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

Question

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?

Answer 1

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.

Answer 2

I would appeal to the laws of exponents. Show for $a,b>0$ that $2^a* 2^b=2^{a+b}$, then extend it to negative $b$, then set $a=b$.

Answer 3

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?)

Answer 4

Why not just take out a calculator, let her punch in any number she likes, and then let her hit square root over and over. The limit is 1, and to show $2^0=1$ will take some explaining, but i think it's the best way to think about this in the long term. Actually this sequence will allow you to later on introduce $log$ and $e$ in a simple way.

Answer 5

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

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

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

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

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

Answer 6

I would try to explain it in terms of exponents

$2=2^{1}=2^{1+0}=2^{1}\times 2^{0}=2\times 2^{0}$

and by a division

$\dfrac{2}{2}=\dfrac{2}{2}\times 2^{0}$

$1=1\times 2^{0}=2^{0}$.

Note: Applying the same argument to any number different from $0$ gives the same result.

Answer 7

I think it's quiet simple to understand:

The base point is 1 (I'm not talking about the base... read on...).

from this initial point of 1, you are applying the base to the result n times:

2^3 = 1 * (2 *2 *2) = 8

2^-3 = ((( 1 / 2) /2) / 2) = 1/8

The base point is always 1 so when you are in the 0 point (n^0) you don't have to multiply the very basic 1 so you have "1"

the starting point is the "1" not the base.

edit: I've just noticed that Zarkonnen already gave this answer, yet I think that the way I presented the answer is easier for a child to understand- So Zarkonnen, I hope you are ok with this post ;)

Answer 8

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.

Answer 9

When I was a kid, I had to teach this concept (and fractions) to some kids in a lower grade. The best method is to work backward with division.

2^3 = 8

2^2 = (2^3)/2 = 8/2 = 4

2^1 = (2^2)/2 = 4/2 = 2

2^0 = (2^1)/2 = 2/2 = 1

And so on. This is also the only way to wrap your head around negative powers.

Incidentally, the best way to teach fractions is by having pizza for dinner.

Answer 10

I think that it is worth pointing out that, strictly speaking, explaining “$2^0$ equals $1$” is not really what people do in this situation. The best we can do is to convince a child of the following facts:

1. If $2^0$ is any number, it makes more sense to consider that $2^0=1$ than considering $2^0$ as any other numbers (such as $0$).
2. It is more interesting to consider $2^0$ to be $1$ than giving up.

Some of the other answers provide good ways to convince a child of these facts.

However, the reason that $2^0$ equals 1 rather than $2^0$ is undefined is really conventions and experiences: it is much more convenient to define $2^0=1$ than leaving $2^0$ undefined. I do not think that it is possible to convince a child of this fact.

Compare this to the following. Let $f(x) = \frac {\sin x} x$. What is $f(0)$?

1. If $f(0)$ were any number, it would make more sense to consider that $f(0)=1$ than considering $f(0)$ as any other values.
2. It would be more interesting to consider $f(0)$ to be 1 than giving up.

However, these “evidences” do not make $f(0)=1$ under the usual way of mathematical writing. I think that the only things that differentiate these two cases are conventions and experiences.

Answer 11

Here is a very prosaic answer. If you go into a grocery store with an order, you expect the cash register to read 0 before any items are rung up. This is because $0 + x = x$ for any number $x$. It is the fact that zero is the neutral element of addition that makes it a valid start for the register; starting there will not cheat you on your order or give you any unfair advantage. To wit: an empty sum is 0.

Likewise, suppose we are multiplying numbers in a multiplying machine. What is neutral there? We know $1*x = x$ for any number $x$. Since an empty product, like an empty sum should be neutral and not affect the product of the numbers coming in after it, the multiplying machine should be set at 1 before it starts work.

Now $2^0$ is an empty product so $2^0 = 1$. Said identity should be true for any base.