The Old Knewton Blog

Be brave, go ahead and divide by zero

with 3 comments

Ethan Hein is a content editor and social media guru at Knewton.

When you learned division in school, the teacher probably brushed off the issue of dividing by zero in one sentence: you can’t do it, moving on. You might feel like you got shortchanged by that explanation. Why not? What happens when you divide by zero?

You can’t ask the computer. Computers fail when you ask them questions with no unambiguous answer. Dividing by zero is just such a question. Folklore suggests that asking the computer to divide by zero makes it spectacularly explode or something. In reality, it returns an error message or the reply Not A Number, or it gives a wrong answer, or the program terminates, or sometimes the machine falls into an infinite loop.

The internet’s favorite divide-by-zero error is the one that temporarily crippled the USS Yorktown, a Ticonderoga-class cruiser that was the test bed for the Navy’s Smart Ship program. When a crew member typed zero into a database field, the computer tried to divide by it, crashing the system badly enough to cripple the ship’s navigation systems for several hours.

Humans are smarter than computers in some ways, and we’re capable of coming up with creative answers to seemingly unanswerable questions. So what do you get when you divide something by zero? My answer draws heavily on the entertaining wikipedia article. For the sake of simplicity, let’s say we’re dividing one by zero. The math people have a crafty method for dealing with problems you can’t approach directly. You can edge closer and closer to the problem and see if you converge on an answer. So instead of dividing one by zero, you could try dividing it by smaller and smaller numbers that approach zero. One divided by one tenth is ten. One divided by one one-hundredth is a hundred. One divided by one one-thousandth is a thousand. Since one divided by one one-gazillionth is one gazillion, logic suggests that one divided by zero is going to be infinity.

It makes sense, but there’s a problem. We’ve been approaching zero from above, but we could just as easily approach it from below. When you divide one by negative one tenth, you get negative ten. One divided by negative one one-hundredth is negative one hundred. One divided by negative one gazillionth is negative one gazillion. So you could just as easily say that one divided by zero is negative infinity. Both infinity and negative infinity are equally valid answers. Here it is as a graph.

Some people interpret this graph to say that infinity and negative infinity are the same number. It’s not as crazy as it sounds. Let’s say that instead of being on the computer screen, the graph was drawn on a globe. Imagine the number line wrapped around the equator. Say the spot where the Prime Meridian crosses the equator is zero. If you’re in a rowboat bobbing in that spot in the Atlantic Ocean, enjoying the warm breeze, you can think of the positive numbers as going off along the equator to the east, and the negative numbers going off to the west. Infinity is the farthest possible point away from you on the equator to the east, and negative infinity is the farthest point away from you to the west. On the Earth, positive and negative infinity are the same place, the International Date Line in the Pacific. For this image to be totally accurate, the Earth would have to be infinitely large, but the math guys bracket that. By this thinking, one divided by zero does have a single, unambiguous answer: this mysterious number called unsigned infinity.


Written by Knewton

October 30, 2009 at 3:06 PM

3 Responses

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  1. What’s bigger thank Infinity ? Infinity + Infinity. Whats bigger than Infinity + Infinity ? Infinity + Infinity + Infinity, and so on….

    For all practical purposes, “Infinity” represents a number that is higher than we can count. Infinity stretches beyond our means of calculation, interpretation, and manipulation.

    Since there are an Infinite number of Infinities AN OBJECT CAN ALWAYS BE DIVIDED INTO SMALLER PARTS BUT THE SIZE OF THOSE PARTS WILL ALWAYS BE GREATER THAN ZERO. So I disagree that the answer to 1/x is simply Infinity because there are Infinite Infinities.

    Current Mathematics deals in FINITE quantities, and the MATHEMATICS of INFINITY are beyond the capabilities of our LIMITED Mathematical system because Infinity is UNLIMITED.


    L Young

    November 2, 2009 at 4:55 PM

    • Here is what Knewton’s resident mathematician, David Y., has to say:

      “Infinity is a broad and complex concept, and the statement ‘1/0 = infinity’ does belie that complexity to some extent. At the same time, the fact that infinity is beyond our computational capabilities doesn’t mean that it hasn’t been an incredibly fruitful concept. Calculus is based on the idea that we can sum up an infinite number of terms and arrive at a finite answer. Set theory and the incompleteness theorems rely heavily on notions of infinity. While a number too big to count may always be, in some sense, beyond us, there is still much to be said about infinity!”

      Ethan Hein

      November 2, 2009 at 8:43 PM

  2. I agree that Infinity is a complex concept, and I’m thankful that Calculus is NOT on the GMAT, even though I loved Calculus in College. (GMAT Math requires lots of Logic,Reasoning & Interpretation as opposed to straight calculations in College Calculus)

    I recall this exact problem in Calculus where the limit of 1/x as x approaches 0 is Infinity. Infinity is an Endless Quantity, and in Mathematics, quantities are always Finite, therefore I found it odd when the accepted answer to a Math Problem was Infinity.

    Perhaps when I’ve completed my GMAT training with Knewton, I’ll explore my Mathematical interests further, but for now, I have my hands full trying to master all the GMAT concepts and improve my timing. Thanks for the debate…keep it coming 🙂

    L. Young-HBS Hopeful

    L Young

    November 3, 2009 at 7:07 AM

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