 6:26pm

ya its not:>

Title: Re: Hard: 24
Post by srn347 on Aug 28^th, 2007, 12:28pm

3/7 involves decimal points which are forbidden. Exponents and sqrts aren't useable either. How about (3-3)/(7-7) or vice-verca. 0/0 is anything so it could be 24.

Title: Re: Hard: 24
Post by pex on Aug 28^th, 2007, 12:36pm

on 08/28/07 at 12:28:07, srn347 wrote:

3/7 involves decimal points which are forbidden. Exponents and sqrts aren't useable either. How about (3-3)/(7-7) or vice-verca. 0/0 is anything so it could be 24.

Where is the decimal point in "3/7"? Okay, we could write it as 0.428571 428571 428571..., but there is not much point to it here. And 0/0 is not equal to anything, but you know now. ;)

Title: Re: Hard: 24
Post by towr on Aug 28^th, 2007, 12:37pm

on 08/28/07 at 12:28:07, srn347 wrote:

3/7 involves decimal points which are forbidden.

How does that involve a decimal points? it's a fraction.

Ootte's solution from 5 friggen years ago is perfectly fine. Just multiplication, addition and division, which are all allowed.

Title: Re: Hard: 24
Post by srn347 on Aug 28^th, 2007, 12:41pm

3/7= 0.4285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714286...

And 0/0 equals anything. And don't evade the word filter system with that word.

Title: Re: Hard: 24
Post by pex on Aug 28^th, 2007, 12:44pm

on 08/28/07 at 12:41:34, srn347 wrote:

3/7= 0.4285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714285714286...

As I said, two posts above yours.

on 08/28/07 at 12:41:34, srn347 wrote:

And 0/0 equals anything.

No, it doesn't. Division by zero is not allowed, so (whatever)/0 is undefined: there is no number which it is equal to.

on 08/28/07 at 12:41:34, srn347 wrote:

And don't evade the word filter system with that word.

???

Title: Re: Hard: 24
Post by srn347 on Aug 28^th, 2007, 12:49pm

The only reason people forbid it is because they can't calculate a random number or multi-answer equation on a calculator.

Title: Re: Hard: 24
Post by pex on Aug 28^th, 2007, 12:56pm

on 08/28/07 at 12:49:35, srn347 wrote:

The only reason people forbid it is because they can't calculate a random number or multi-answer equation on a calculator.

No: it was forbidden long before calculators were invented.

Title: Re: Hard: 24
Post by srn347 on Aug 28^th, 2007, 12:59pm

Well it shouldn't be forbidden. Just like square roots of negative numbers were once forbidden, which is why they invented complex numbers.

Title: Re: Hard: 24
Post by pex on Aug 28^th, 2007, 1:02pm

on 08/28/07 at 12:59:10, srn347 wrote:

Well it shouldn't be forbidden. Just like square roots of negative numbers were once forbidden, which is why they invented complex numbers.

Correct.

Thus, within the complex numbers, we can say "sqrt(-1) = i"; but within the real numbers, we have to say "sqrt(-1) has no value". Similarly, working within the real or complex numbers, x/0 has no value, whatever x is. If you want to divide by zero, you have to come up with a number system in which division by zero works.

Title: Re: Hard: 24
Post by towr on Aug 28^th, 2007, 1:04pm

on 08/28/07 at 12:59:10, srn347 wrote:

Well it shouldn't be forbidden. Just like square roots of negative numbers were once forbidden, which is why they invented complex numbers.

Well then, invent a number system in which 0/0 makes sense.
There is a good reason why there isn't one.

(And squareroots of negative numbers are still 'forbidden' in arithmetic on real numbers; actually, they're not so much forbidden as that they simply can't be applied there.)

Title: Re: Hard: 24
Post by srn347 on Aug 28^th, 2007, 1:07pm

In the future they'll have a way to define it. I'm already in that process and I'm 13.

Title: Re: Hard: 24
Post by towr on Aug 28^th, 2007, 1:29pm

on 08/28/07 at 13:07:24, srn347 wrote:

In the future they'll have a way to define it. I'm already in that process and I'm 13.

