Does division by zero = infinity?

Why should it? Well, try this:

  • 5 ÷ 10 = 0.5
  • 5 ÷ 1 = 5
  • 5 ÷ 0.1 = 50
  • 5 ÷ 0.01 = 500
  • 5 ÷ 0.001 = 5000

As we divide by smaller and smaller numbers the result gets ever bigger. Logically, then, as the divisor tends to (ie gets closer to) zero, so the result tends to infinity. But this is not the same as saying that division by zero actually is infinity, is it? What about drawing a graph with an asymptote?

Here’s the graph of {y}={{12} \over {x-2}}

We can see that as x tends to 2, so the graph gets closer to the line x = 2 (the line is called an asymptote), with values tending towards \pm \infty , but again, it doesn’t actually happen – there is no point on the graph (2, \infty ).

Let’s look at it another way. As an example, 12 ÷ 4 means “how many 4s make 12? Answer: 3. So 12 ÷ 0 must mean “how many zeroes make twelve?” Answer: impossible. Even an infinity of zeroes still makes zero. All we can say is that if we divide by a number which gets ever closer to zero, the result gets bigger and bigger – for ever!

Of course, dividing into zero is another matter. 0 ÷ x = 0, whatever the value of x. But this leads us to another knotty problem – what is the value of 0 ÷ 0? We know that:

\frac{0}{x} = 0

\frac{x}{x} = 1

\frac{x}{0} is undefined.

So could \frac{0}{0} be any of these? Let’s pose the problem another way. Suppose \frac{0}{0} = k. It would then follow that 0 = k × 0, but that means that k could take any value. So we must conclude that \frac{0}{0} is also undefined.

A well known fallacy

Division by zero is at the heart of this fallacious argument which proves that 2=1!

Suppose we have two numbers a and b, such that a = b
Then:

  • ab = b2
  • aba2 = b2a2
  • a(ba) = (b + a)(ba)
  • a = b + a
  • a = 2a (Since b = a)
  • So 1 = 2.

The fallacy lies after the third line of working where I divided both sides by (b – a); since a = b this means ba = 0, and that makes everything go horribly wrong! Effectively, it’s like saying 2 × 0 = 3 × 0, so divide both sides by 0 to get 2 = 3.

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