Using Visualization for Learning: $a^0=1$

Question

Honestly, this was a question from a student. I show $\forall a\in \mathbb{R}-\{0\}:a^0=1$ for the class. $$1=\frac{a^m}{a^m}=a^{m-m}=a^0$$or $$a^{m}=a^{m+0}\to a^m=a^m\times a^0\\(a\neq 0) \to cancel \space a^m\\1=a^0$$ after that work, one of students asked for a visual proof, to make sense more for the proof. I can't go further more. Can you help me if there is a visual sensation for that proof? Thanks in advance for any idea.

Answer 1

If you want a visual proof of $a^0=1$, try this one:

Let $a$ be any arbitrary integer, in this case let's use $a=2$: \begin{align} & ~~\vdots \\ 2^3 &= 8 \\ 2^2 & = 4\\ 2^1 & = 2 \\ 2^0 & = ? \end{align}

We keep dividing by $2$ to find the next lower power of $2$. Then it is only logical to follow the pattern for $2^0$. Hence we find that $2^0 =1.$ Try this for any integer.

In general,

\begin{align} & ~~ \vdots \\ a^3 &= n^3 \\ a^2 & = \frac{n^3}{n} = n^2 \\ a ^1 &= \frac{n^2}{n} =n^1\\ a^0 & = \frac{n^1}{n} = n^0 =1\\ \end{align}

Answer 2

Picture one:

A sea full of $2$s. A $5$ and a $7$ jump in and become wet with $2$s and turn into a $10$ an $14$. All the other numbers run away screaming.

Picture two:

A sea full of $0$s. A $5$ and a $7$ jump in and become wet with $0$s and turn into $0$s themselves. All the rest of the numbers are terrified and run around panicking and wetting themselves in abject horror. A few are vomiting because it was so shocking and disgusting.

Picture three:

A sea full of $?$ marks. The numbers look at it fearfully. A $13$ sneaks up behind an unaware $6$ and sacrificially pushes him in. The $6$ remains a $6$. A huge sigh arises from the crowd and all the numbers jump in. The numbers all stay themselves and and they frolic in the surf. A curious $9$ cautiously picks up a question mark and looks at it with a magnifying glass. Under close examination it is see that the question mark is made out of $1$s. Everyone is happy.

Picture four:

A $5$ is swimming in a sea full of the $?$ marks but now it is clear that they are actually $1$s. But they don't look like the numbers $1$ but there is subtle shading and hints that we know they are little tiny drops of $1$s.

Another $5$ jumps in and the hold hands. Hold hands the appear to be a chain of $5 \times 5$. Above their heads a phantom writing appears stating $5^2$.

Picture five:

A third $5$ jumps in and joins the other two $5$s and holds hands so that form a chain: $5\times 5\times 5$ then phantom writing fades and reappears as $5^3$.

A fourth $5$ jumps in and joins and makes a chain $5\times 5\times 5\times 5$ and the phantom writing wifts away and becomes $5^4$.

Picture six:

The four $5$ breaks away and goes ashore, the writing over the remaining chain of $5\times 5\times 5$ reverts to $5^3$.

The third $5$ breaks away leaving a chain of $5\times 5$ and above it the smoky writing morphs into $5^2$.

The second $5$ breaks away leaving just a single $5$. The writing transfigures into $5^1$.

Picture seven:

The last $5$ leaves the sea but the smoky writing remains. It reads $5^0$.

Picture eight:

We zoom in close the little drops of $?$ mark/$1$ hybrid little particles of water. The writing $5^0$ becomes gradually more solid. Every but the $5^0$ and a single drop of $1$ water fades away.

We are left with a blank background, the writing $5^0$ now as solid as stone and black as onyx, and single tiny but crystal-clear blue drop of $1$ water.

Answer 3

For the sake of the visual, let $a$ be some small, workable number. Let's say, $4$.

Draw $a^3$ as an $a\times a\times a$ rectangular solid. Show the $64$ unit cubes within. And explain that $a^3$ is counting how many unit cubes are inside.

Now animate the big cube flattening in one dimension so it becomes an $a\times a$ square. Show the $16$ unit squares within. And explain that $a^2$ is counting how many unit squares are inside.

Now animate the big square flattening in one dimension so it becomes a line segment of length $a$. Show the $4$ unit sub-segments within. And explain that $a^1$ is counting how many unit sub-segments are inside.

Now animate the big line segment flattening its last dimension so it becomes a point. Show the $1$ point within. And explain that $a^0$ is counting how many points are inside.

Answer 4

The curves of $a^x$ and $a^{-x}$ meet at $(0,1)$ because $1$ is its own inverse.