0

TL;DR: How do I prove x0=1 without proving by division of x?

I'm somewhat new to calculus, so I don't have much experience with complex proofs. That being said, I ran into someone recently who argues that x0=0 (where x is any number). I showed this person several videos by Eddie Woo and blackpenredpen, but they say that the explanations used "patterns instead of straight-foward thinking." I want to prove that x0=1, but I can't find any videos that prove this without using proof-by-reversibility patterns (dividing by the base to get 1).

I've tried using the Fundamental Theorem of Calculus to prove that even complex math is reversible (and can be used to prove by patterns), but this person just says that this thinking is flawed and is based on circular reasoning (they also suggested that calculators are programmed to display x0=1 to affirm and satiate mathematitians).

Any help is appreciated!

EvanderIV
  • 13
  • 1
    You can easily argue that $x^0=1$ for any $x\neq0$, but there is much more conflict about $0^0$. this is because in most discreet branches of math, it makes more sense to call $0^0$ equal to $1$ rather than undefined. In other branches, there is no correct answer, because $\lim_{x \to 0} x^0=1$, while $\lim_{x \to 0} 0^x=0$, and there are many more functions that approach $0^0$ that don't equal either. – Jack Phillips Feb 10 '23 at 20:01

2 Answers2

2

There can be problems with $x^0=1$, so someone who merely disagrees with that could have a point. But someone who claims conversely that $x^0=0$ has to have an explanation that deals with

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

I'd ask them which equality step they want to give up on, or if they think $2$ does equal $0$.

(BTW, the weakest step in that chain is where we used $a^{bc} = a^b a^c$, as that requires us to put some sort of restrictions on $a, b$, and $c$ to be valid.)

JonathanZ
  • 10,615
  • Your example is very clever! I never thought of using simple algebra to prove such a confusing point. Thank you so much! – EvanderIV Feb 10 '23 at 22:09
  • Sure. And if you do present this to your friend, I'd be curious to see what their reaction is. Feel free to update us here. – JonathanZ Feb 10 '23 at 22:40
  • I showed this to my friend as soon as I saw it, but they just said that it's not true because exponents are a synthetic operation instead of the "fundementals of math", and therefore is flawed in some way. They seem very convinced that math fundementally changes when the number zero is involved, so I suppose there is little I can do to convince them. – EvanderIV Feb 10 '23 at 22:51
  • Ha! Well, I guess if exponents are a "synthetic operation", then the $1$ you get from raising $2$ to the power of $0$ is a "synthetic $1$", which must be different from the "natural $1$" you get when you subtract $3$ from $4$. Or maybe $2^0 = $ a slice of cheesecake? I hope you enjoy these debates with your friend, because it seems pretty obvious by now that have no interest in understanding things; they just enjoy disagreeing and arguing. – JonathanZ Feb 10 '23 at 23:10
0

they also suggested that calculators are programmed to display x0=1 to affirm and satiate mathematitians

This is essentially correct. Similarly, English is written left-to-right to satisfy and satiate English speakers, while Hebrew and Arabic are written right-to-left to satisfy and satiate Hebrew and Arabic speakers. That $x^0 = 1$ is a convention, not something you can prove. The powers of $x$ can be defined as $$x^n = \begin{cases} 1 & \text{ if $n = 0$,}\\ x * x^{n-1} & \text{ if $n > 0.$} \end{cases}$$ This is an inductive definition. Similarly, if you are familiar with the Fibonacci numbers, they are defined $$F_n = \begin{cases} 1 & \text{ if $n = 0,1$,}\\ F_{n-1} + F_{n-2} & \text{ if $n > 1.$} \end{cases}$$

So how do you prove $F_0 = 1$? Well, you don't. It's a definition. In fact, sometimes the convention is that $F_0 = 0$. It depends on what text you are reading. But once we decide on the definition above, we don't get to argue that $F_2 = 7$. The statement $F_2 = 1+1 = 2$ is then provable, and we don't get to decide it.

So, tell your friend that they are in fact free to decide that $x^0 = 0$. It's just that this will confuse everyone else they talk to, because virtually all mathematicians agree on the convention $x^0 = 1$. They're free to write a whole book in which $x^0 = 0$, developing this as an alternate form of math - just like there is a book out there that defines $-1 * -1 = -1$. It's just that they will probably have a hard time finding readers. You could also publish a map that refers to New York City as Moscow, if you so wish.

But now why do most mathematicians write $x^0=1$? Well, that's a more interesting question. As I said, there is no proof, but there are many explanations as to why this is a good choice, as in the demonstrations you have alluded to. It turns out to be very convenient.

Jair Taylor
  • 16,852