Well, in all fairness, someone already has a number system that gives 1/0 and 0/0 a value. One version is used in computers, try the following in your browsers location bar and it'll give it's values for both.
javascript:alert(0/0)
javascript:alert(1/0)

0/0=NaN, and for any number x, x*NaN =NaN
1/0= +Infinity, and 0*Infinity=NaN, and for any other number x, x*Infinity=Infinity
NaN stands for "not a number"

And there was some noise a little while ago from someone coming up with a slightly different system, which has absolutely no advantage as far as anyone could tell. Even made the news as a great breakthrough, because the media like to pretend they have a clue. (But if they had, they wouldn't have wasted two second on it, probably)

Title: Re: Hard: 24
Post by pex on Aug 28^th, 2007, 1:42pm

That has its own complications, though: NaN is not equal to NaN, for example. (Try javascript:alert(NaN==NaN), if you like.)

Title: Re: Hard: 24
Post by SMQ on Aug 28^th, 2007, 1:52pm

on 08/28/07 at 13:29:05, towr wrote:

And there was some noise a little while ago from someone coming up with a slightly different system, which has absolutely no advantage as far as anyone could tell.

Ahh, yes, Nullity (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_general;action=display;num=1172523407). Just like NaN except http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cphi.gif = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/cphi.gif... which, as Icarus points out in the linked thread, actually ends up leading to more special cases rather than fewer.

--SMQ

Title: Re: Hard: 24
Post by srn347 on Aug 28^th, 2007, 4:24pm

Not a number?! That doesn't define it. At least it uses infinity.

Title: Re: Hard: 24
Post by towr on Aug 29^th, 2007, 12:40am

on 08/28/07 at 16:24:55, srn347 wrote:

Not a number?! That doesn't define it.

Sure it does.
You could also define it as chicken soup; as long as you provide the right axioms to deal with the value, it works.
You could make a calculus out of fruit if you wanted to. All that matters are the axioms and inference rules.

More to the point, for all the reasons already discussed it makes sense to define 0/0 (and expressions involving it) as 'not a number', to define it as a special value outside the number line. Heck, if it didn't make sense it wouldn't be used in every computer.

Title: Re: Hard: 24
Post by mikedagr8 on Aug 29^th, 2007, 12:51am

Quote:

In the future they'll have a way to define it. I'm already in that process and I'm 13.

If your honestly in the process, please share your thoughts, it'd be quite interesting to see how much I have changed since I was a 13 year old. :)

Title: Re: Hard: 24
Post by srn347 on Aug 29^th, 2007, 9:41am

1/0 equals infinity(positive, negative, complex, or negacomplex). It can be simplified with 0 being replaced by +0, -0, +0(i), or -0(i). 0/0 can equal anything(but only one thing at a time). Also when in an equation it sometimes shifts value.

0/0=5
2(0)/0=10
0/0=10
(that doesn't make 5 equal to 10).

Title: Re: Hard: 24
Post by Obob on Aug 29^th, 2007, 10:06am

The idea of defining 0/0 to be "anything, but only one thing at a time" does not prove to be very useful. Rather, it only further justifies why we do not define what 0/0 means.

So long as we are only working with real numbers, the expression 1/0 could possibly mean positive infinity or negative infinity, or neither. The idea of associating a value to 1/0 commonly arises in calculus, with the notion of a limit. So instead of talking about 1/0 directly, we would talk about 1/x, in the limit as x goes to 0 (or something like that). But 1/x as x goes to 0 could be either +infinity or -infinity, depending on if x approaches 0 through negative numbers or if x approaches 0 through positive numbers. So in this case we say that the limit does not exist, and do not define it. Similarly, the limit of 1/x^2 as x goes to 0 is indeed positive infinity, since it gets larger and larger whether x approaches 0 through positive numbers or through negative numbers.

The idea of a complex or negacomplex infinity is not useful at all, since either there should be a single infinity in the complex plane, or infinitely many of them. One construction of infinitely many infinities puts a single infinity in every direction from the origin of the complex plane. But by far the more common mathematical usage of infinity in complex arithmetic is that there is just a single infinity in the complex plane. An analogy is this: if we let a complex number z get closer and closer to zero in the complex plane, then 1/z goes farther and farther away from zero, but starts spinning around the origin like crazy. So there is no particular direction in which all the values are going, and we could not meaningfully assign a limit value if we were to put an infinity in every direction. But if we just have one infinity, we could in fact say that the limit is infinity: all the values are going farther and farther away from the origin, so they must be approaching the single point that is infinitely far off in the distance.

