Anything multiplied by 0 is 0 right so why does x number raised to the power of 0 = 1? isnt it x^0 = x*0 (im turning grade 10 and i asked my teacher about this he told me its because its just what he was taught 💀)
In: Mathematics
Here is a second pass at why the question points to a deep puzzle, I think.
The **usual explanation** for why x^0 = 1 points to something like the relationship between x^m and x^(m-1), extrapolating from there. So, e.g., if 10^3 = 1,000 and 10^ 2 = 100, we see that reducing the exponent by 1 is done by dividing by the base once. And once we notice that, we get a tidy path to 10^0, which is 10^1 / 10, and so on well into the negative numbers.
**But** notice that exponents are often defined in very similar, algorithmic terms:
“The exponent of a number says how many times to use that number in a multiplication.” [https://www.mathsisfun.com/definitions/exponent.html](https://www.mathsisfun.com/definitions/exponent.html)
“An exponent refers to the number of times a number is multiplied by itself.” [http://www.mclph.umn.edu/mathrefresh/exponents.html](http://www.mclph.umn.edu/mathrefresh/exponents.html)
“An exponent refers to how many times a number is multiplied by itself.” [https://www.turito.com/learn/math/exponent](https://www.turito.com/learn/math/exponent)
Of course, on those definitions, answers to things like n^0, n^-1, and n^e are very puzzling! The instruction “multiply 5 by itself never” does not seem to lead to 1. Imagine I asked “What is the difference between no numbers?”. Our most familiar arithmetic operators need some operands! And why would we get a math answer from *not* doing any math (which is what the ordinary definition suggests we should do in the x^0 cases).
So what to do about that puzzling? Well, as the wikipedia for exponentiation, [https://en.wikipedia.org/wiki/Exponentiation](https://en.wikipedia.org/wiki/Exponentiation), explains, one way to figure the rest is to start to use properties of the natural number exponents and extrapolate from there. That’s great (and gets us something like the usual explanation). But it leaves our original understandings of what an exponent is high and dry. Which might be why the high schooler’s math teacher balked.
What to do then? One answer is to appeal to **special cases**, as the University of Minnesota link above does. But we still need an explanation for why we want to admit those special cases. Especially because this means that our definition of exponentiation is now branched or disjunctive.
As far as I know, there are two related answers:
1. A pragmatic answer. Having the disjunctive definition allows us to do more math more easily. Undefined bits gum up the works. We’d like to do things like figure out how to make sense of (x^m)*(x^n), and if we have to start adding all sorts of qualifications, that’s really going to undermine how **usable** math is.
2. A deeper principled answer. It turns out that the original, familiar definition was overly simple. There is a deeper, principled, and **unifying** definition of exponentiation which explains all of the ostensible special cases. When you understand that definition, everything comes into focus. Of course, one reason to choose that unifying definition over the original but simple definition might be that it is more useful part of our math practice. But still, that we have a unified definition might be very handy!
So, I take it that the deep answer to the student’s question will pick up either or both of usability theme or the unifying theme. But ymmv!
Latest Answers