Why can’t we divide by zero?
The calculator shows an error
Type 6 ÷ 0 into a calculator and instead of an answer you get an error. Is it just a rule someone made up? Lots of people would guess the answer is zero, others infinity.
How many times does it fit?
Division is a question: how many times does one number fit into another? 6 ÷ 2 means how many times you can take two sweets from a pile of six before none are left. Three times, so 6 ÷ 2 = 3.
Checking by multiplying
Every division can be checked by multiplying: the answer times the number we divided by must give back the original number. 6 ÷ 2 = 3 is right, because 3 × 2 = 6. Three pairs of sweets make six again.
How many times does zero fit?
Now 6 ÷ 0. How many times can you take zero sweets from six before none are left? The pile never gets smaller, and a thousand or even a million times isn’t enough: you never finish.
The check fails
Let’s try the check: we’re looking for a number that gives six when multiplied by zero. But anything times zero is zero: 7 × 0 = 0, 100 × 0 = 0 and 0 × 0 = 0. No such number exists, not even zero.
Zero divided by zero
With 0 ÷ 0 it’s the other way round. We want a number that gives zero when multiplied by zero, and every single one does: 5 × 0 = 0 and 1,000 × 0 = 0. There are infinitely many answers, and none has a better claim than the others.
What about infinity?
No ordinary number works, but what about infinity, something bigger than any number? Try dividing by smaller and smaller pieces: 0.1 fits into one ten times, 0.01 a hundred times and 0.001 a thousand times. The closer the piece gets to zero, the more times it fits, without any limit.
From the other side
But you can also approach zero from the negative numbers: 1 ÷ (−0.1) = −10, 1 ÷ (−0.01) = −100. There the answer plunges into the negative numbers, again without limit. So you can’t even say whether 1 ÷ 0 would be a huge positive number or a huge negative one.
Try it yourself
Open the calculator on your phone and divide one by 0.001, then by −0.001 and finally by 0.000001. You get 1,000, −1,000 and a million. But try dividing by zero and the calculator shows an error again.
What if we allowed it anyway?
2 × 0 = 3 × 0 is true: both sides are zero. If we were allowed to divide by zero, we could divide both sides by it and be left with 2 = 3. As soon as we allow dividing by zero, we can “prove” anything this way.
So why can’t we divide by zero?
The answer to a division is the number that, multiplied by what we’re dividing by, gives back the original number. For 6 ÷ 0 there isn’t one, for 0 ÷ 0 every number fits, and with infinity we can’t even tell if it’s positive or negative. It isn’t an arbitrary ban: there’s simply no answer that makes sense.