I’m too dumb to get it
In: Mathematics
Exponents of integers is just a shorthand notation for a number multiplied by itself a number of times.
So we define the following rules
b^(1) =b
b^(n+1) = b^(n) * b
That mean that bn = b*b*b*b… (you should have n b:s on the right side.)
From that and associativity of multiplication ie that a*(b*c) =(a*b)*c =a*b*c the following line follows if m and n are positive integers.
b^(m+n) =b^(m) * b^(n)
That is the fundamental rule of positive integer exponent. If you look at what happen if the exponent is 0 you can use the face that 4=3+1=4+0 so b to the power of 4 should be identical regardless of how we write it.
b^(4)=b*b*b*b
b^(3+1) = b^(3) * b^(1) =(b*b*b)* b= b*b*b*b
b^(4+0)=b^(4) * b^(0) = (b*b*b*b) * b^(0)
So becaus b^(4) should be the same as b^(4+0) you get the relationship
b^(4) = b^(4+0) => b*b*b*b =(b*b*b*b) * b^(0) => b*b*b*b/(b*b*b*b) = b^(0) =>1=b^(0) So b^(0) has to be 1 for the rules to make sense. That is for any b ≠0 because we divided by b in the equation. what you define 0^(0) as depend on the context.
For the arithmetic operation we have agreed on b^(0) has to be 1 if b≠0 if not the rules do not work. Why for rules that look initially strange in math is generally because that is needed for the rules we have defined to work as smooth and simple as possible. You might say that b^(0) is not allowed but because we can define is as b^(0)=1 and it works fine with other rules we use that definition.
Mathematicians *define* x^0 = 1 in order to make the laws of exponents work even when the exponents can no longer be thought of as repeated multiplication. For example, (x^3 ) (x^5 ) = x^8 because you can add exponents. In the same way (x^0 ) (x^2 ) should be equal to x^2 by adding exponents. But that means that x^0 must be 1 because when you multiply x^2 by it, the result is still x^2. Only x^0 = 1 makes sense here.
Essentially x^0 MUST equal one for all other exponents to work.
Edit: thank you kind strangers for your gifts of silver and gold!
WE HAVE 13 axioms in number theory. Things that everyone agree are taken by true. They don’t need proof. There’s nothing magic about these axioms, other set could have been chosen. But it was decided to have those 13.
As an example, one of them is: x * 1 = x . This is accepted by a true not needed to be proved.
And you have de definition: x ^ y = x * x * x… * x a number of y times, right?
Now we have: x ^ (y+z) = x^y * (x^z). It can be proved by induction, but it makes sense by the above definition.
and x ^ (y-z) = x^y / x^z. It also can be proved by induction.
So x ^ (y-y) = x^y / x^y .
and you have
x^0 = 1.
For multiplication, 1 is the neutral number, just like 0 is the neutral number for addition. When you take the take the 0th power of a number, you bring it back down to neutral, and so that makes it 1. It’s not supposed to make sense, it’s just an arbitrary base case to make the rest of algebra work.
*x^n* is the number of times you have to divide *x* out of *n* to get *1*.
125 = 5^3
25 = 5^2
5 = 5^1
1 = 5^0
1/5 = 5^(-1)
1/25 = 5^(-2)
Can I get some ELI2?
There were better explenations here but here is another angle. You can also look at it in terms of an empty product.
If you look at an empty sum, then it should be the neutral element for addition, so 0. (Add 0 to any number and it stays the same). So if add up a number 0 times to it self, it’s 0. Look at 0*x = 0 for any number x.
By the same logic an empty product should be the neutral element of multiplication, so 1. (Multiply any number with 1 and it stays the same). So if you multiply a number 0 times by it self, It’s 1. Look at x^0 = 1 for any number x.
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!
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