Is 0/0 Undefined?

Dividing by zero might end the world. But what if you divide ZERO by zero? http://mathfour.com/?p=10385 #mathchatI got myself in a jam the other day in class. I said “0/0 is undefined.”

Since I encourage students to always question the (rather confident) stuff that comes out of my mouth, they did.

So I proceeded to explain it… And got hung up!

0/0 is not undefined!

Turns out I was wrong. Infinitely wrong.

Let’s say 0/0 is 85. (Just join me on this journey – don’t freak yet.)

So 0/0 = 85, which means 85 x 0 = 0 and 0/85 = 0.

Which is just fine!

But the guy down the road might want 0/0 to be -72π instead. And it’d be just fine, too.

But then 0/0 = 85 and 0/0 = -72π.

And then you come along and want 0/0 to be 1,000,000.

Which means if I have $85, I also owe someone $72π, and I also have $1,000,000.

I assure you, I don’t have $1,000,000.

So 0/0 is indeterminate.

0/0 doesn’t not exist. It just exists as too many things. And when that happens, I’m a millionaire. And so are you.

In reality, this isn’t true.

At least about me.

Your turn…

Did you ever get this one confused? Share your thoughts in the comments!

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4 Responses to Is 0/0 Undefined?

  1. The most fun that I had last year with my Pre-Calculus class was looking at the limit as x approaches one of
    (x^2 – 1)/(x-1).

    We were just introducing limits, so the first thing they tried was to plug in one. Hearing them tackle with what 0/0 equals was great. Is it one, is it zero, is it undefined? It passes all of those definitions, but it can’t be all three.

    So much fun.

  2. I saw your tweet and confidently thought, “Of course it is!” I’m sure glad I read the article before tweeting all high and mighty. Good stuff.

    It reminds me of when me honors class asked, “Is 0^0 still 1?” and I responded, “Ye… Well, no… I have no idea. Let’s call Ms. Zick.”

  3. I have a video that deals with this very issues in approachable but meaningful terms. I show it to my calculus students each year at the start of class. It deals with a drive to visit my grandmother (complete with a stopover at Wendy’s), that shows us where indeterminate comes from, and why it is at the crux of our mathematical journey in calculus.

    This is it if you’re interested.

    http://www.youtube.com/watch?v=PXXitoq6_pQ

    hope this helps,
    – Scott

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