0 divided by 0?

[Pen]

Do you consider this "Undefined"? or "Infinite"? I've heard both as answers. Which answer is considered the preferred (or correct) one, and why?

[quark]

Do you consider this "Undefined"? or "Infinite"? I've heard both as answers. Which answer is considered the preferred (or correct) one, and why?

 

You might find

helpful. (YouTube link). We love the Numberphile channel.

[Pen]

Well, how can one cut nothing by something? Yet, 0/x=0

 

I think the argument for Infinite is that any value of x for x(0)=0

 

OTOH, x/0=undefined, so that should include 0/0, where x=0

[Pen]

You might find

helpful. (YouTube link). We love the Numberphile channel.

 

Thank you! I'll look as soon as I'm where I can get high speed access. Will you give me a hint as to its answer? This type of thing can get into my head and refuse to budge till satisfied.

[stripe]

Well, how can one cut nothing by something? Yet, 0/x=0

 

 

If you have zero cookies, and share with three friends, everyone gets zero cookies.

[quark]

Thank you! I'll look as soon as I'm where I can get high speed access. Will you give me a hint as to its answer? This type of thing can get into my head and refuse to budge till satisfied.

 

But that won't be fun! Besides, ogling these math geeks is 99% why I like Numberphile. The answer (or the lack of a clear answer in some cases) is just a fun bonus. :001_smile:

EETA: OK, I hope I did not come off as being cheeky OP. I'm watching it again just to be sure and if you forward to 10:32, you'll be able to see what I mean.

Edited November 1, 2012 by quark

[Chrysalis Academy]

You might find

helpful. (YouTube link). We love the Numberphile channel.

 

Love, love, love this!! Why are British geeks so much cooler than American geeks? I can't wait to show some of these videos to the girls, especially the googol one. My dd6 is obsessed with googols right now!

[quark]

If you have zero cookies, and share with three friends, everyone gets zero cookies.

 

From the practical POV, I prefer this.

[quark]

Love, love, love this!! Why are British geeks so much cooler than American geeks? I can't wait to show some of these videos to the girls, especially the googol one. My dd6 is obsessed with googols right now!

 

I happen to like this American geek. :001_wub:

[Arcadia]

You might find

helpful. (YouTube link). We love the Numberphile channel.

 

Thanks. Now my boys have another fun channel to watch beside AOPS :)

[kiana]

0/0 is indeterminate.

 

Division is the inverse operation of multiplication. When we say n/d = q, what we are really saying is that dq = n. (n for numerator, d for denominator, q for quotient.)

 

If n is not zero, then to evaluate n/0, we need to solve q*0 = n. But since q*0 is 0, this means solving n = 0. But since we already said n wasn't 0, there isn't any (real) number we can define as the value for n/0. (On the Riemann sphere, we can define it as infinity, but infinity is not a real number.) We do know, however, that if x is a variable and n is not 0, that n/x gets quite large (either positive or negative) as x gets very close to 0.

 

Now, if both n and d are 0, then we have q*0 = 0. This is true for any value of q. This means that we can't really say anything about what happens at n/d. Here's some algebra to illustrate how weird this can be.

 

a) Look at 7x/x. If x = 0, this is 0/0. But if we cancel the x's, we get 7.

b) Look at -5x/x. If x = 0, this is 0/0. But if we cancel the x's, we get -5.

c) Look at x^2/x (x squared over x). If x=0, this is 0/0. But if we cancel an x, we get x/1 = x, and THEN if x = 0, that's 0.

Look at x/(x^3). If x = 0, this is 0/0. But if we cancel an x, we get 1/(x^2), and THEN if x=0, we get 1/0.

 

In calculus, you learn this as the concept of limit, and you learn that the limit as x goes to 0 of a) is 7, of b) is -5, of c) is 0, and of d) is (depending on your textbook's notation) either infinity or undefined.

 

I hope I haven't perpetrated any errors in this post; if something seems odd, do ask, sometimes I typo :P

[stripe]

(On the Riemann sphere, we can define it as infinity, but infinity is not a real number.)

 

I wish this post had some sort of "love" button.

[Pen]

0/0 is indeterminate.

 

Division is the inverse operation of multiplication. When we say n/d = q, what we are really saying is that dq = n. (n for numerator, d for denominator, q for quotient.)

 

If n is not zero, then to evaluate n/0, we need to solve q*0 = n. But since q*0 is 0, this means solving n = 0. But since we already said n wasn't 0, there isn't any (real) number we can define as the value for n/0. (On the Riemann sphere, we can define it as infinity, but infinity is not a real number.) We do know, however, that if x is a variable and n is not 0, that n/x gets quite large (either positive or negative) as x gets very close to 0.

 

Now, if both n and d are 0, then we have q*0 = 0. This is true for any value of q. This means that we can't really say anything about what happens at n/d. Here's some algebra to illustrate how weird this can be.

