why does 0 to the power of 0 equal 1?
In: 97
It doesn’t; that expression is undefined. For real numbers other than 0, n^0=1 because 1 has the same relationship to 1/n as n has to 1. That is to say they are separated by a common factor (n). For real numbers other than 0, 0^n=0. Zero multiplied by itself an arbitrary number of times will always equal zero. 0^0 can’t be both 0 and 1 and it doesn’t make sense to assign an arbitrary value like 1/2 so the value has to be left undefined.
Here’s a really good video on the topic by [Numberphile](https://www.youtube.com/watch?v=Mfk_L4Nx2ZI).
In set-theory context, the number m^(n) can be interpreted as the number of maps (functions, putting one element of the second set in correspondence to each element of the first set) from an n-element set to an m-element set. A zero-element set is an empty set, and there is exactly one map from an empty set to itself (We don’t need to put anything in correspondence to any element to define that map).
0 to the power of 0 means 0 is divided by itself. 0 divided by 0 can be absolutely any number. Any number, when multiplied by 0, gives 0. Therefore, we can say 0/0 = 0, 0/0 = 1, 0/0 = 2 or 0/0 = 1234567, and all of these will be right.
To prevent that, mathematicians say 0/0 and 0^0 are undefined.
It’s undefined as there’s no clear answer.
* x^0 = 1
Any number (except 0) to the power of 0 is 1.
Why 1? Because that’s the neutral element. 2^2 isn’t just 2 times 2, it’s 1 times 2 times 2. 2^1 is 1 times 2 so 2^0 is 1
* 0^x = 0
Taking 0 to any power is 0, as 0 times 0 times 0 will always stay zero.
* 0^0
You could argue that both answers 1 and 0 are equally valid, but ‘equally valid’ in this sense means that both are invalid. The expression itself just doesn’t make any sense because you can’t take 0 to the power of 0, just like you can’t divide by 0.
There’s already been some posts describing why defining 0^0 = 1 is convenient in power series, Taylor series or some combinatorics applications.
But there’s also situations where 0^0=0 is more convenient, such as defining the discrete metric or the L^0 ‘norm’ (not really a norm). One such example would be to note that the mean is the number that minimises the squared distance from all the points, and the median minimises the distance from all the points, then the mode would minimise the ‘0-th power’ of the distances from all the points, provided that you define 0^0=0.
It isn’t equal to 1. It is undefined. That is, it isn’t well-defined as a number (having a set address on the number line.)
One good way to see this is to complete the following sequences:
81, 27, 9, 3, __;
16, 8, 4, 2, __;
1, 1, 1, 1, __;
0, 0, 0, 0, __;
It should be obvious that the answers are 1, 1, 1, and 0.
Now I’ll write the exact same sequences in a different format. (Remember that ^ means “raised to the power of.”)
3^4, 3^3, 3^2, 3^1, ___;
2^4, 2^3, 2^2, 2^1, ___;
1^4, 1^3, 1^2, 1^1, ___;
0^4, 0^3, 0^2, 0^1, ___;
Now the answers are obviously 3^0, 2^0, 1^0, and 0^0. The first three examples seem to suggest that “any number raised to the 0 power equals 1.” But the fourth example suggests that “0 to any power equals 0.” Since a number cannot be both 1 and 0 simultaneously (quantum superposition aside), 0^0 is not a defined number.
Let me try an alternative way of answering this by dropping the mathematical formalism for a second:
Start with the question ‘what is zero?’ A key thing to grasp is that zero isn’t a quantity: it’s the absence of quantity. It’s nothing.
Certain mathematical operations involving zero are meaningful because they correspond to something in reality. The equation 3 + 0 = 3 is meaningful because it represents the real phenomenon that adding nothing to the quantity 3 leaves you with 3.
An exponent is a formal way of designating an iterative multiplication operation. What in reality is designated by “zero to the power of zero”? If translated into an actual operation, it would mean “multiply nothing by itself no number of times.” What would it mean to perform iterative multiplication for no number of iterations? It’s meaningless.
We can formulate all sorts of equations — and even sentences in ordinary language — that are formally / syntactically / grammatically valid but are meaningless as statements (consider the sentence “Purple hope throws humidification at adverbs swimmingly.”). But all that is just talking nonsense. Saying something in mathematics is ‘undefined’ is basically a polite way of saying it is like this.
Let’s start by showing aything raised to the power of 0 is equal to 1 under the following logic.
When you divide like bases raised to an exponent, you subtract the exponent. Like so
A^b / A^c = A^b-c. So what if you divide like base and like exponent?
A^b / A^b = A^b-b
b-b is equal to 0, but also, any expression divided by itself is 1. So A^b / A^b is equal to A^0, but it’s also equal to 1. Therefore, anything (A) raised to the 0th power is 1.
However, 0 is an exception, because following the logic above, 0^0 is the product of 0^b / 0^b resulting 0/0 which is undefined.
It isn’t. zero to the zeroth power should be undefined. The Ti89 and some other texas instruments calculators output 1 but that may be a limitation with the hardware software of that calculator.
Latest Answers