Division by Zero

What is any number divided by 0? Undefined. What is 0 divided by 0? Indeterminate.

How are the two different? Here’s a math forum page informally and discreetly explaining “undefined” as something where the answer does not exist while “indeterminate” as something that “if it pops up somewhere, you don’t know what its value will be in your case.”

Either take that or go read what Prof. Arsham has to say about the ordeal that is division by zero. The article presents cultural, historical, mathematical and psychological perspectives on the number representing nothing. Aptly entitled the Zero Saga, he summarizes the whole point of going at length about the common misconceptions involving dividing by zero:

If one does allow oneself dividing by zero, then one ends up in a hell. That is all.

And he holds his case against others well, such as when he retorts the following when a reader writes of the fact that dividing 1 apple into 0 equal parts still yields an apple:

Have you really attempted in doing so? I am sure you failed, Right? So do not conclude anything.

And in quite the similar vein, someone asks:

If you interpret 20/5 =4 means that if you take 5 oranges from a total of 20 oranges in your fridge you can do it 4 times. Then if you take 0 oranges from 2 oranges (2/0) you can take it infinite number of times (that is, it does not end but surely exists/continues).

He then points out the absurdity of the operation in his response:

If you “take 0 oranges from 2 oranges”, it means 2-0=2. Repeating this operation again and again is nonsense. Once is enough, right? Otherwise eventually you get tired of counting this repetition, beyond that is infinity which you have never reached.

Prior to encountering the read, I never realized how much zero made things much more difficult for us with its introduction in the 13th century. Sure I still remember the additive inverse and difficult stuff stemming from consequences brought about by 0’s existence back in college. But it still took that piece touching lots of the higher math topics to have me pondering how much a seemingly insignificant number caused that much trouble in the number system.

One Response

  1. “Because the universe will blow up,” was the usual answer I got when my teachers tried to explain why we couldn’t divide by zero. From a young age, I was a sort of anti-Pythagorean in that I believed people created numbers, not that the universe was ruled by them. So why then did we create the divide-by-zero bomb?

    The best way I’ve found to describe why dividing by zero will destroy everything is to go back to translating fractions. What does “1/2” really mean? “1/2” translates to “1 out of 2” or “I have one piece of candy out of the two pieces on the table, so I have half of what is on the table. My sister is a good sharer.”

    Now try this with “0/2”. This translates to “zero out of 2” or “I have zero pieces of the two that are on the table. My sister’s cheap!”

    Both of these situations are real. You can have one piece of candy out of two. You can have none of the pieces of candy. Even if the fraction is an improper fraction, like “3/2”, certainly you can’t have three out of two pieces of candy; this makes no sense at all. But then we remember that improper fractions can be written into mixed fractions, so “3/2” becomes “1 and ½”, and we sure can have 1 and a half of the pieces of candy on the table [leaving our cheap sister with just ½! Haha!]!

    So then comes “2/0”, which would translate to “2 out of zero” or “I have two pieces of candy out of the zero that are on the table.” HUH?? This obviously doesn’t make sense! Despite what Little Orphan Annie and Jay-Z may lead us to believe, you can’t make something out of nothing. It’s just basic physics.

    Once a student begins learning about slope and functions, the impossibility of “2/0” becomes even more obvious. Let’s think of a graph that measures your height against your age. “2/0” represents a rise (y-value or “height”) of 2 and a run (x-value or “time”) of 0. This is to say that, for example, at time 0 you are 2 feet tall. Ok, so maybe you were born 2 feet tall. That’s possible. Now let’s move up from coordinate (0, 2). The slope of “2/0” tells us to move up 2 and over 0. We move up two spaces to 4 feet tall and over to… over to nothing! We stay at zero! So a slope of “2/0” says that you can be 2 feet and 4 feet tall at the same point in time. This is impossible!

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