Let us take an example of cake.
I have 1 cake.
Multiplication by 2 is 2 cakes
Multiplication by n is n cakes
Division is making parts of that cake.
Division by 1 is 1 cake
Division by 2 is making 2 equal parts of cake
Division by n is making n equal parts of cake.
Now you cannot split a cake into 0 equal parts.
Unless you turn it into many many many tiny crumbles such that each part is as good as none and each crumble cannot be counted in any meaningful manner.
Hence division by 0 is infinity
I always try to imagine dividing candy among a group. 5 piece to 5 people is 1 piece per person. 5 pieces to 2 people is 2.5.
But how would you divide 5 pieces of candy to 0 people? You can’t, the whole premise falls apart. You would just be some weirdo standing on your porch with a fistful of candy.
Let’s start with 2×6=12. Then 12/6=2, and 12/2=6. This is always true.
Let’s try with another number. 12/0. That means we have to find the answer for x×0=12. Which has no answer: there is no number which gives you 12 when multiplied by zero.
How about zero then? Surely we could at least divide zero by zero: 0/0. But then
x×0=0. Since every number multiplied by zero gives you zero, any number could be the answer.
Somebody already gave this answer, but I’ll give it a go:
x/0 is defined as the number y such that y*0 = x.
But y*0 is always zero. Thus if x is not 0, then x/0 is “undefined.” It’s not a number.
Now what about when x is 0?
0/0 is defined as the number z such that z*0 = 0.
But this is true for *any* number z. Thus 0/0 is “indeterminate.” It’s not a number.
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