Why x*0 = x

[robvalue]

By the way, there's nothing stopping people making up their own "number systems" with whatever rules they want. Abstract mathematics doesn't even have to correlate with reality in any way at all, it generally just has to be internally consistent. But if you want to make up your own system that isn't internally consistent, that's up to you really. Although a lot of the concepts obviously are inspired by reality, and hence applicable to them, the maths itself remains entirely abstract.

However, there are more ways you can visualize why these things should be true (apologies if I'm repeating other examples):

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A bucket holds 10 apples. I have no bucketfulls. How many apples do I have? I have none. There is no bucket, hence no apples. 0 * 10 = 0

I have 10 buckets, each with 0 apples in them. How many apples do I have? I have none. No apples anywhere. 10 * 0 = 0

What is the area of a shape which is 10cm long by 0cm wide? It has no area, it takes up no 2D space.

Or a pattern...

4 * 5 = 20
3 * 5 = 15
2 * 5 = 10
1 * 5 = 5
0 * 5 = ?

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I really like the example of putting 10 books into 0 piles. I never thought of that as a teacher!

You can think of X / 0 as the limit of X / N as N tends to zero (say X and N are positive for simplicity). For any finite number K, we can find an N small enough so that X / N is larger than K. The limit is infinite.

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A ninth, 1/9 = 0.11111111.....

1 = 9 * (1/9) = 9 * 0.1111111.... = 0.999999.....

Or

X = 0.99999......

10X = 9.99999.....

10X - X = 9.99999..... - 0.999999....

9X = 9

X =1