Topics Mathematics

Question 5 divided by 0? Personally, I believe that it is infinite based on the idea that division is just repeated subtraction just like multiplication is repeated addition. For example, in 4/2, it's pretty much like saying how many times can you subtract 2 from 4 before you get to 0.

Accepted:
October 23, 2005

Comments

I'm going to answer this question indirectly, by means of a simple algebaic argument that you may already know. Suppose we begin by assuming that

A = B

Now consider the following argument from that assumption:

A2 = AB

(Both sides multiplied by A.)

A2 - B2 = AB - B2

(B2 substracted from both sides.)

(A + B)(A - B) = B(A - B)

(Each side rewritten.)

(A + B) = B

(Both sides divided by (A - B).)

(B +B) = B

(A replaced by B, since assumed equal.)

2B = B

(Left side rewritten.)

2 = 1

(Both sides divided by B)

Pretty neat, eh? But there had better be something wrong with this argument, since the assumption is fine and the conclusion is crazy. If you want to figure it out for yourself, stop reading now; otherwise continue below.

The fallacy occurs in the line where I divided both sides by (A-B), and that is because my starting assumption that A=B makes that dividing by zero. The moral of the story is that dividing something by zero does not give you infinity, a number that may be well defined. The moral is rather that division is an operation that has no meaning for zero: it's illegal.

By the way, I think this algebraic example is a pretty good model for sceptical arguments in philosophy, arguments like those that seem to show that we know almost nothing because almost everything could just be a dream. Those arguments start from what appear to be perfectly innocent assumptions and end up with crazy conclusions. If only it was as easy to pinpoint the 'trick' in those philosophical arguments as it is in the algebraic case!

I would give a slightly different moral to Peter's story. Mathematicians could have defined 5 divided by 0 to be infinity--one of the wonderful things about mathematics is that we can define things however we want. However, what Peter's proof shows is that if you define division by 0, then some of the familiar algebraic laws aren't going to work anymore. (It is an interesting exercise to identify the algebraic law used in the proof that would stop working if we defined division by 0.) So it would actually be quite inconvenient to change the usual definition of division, according to which division by 0 is undefined.