 

a) Look at 7x/x. If x = 0, this is 0/0. But if we cancel the x's, we get 7.

b) Look at -5x/x. If x = 0, this is 0/0. But if we cancel the x's, we get -5.

c) Look at x^2/x (x squared over x). If x=0, this is 0/0. But if we cancel an x, we get x/1 = x, and THEN if x = 0, that's 0.

Look at x/(x^3). If x = 0, this is 0/0. But if we cancel an x, we get 1/(x^2), and THEN if x=0, we get 1/0.

 

In calculus, you learn this as the concept of limit, and you learn that the limit as x goes to 0 of a) is 7, of b) is -5, of c) is 0, and of d) is (depending on your textbook's notation) either infinity or undefined.

 

I hope I haven't perpetrated any errors in this post; if something seems odd, do ask, sometimes I typo :P

 

 

Okay. So far so good. What is "Indeterminate" that you started with? Is this what we call something that is either infinity or undefined? Or a different term in place of undefined? Or something else?

 

ETA: another problem I am having is the idea of putting x/x and canceling x's makes it, where x=0, seem like 0/0=1, which causes my head to feel dizzy. I did have calculus both integral and differential, but do not recall anything particular about 0/0...nothing beyond geometry got used much in the path I chose, at least so far. Being a homeschool parent may change that, but if/when ds gets to calculus I think I'll be using a good bit of Sal Khan and/or Dana Mosely.

Edited November 2, 2012 by Pen

[Guest]

You might find

helpful. (YouTube link). We love the Numberphile channel.

 

This look great! Thank you :)!

[Chrysalis Academy]

I happen to like this American geek. :001_wub:

 

rrrrrrrrrrrrrrrowr! :D;)

[quark]

I wish this post had some sort of "love" button.

 

:iagree: What wouldn't I give to have you mentor my kiddo (and me too!) kiana. Thanks so much. I need to look at it again and again though. Just like I needed to watch that numberphile video over and over. I'm not a math head...just an amateur math enthusiast. And an ogler of math geeks. :D

[wapiti]

You might find

helpful. (YouTube link). We love the Numberphile channel.

 

Thanks, quark! (These videos were very useful for 9 y.o. boys procrastinating on going to bed. They were fighting over which one to click on next. One led to another and another...)

[kiana]

Okay. So far so good. What is "Indeterminate" that you started with? Is this what we call something that is either infinity or undefined? Or a different term in place of undefined? Or something else?

 

Undefined just means that we can't assign a real number as 'the answer'. 0/0 is undefined. We usually use indeterminate form to indicate something where it's not just undefined, but we don't really have a clue about the behaviour based on that form. For example, if we are taking the limit of a rational function, and we get '7/0', we can say the function diverges to infinity. We can't say anything like that with respect to 0/0.

 

ETA: another problem I am having is the idea of putting x/x and canceling x's makes it, where x=0, seem like 0/0=1, which causes my head to feel dizzy. I did have calculus both integral and differential, but do not recall anything particular about 0/0...nothing beyond geometry got used much in the path I chose, at least so far. Being a homeschool parent may change that, but if/when ds gets to calculus I think I'll be using a good bit of Sal Khan and/or Dana Mosely.

 

You would actually have seen this during the limits section of calculus and when you did derivatives as limits of difference quotients. Every time you find the derivative as lim_{h->0} (f(x+h)-f(x))/h, if you evaluated at the first step you would get 0/0. But when you simplify, cancel some h's, and *then* take the limit, you're getting an expression for the slope of the tangent line.

[kiana]

:iagree: What wouldn't I give to have you mentor my kiddo (and me too!) kiana. Thanks so much. I need to look at it again and again though. Just like I needed to watch that numberphile video over and over. I'm not a math head...just an amateur math enthusiast. And an ogler of math geeks. :D

 

Quark, thank you so much for the kind words; they're much appreciated. I'm feeling terrified as I send out applications for employment (I'm finally getting my degree!)

[Seasider]

Great video! I'll have to look for more Numberphile offerings.

[Pen]

THANK YOU!!!!

Undefined just means that we can't assign a real number as 'the answer'. 0/0 is undefined. We usually use indeterminate form to indicate something where it's not just undefined, but we don't really have a clue about the behaviour based on that form. For example, if we are taking the limit of a rational function, and we get '7/0', we can say the function diverges to infinity. We can't say anything like that with respect to 0/0.

 

 

 

You would actually have seen this during the limits section of calculus and when you did derivatives as limits of difference quotients. Every time you find the derivative as lim_{h->0} (f(x+h)-f(x))/h, if you evaluated at the first step you would get 0/0. But when you simplify, cancel some h's, and *then* take the limit, you're getting an expression for the slope of the tangent line.

 

I think calculus went into a black brain hole. Good luck with job hunting! Is it math related work you seek? It looks like you not only know your stuff there, but can explain it too!

[quark]

Quark, thank you so much for the kind words; they're much appreciated. I'm feeling terrified as I send out applications for employment (I'm finally getting my degree!)

 

:grouphug: Good luck and :hurray:! Congratulations!