My daughter is stuck on the concept that 20=1, having the intuitive expectation that it be equal to zero. I have tried explaining it, but I guess not well enough.

How would you explain the concept to a child, other than the teachers “that is just the rule” approach?

**Answer**

I will give a different answer than the answer I gave in the other thread which tries to appeal to intuition. I am sure your daughter has no problem accepting that 2×0=0. Intuitively this is because if you add 2 to itself zero times, you get zero. Or, to be concrete, if someone gives you two apples zero times, you have zero apples.

For repeatedly adding 2, talking about collections of apples is a good model. But for repeatedly multiplying by 2, it isn’t necessarily, since you can’t multiply apples and apples (at least, not in a way that makes sense to a child). But you can multiply apples by numbers; that is, you can start with 1 apple, then double the number of apples you have to get 2 apples, then double the number of apples you have to get 4 apples, and so forth. In general if you double your apples n times, you have 2n apples.

What happens if you double your apples zero times? Well, that means you haven’t started doubling them yet, so you still have 1 apple. If you want your notation to be consistent, then you should say 20=1.

This is a subtly different argument from the argument I gave before. It’s intuitive what it means to add different amounts of apples, and it’s intuitive what it means to have zero apples. But the twos I am now working with aren’t numbers of apples, they’re just abstract numbers; in other words, they’re unitless, so it’s harder to get a grip on them. What 2n really represents above is an endomorphism of the free commutative monoid on an apple, which is much less concrete than an apple.

There is a way to gain intuition here which sort of involves units, but I don’t know if you can convince your daughter that it makes sense. One way to interpret 2n is that it is the “size” of an n-cube of side length 2 in dimension n. For example, the length of a segment of side length 2 is 2, the area of a square of side length 2 is 4, and so forth. One way to say this is that 2n is the number of n-cubes of side length 1 that fit into an n-cube of side length 2.

To get a meaningful interpretation of the above when n=0 we need to decide what 0-dimensional objects are. Well, if 2-dimensional space is a plane and 1-dimensional space is a line, then 0-dimensional space must be… a point. In particular, a 0-cube, of any side length, is a point, and so exactly one 0-cube of side length 1 fits into a 0-cube of side length 2. Hence 20=1.

(I’m really curious what her response to this argument will be, actually. Could you report back on this?)

**Attribution***Source : Link , Question Author : Yishai , Answer Author : dfeuer*