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A review of the rule that $n^0$ is always 1, when n is not 0, has made me question if all math rules can be visualized or conceptualized with a general intuition gained from viewing the natural world. It seems that me that there is no natural representation of this rule or a representation that can be conceptualized in terms gained from experiencing reality. I have come across three methods of justification that are used to determine that $n^0=1$. One uses a somewhat altered, or added to, definition of exponents, the other relies on maintaining internal consistency with other math rules, and the last relies on a pattern of division seen with single terms exponentiated. I’ll list the justifications now:

  • 0 power rule justified with exponent subtraction

  • The second justification has to do with assuming that the definition of exponents always starts with a multiplication of 1. Therefore $x^3=1*x*x*x$ and $x^0=1$

  • The last justification tries to justify the rule from a pattern: $3^3=27$, $3^2=9$, $3^1=3$. Each time the following result is the former result divided by 3. If we keep going, we get $3^0=1$.

None of these methods make very much conceptual sense because none of the justifications are attempts to show conceptual justifications, instead they are abstracted away from real life examples and intuition. If we think about exponentiation as repeated multiplication, similar to how multiplication is repeated addition, I cannot think of an everyday situation which would repeat multiplication 0 times and get 1. As a result, I cannot grasp an intuitive sense of the rule. With multiplication by 0, it is easy to see that when you repeat addition on X 0 times, you get 0.

So, my question is, is there a more intuitive explanation that I am missing or not understanding? If not, then I feel like I have been treating my math studies in the wrong way. I have been trying to intuitively understand mathematical concepts in the same way a philosopher mathematician might have conceptualized them early on in their brains. Is this the wrong way to go about thinking about math? Should I instead just treat math as a series of arbitrary rules that are built upon each other to create a system? Edit: Or as LittleO said, "convenient rules".

Edit 1: Called the three justifications "rules" on accident.

Also thank you to the kind admin/moderator who helped clean up the math formatting.

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    $\begingroup$ Welcome to Mathematics Stack Exchange. An empty product is $1$ $\endgroup$ – J. W. Tanner Jul 23 '19 at 0:39
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    $\begingroup$ "Should I instead just treat math as a series of arbitrary rules" Defining $n^0$ to be $1$ is just convenient, because it makes exponent notation more expressive / powerful while preserving the basic rules of exponents that we know and love. I don't have any kind of physical or "real-world" picture or intuition behind $n^0 = 1$, but that does not mean it is an "arbitrary rule". It is rather a convenient definition. $\endgroup$ – littleO Jul 23 '19 at 1:07
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Here is a somewhat contrived, but real-life example, for positive, integral $n$ that just applies your third rule. Nonetheless, I hope showing this in a physical model provides some intuition.

Assume there is a species where each organism at each next time step reproduces to form $n$ children and then immediately dies itself. Also, assume you start with some positive, integral $k$ organisms at $t = 0$. Then at time $t = 1$, there will be $n$ organisms for each of the original $k$ for $kn$ organisms in total. Next, at $t = 2$, there will be $kn\times n = kn^2$ organisms. In general, at time step $t \gt 0$, there will be $kn^t$ organisms. If you want to have the same formula also be used for time step $t = 0$ requires that $k = kn^0$, so $n^0 = 1$. Otherwise, you'd need to create a special case for handling $t = 0$. However, if $k$ is actually an integral multiple of a power of $n$ (e.g., $k = 2n^3$), there may have been other time steps before this where the organism was reproducing, so you should then also create a special case for each of these earlier, original start times if they were used instead. To me, this would just add unnecessary complexity with no benefits.

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Here's the simplest conceptualization I can think of for why $n^0 = 1$ for all $n \ne 0$:

Suppose you deposit some money into a bank account that earns interest. Then $0$ years after the deposit, you will have $1$ times as much money as you started with. This is true regardless of what the interest rate is.

That said…

I have been trying to intuitively understand mathematical concepts in the same way a philosopher mathematician might have conceptualized them early on in their brains. Is this the wrong way to go about thinking about math? Should I instead just treat math as a series of arbitrary rules that are built upon each other to create a system?

Well, some mathematical concepts have easy conceptualizations; others do not. It should be possible to come up with some kind of intuition for pretty much every mathematical concept you come across, but unfortunately, some concepts have a much more meager intuition than others. As an example, the $3 n + 1$ rule of the Collatz conjecture really doesn't seem to have any intuitive meaning at all.

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The OP writes

I have been trying to intuitively understand mathematical concepts in the same way a philosopher mathematician might have conceptualized them early on in their brains.

I recommend reading the following links:

Who Invented Exponents?

History of logarithms


One Beginning to the Story...

Let the integer $b \gt 1$ be given. Every integer $a \ge 0$ has the familiar Base-$\text{b}$ representation,

$$\tag 1 a = a_0 + \sum_{k=1}^n a_k b^k$$

Wait a minute! Let's try that again,

$$\tag 2 a = \sum_{k=0}^n a_k b^k$$


... to The Glorious Pinnacle
(copied from the history link)

Euler

Around 1730, Leonhard Euler defined the exponential function and the natural logarithm by

${\begin{aligned}e^{x}&=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n},\\[6pt]\ln(x)&=\lim _{n\rightarrow \infty }n(x^{1/n}-1).\end{aligned}}$

In his 1748 textbook Introduction to the Analysis of the Infinite, Euler published the now-standard approach to logarithms via an inverse function: In chapter 6, "On exponentials and logarithms", he begins with a constant base a and discusses the transcendental function ${\displaystyle y=a^{z}}$. Then its inverse is the logarithm:

$z = log_a y$

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