How do I explain 2 to the power of zero equals 1 to a child

My daughter is stuck on the concept that $$2^0 = 1,$$ 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?

Solution 1:

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 $2\times 0 = 0$. 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 $2^n$ 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 $2^0 = 1$.

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 $2^n$ 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 $2^n$ 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 $2^n$ 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 $2^0 = 1$.

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

Solution 2:

How about this: There's always an implicit 1 in the expansion:

$$2^{3} = 2 \cdot 2 \cdot 2 \cdot 1 = 8$$

$$2^{2} = 2 \cdot 2 \cdot 1 = 4$$

$$2^{1} = 2 \cdot 1 = 2 $$

$$2^{0} = 1 = 1 $$

Solution 3:

I'd demonstrate this using a pattern.

$2^3 = 8$

$2^2 = 4$

$2^1 = 2$

$2^0 = 1$

$2^{-1} = 1/2$

$2^{-2} = 1/4$

When you decrease the exponent, you divide by 2. So, when you go from 2¹ to 2⁰, of course you divide by 2, which gives you 1.

From there, you can segue into negative exponents, if you'd like. Just keep dividing by 2.

Solution 4:

I want to extend the answer by @Qiaochu Yuan.

I assume the kid accepts $2\times 0 = 0$. In other terms:

"Some number times $0$ yields the no-changer of plus."

Analoguously:

"Some number to the power $0$ yields the no-changer of times."

By no-changer I refer, of course, to the unit element. That this can be added/multiplied to anything without resulting in a change should be accepted. I am unsure wether this approach helps understanding the hierarchy of arithmetic operators or wether you need the hierarchy for understanding the approach.