The reason that arithmetic involving division by zero is not usually defined is that it inherently leads to either contradictions or tons of special cases which don't actually make things simpler.

I suppose the moral of the story is that when it comes to simple arithmetic, if you weren't taught it in school then there is more likely than not a good reason.

Title: Re: Hard: 24
Post by srn347 on Aug 29^th, 2007, 10:20am

0 is positive, complex, negacomplex, and negative. Infinity in the case of 1/0 takes the reciprocal sign of 0(changed if you change 1 to another sign, which you multiply into infinity's sign).

Title: Re: Hard: 24
Post by Obob on Aug 29^th, 2007, 10:33am

Thank you for reading my previous post and understanding the point...

Title: Re: Hard: 24
Post by towr on Aug 29^th, 2007, 10:47am

on 08/29/07 at 10:20:54, srn347 wrote:

0 is positive, complex, negacomplex, and negative. Infinity in the case of 1/0 takes the reciprocal sign of 0(changed if you change 1 to another sign, which you multiply into infinity's sign).

You can say this as often as you want, but that doesn't make it true.

Why do you refuse to even look into the the possibility that you might not actually know what you're talking about; I mean, considering that everyone disagrees with you. You can google "is zero positive" and you will find everyone explaining why it isn't.
Did you get some personal revelation from the God of Mathematics or something to convince you everyone in the whole world is wrong and you are by definition right and infallible?

Questioning everything is one thing, but refusing all evidence and discourse does not speak well for you.

Title: Re: Hard: 24
Post by srn347 on Aug 29^th, 2007, 10:51am

http://en.wikipedia.org/wiki/-0

Title: Re: Hard: 24
Post by towr on Aug 29^th, 2007, 10:55am

on 08/29/07 at 10:51:42, srn347 wrote:

http://en.wikipedia.org/wiki/-0

To quote that page:
"In mathematical terms there is no concept of a negative (or positive) zero, and -0 is equivalent to, and represented as, zero."

Do you even read the pages you link?
No, of course not..

Title: Re: Hard: 24
Post by srn347 on Aug 29^th, 2007, 11:28am

Read the rest of it, why not.

Title: Re: Hard: 24
Post by Obob on Aug 29^th, 2007, 12:51pm

What that page describes is merely a convention used by computers, however, to describe the effects of rounding errors. In essence, it is talking about the limiting process I described earlier. Its usefulness as far as actually helping mathematics, however, is nonexistent. Any time you need to talk about concepts like "approaching zero from the left" what you really mean is that you are taking a limit. It is better then to actually leave the expression as a limit, since the limit might depend on how you approach zero. If you take calculus someday you will learn all about this, and your urge to divide by zero will no longer exist: we can't really do it, but there are good substitutes in cases where it is important.

Granted, there are theories that develop the meaning of infinitesimals and such. However, they are not nearly as transparent as your idea of "positive zero". In fact, any theory which makes use of infinitesimals has not only infinitely many such infitesimals, but also infinitely many infinities!. For in order to be able to invert a really tiny infinitesimal, you need a really huge infinity. In these theories, you still cannot compute 1/0; instead, your concept of 0+ is replaced by a tiny infinitesimal bigger than zero (so it is actually a positive number, and entirely different from 0), and then you could take the multiplicative inverse of that infinitesimal to get some positive infinity. Which positive infinity depends on the size of the infinitesimal you started with.

I would encourage you to only post when you really know what you are talking about, and even then only when you have read and understood most of the thread up to that point. You will gain a lot more respect around here if your posts are well thought out.

Title: Re: Hard: 24
Post by Obob on Aug 29^th, 2007, 1:38pm

As an example of why your idea fails, what is (0+)+(0+)? The only logical intuitive answer is that it would again be 0+. But then subtract 0+ from both sides of the equation (0+)+(0+)=0+ to get 0+=0. Right? So we can't possibly have (0+)+(0+)=0+. The consequence of this is that there must be another extra number, say 0++, which equals 2(0+). By using similar tricks, you will find that there have to be tons and tons of extra numbers, like 5+, 10.3++, pi+++, etc., and this is roughly the theory of infinitesimals that I alluded to earlier.

Title: Re: Hard: 24
Post by srn347 on Aug 29^th, 2007, 4:46pm

Except they're not infinitesimal. Interesting theory though. Although, +0 can also be attained by getting to 0 from the positives or by rounding from them.