I’m too dumb to get it
In: Mathematics
i like to think of the exponent as the number of times you have to multiply one by the base.
ex. 2^3 means you multiply 1×2, then multiply that by two, then that by two
so a 0 exponent means you don’t multiply one by anything, so you always get 1.
a 1 exponent means you multiply 1 by the base, which is why you always get the base.
10^3 = 10* 10* 10 = 1000
10^2 = 10*10 = 100
10^1 = 10*1 = 10
10^0 = 1
10^-1 = -10*1 = -10
10^-2 = -10*-10 = -100
10^-3 = -10* -10* -10 = -1000
See the pattern?
Some things in math are messed up.
The whole “x^m / x^n” argument is a great way of showing that it must be so, but there is a much more relatable, “real-world” explanation for x^0 = 1. As such, the “no! It’s just because we defined it that way!” response is wrong – this time. It is true that that is sometimes the answer, but not here.
How about an ELI10? This operation (powers) describes the number of possible outcomes. So, if you were to flip a coin once, how many possible outcomes are there? Two – heads or tails, and 2^1 = 2. If you were to flip it 7 times, how many possible strings of heads and tails are there? 2^7. If you flip the coin 0 times, how many possible outcomes are there? Well…1, right? You get nothing. Thus, 2^0 = 1.
This of course ignore the interesting philosophical discussion of how to identify and count nothingness, but at a real-world level it is correct.
This understanding of exponentiation is of course restricted to non-negative integers in both the base and the exponent. By extension you then define the operation for all complex numbers in the base, and then you properly have the stated result. You can also extend to all complex numbers in the exponent. If you prefer to cut back down to real numbers, you of course get weird looking rules that come with that restriction.
Lastly, always remember – don’t drink and derive!
Don’t think of exponents as simply being a number multiplied n times itself (If n=3, then 2x2x2). Think of exponents using one extra number: 1 (allowed under Identity Property of multiplication – not important here). So 2 squared is 1x2x2=4. 2 cubed is 1x2x2x2=8. So if n=0, then 2ⁿ is 1x(zero 2s) which equals 1. Keep in mind that zero 2s means it doesn’t exist, not 2×0. So any number with an exponent of 0 is 1 times nothing else. 1.
x^n means “Multiply 1 by x n times”
Math|Math Expanded|Words
:–|:–|:–
2^4|1x2x2x2x2 = 16|Multiply 1 by 2 four times
2^0|1|Multiply 1 by 2 zero times
How many times can you count a number 0 times? No times. But that set of nothing is still has nothing. Which is counter as 1. You count nothing
new problem:
why is 0! == 1?
How bout 0⁰?
0 to the power of anything is 0 but anything to the power of 0 is 1.
x^0 is something called an [empty product](https://en.m.wikipedia.org/wiki/Empty_product), that is, the result of multiplying zero numbers together. It is defined to be 1, the multiplicative identity (x • 1 = x), because that is the most useful and intuitive way of defining it. In any multiplication problem, let’s say (2 • 3), you can say there’s an implicit (2 • 3 • 1 • 1 • 1 • 1. . .) at the end. If you multiply zero number together, you’re left with only the (• 1 • 1 • 1 • 1. . .), so you get one.
Here’s another deeper angle I haven’t seen explored yet:
Consider that “fractional powers” can mean square/cube/etc roots, so 2^(1/2) = about 1.41. 2^(1/3) = about 1.26. Even when you take the 10th root of 2, it’s 1.07. So if you take the general case, N^(1/M), you can think of it as if you’re taking the M^(th) root of a number N. 1/infinity is 0, so it’s describing what happens if you take the “infinite root” of a number, and it just so works out that it’s always 1, even with fractions.. the 10th root of 0.1 is 0.99. It’s not something that makes practical sense, per se, so that’s probably why we’re just given the rule without much explanation.
Latest Answers