Division by and division into zero can cause conceptual difficulties for students. Use your calculator now to do:
  • 5 0
  • 0 5
Note:

Calculator entry can also be an issue. For 5 0: first enter 5, then press , and finally enter 0.
For 5 0, the display should show "E" or "ERROR". Why?

For 0 5, the display should show 0. Why?

Before considering the following explanation, refer to:
Meanings of arithmetic operations (division)
Division means splitting up into equal groups. Two two kinds of questions are possible in relation to the answer to the division:
How many in each group?
How many groups?
Let us consider 0 5 and 5 0 for each kind of question:

Imagine the context for splitting up into equal groups is emptying a large barrel by using a bucket:
For the question "how many groups?", 0 5 = ? means the barrel has 0 litres of water in it (is empty) and you use a bucket that holds 5 litres to empty the barrel. The question becomes:

"How many 5-litre bucketfuls (groups) do you need to empty the barrel?"

Clearly, the answer is that you need zero 5-litre bucketfuls, because the barrel is already empty.
0 5

For the question "How many in each group?", 0 5 = ? means the barrel is empty and you use 5 bucketfuls to empty the barrel. The question becomes:

"How many litres in each bucketful?"

Since the barrel is already empty, the bucket needs to hold 0 litres of water.
Thus, it makes sense that 0 5 = 0.
5 0 The answer to 5 0 is more interesting. Again, let us imagine a large barrel and a bucket to empty it.

For the question, "how many groups?", 5 0 means the barrel has 5 litres of water in it and you use a bucket that holds 0 litres of water to empty it. The question becomes:

"How many bucketfuls do you need?"

Clearly, you are setting yourself up for a futile task. You will never be able to empty the barrel because your bucket holds 0 litres of water. You will be at the job forever. This is why the answer to 5 0 is sometimes referred to as (infinity).
For the question "how many in each group?", 5 0 means the barrel has 5 litres of water in it and you must use 0 bucketfuls of water to empty it. The question becomes:

"How many litres must each bucketful hold?"

This time the entire matter is silly. You are trying to empty the barrel by not using a bucketful at all. It doesn't matter how many litres your bucket can hold because you are not going to use the bucket.
Now let us return to the calculator results:
  • For 0 5, the display showed "0". This should now make sense.
  • For 5 0, the display showed "E" or "ERROR". What the calculator is telling you, in a politically correct way, is "Idiot, you can't divide by zero because the division task doesn't make sense!"

In somewhat more formal talk, 5 0 = , in other words, not possible.

You can consider 5 0 from another point of view by thinking about a sequence of arithmetic results.

    • 5 1
    • 5 .1
    • 5 .01
    • 5 .001
    • 5 .0001
    • 5 .00001
    • 5 .000001
Use your calculator to do this arithmetic:
You should notice that the answer is getting bigger and bigger (closer to infinity) as the divisor gets smaller and smaller (closer to zero).
Thus, a number value for 5 รท 0 is NOT POSSIBLE.