My wife started the Math Academy diagnostic on Sunday night. She couldn’t remember that if you raise a number to the 0th power, you get 1.
Her frustration instantly reminded me of being a kid and getting mad at math.
She was annoyed because she couldn’t see the intuition behind “if I multiply something by itself zero times, I get… one?”
I don’t get that either. Rather than look it up, I figured we could “prove” that this must be true based on rules that feel more visible.
Here’s how I tried to make sense of it from basics:
I asked her what 2² * 2³ was — something she could manually see as 4×8=32.
Then I said “Represent 32 using the same base of 2”.
She got 2⁵.
I had her write down what we did so far:
2² * 2³ = 2⁵
So what’s the pattern?
“That you add the exponents when you multiply?”
Right — as long as the base is the same.
But here’s the key part: she basically derived the rule herself from observation. And that matters. Most of us don’t like to accept rules “just because.”
So from there I ask what’s: 2³ * 2⁻³?
Add the exponents… 2⁰
But we know, by definition, that 2⁻³ = 1/2³
So:
2³ * 2⁻³ = 8 * 1/8 = 1
So 2⁰ MUST also be 1.
There’s no intuition here—but it follows inevitably from the basic definitions we already accept.
Back to sympathy for the learner…
She still felt annoyed even though she followed the chain. I remember being frustrated as a kid: you can see how it works, but it’s not intuitively satisfying.
As I’ve gotten older, I’ve grown more patient with “I don’t get the intuition, but I see why this rule must be true given the rules I know are inviolable.”
But when you’re young, or new to a topic, it’s easy to get bogged down by “not getting the why.” You don’t yet have the faith that a little time, a few reps, or a fresh look on another day will eventually give you that satisfying resolution in your head.
Without that faith, you feel like you’re following a recipe blindly. Which is, of course, why my wife was annoyed. She doesn’t want to rely on arbitrary memorized rules. When you feel that way, you don’t feel confident — you don’t feel like you could re-derive the rule if you needed to.
It feels like an assault on your independence or intelligence.
Anyway, this is just a thought I had because I totally commiserate with that frustration in numeracy, and this little back-and-forth dominated Sunday night’s family dinner conversation.
By the way, I still haven’t looked up “intuitive explanations for why raising numbers to the zero power equals 1,” but the proof-by-necessity — the “how could it be otherwise?” style — is satisfying enough for me not to care.