Is 0/0 Undefined?

Dividing by zero might end the world. But what if you divide ZERO by zero? #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.

    hope this helps,
    – Scott

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