I 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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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.
It’s a weird one, Chris. Thanks for dropping in and sharing!
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.”
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