But What Is ∞ ^ 0

In this video, we will discuss the concept of infinity to the power of 0. Contrary to popular belief, this mathematical expression is undefined, not equal to 1. While some may argue that it equals 1, the general consensus is that it cannot be defined due to the nature of infinity and its properties in exponentiation. Join me as we delve into the intricacies of this concept.

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information. Viewers should always verify the information provided in this video by consulting other reliable sources.

[BriTheMathGuy]

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[BenjaminMellor]

As Vsauce said, infinity is not a number, but a kind of number. Under certain axioms, there are infinitely many infinities.

[Bodyknock]

I think a slightly modified approach that I like would be to define the symbol ∞ to be shorthand for "the limit as x grows to infinity". That fits the informal definition at the end of the video (∞⁰ = lim x⁰ = 1). The indeterminate forms would be more like ∞ ^(1/∞) or something similar with the exponent is some function that includes a limit approaching infinity in it as a parameter.

[aaryananand]

2:17 wtf is wrong with that calculator?!

[iamthekittycat]

The calculator: *domain error*
What it actually did:

[antoniusnies-komponistpian]

This reminds me of Bertrand Russell writing that numbers are actually variables themselves. Infinity is a variable too and how it is to be used simply depends on what it actually stands for.

[xwddydfy_]

guess what i didnt understand a single thing

[NOT_A_ROBOT]

0:09 it's also true for negative real numbers as well.

[sanszeesingh]

I take this figure and expand it. Intuitively, let infinity be a number divided by zero. Now using properties of exponent, 1^0/0^0 is what we get. As you said at the beginning of the video, 1^0 is 1, and you also said 0^0 is undefined. This means that the form whose value we try to find is undefined. We can settle by saying infinity^0 is an indeterminate form. This is my intuition. Please correct me if my intuition needs to be updated.

[kuroshite]

We should have more videos with completely black backgrounds. It looks so cool and futuristic like a hologram on OLED screens.

[aldenstromberg]

x^0 = x^-1 * x^1 = x / x
infinity^0 = infinity / infinity
im not taking about the type of number infinity think of it like this
sum from n=0 to infinity of 1 = infinity. Now lets divide sum from n=0 to infinity of 1 by itself that it essentially infinity / infinity
and if you think about it youre dividing something by itself for every term in the top infinity there is that same term in the bottom infinity so it will always be 1 no matter how far out the sum is written out, even to infinity. The exact same type of infinity divided by itself is 1. If they were different types of infinity it would come out to be either 0 or infinity.
However, when you take x^0 = x / x
x is the only variable in the equation. if x = infinity, the infinities youre dividing are exactly the same, so infinity^0 should be 1.

theres something wrong with that feel free to tell me what it is

[glutentag]

You could use any positive number, set its power to 0, and voila! 1. Because infinity is every positive number, then it itself cannot have an exponent. However the infinite things inside of it can. As a result, it is safe to say, it's 1.

[gitanjalideb]

This channel just makes videos on my thoughts which like his thoughts.

[NOT_A_ROBOT]

2:16 that is an unusual-looking calculator, to say the least.

[JoeCMath]

I really appreciate your marriage of Manim with talking head Brian that uses markers on a mirror or glass. You have your own style and I love it! Keep it up Bri!

[niccolopaganinifranzliszt]

Also, if you take the limit as x approaches infinity at x^(1/x) you get infinity^0, and it results also at this function as 1.

[isjosh]

00:08 is it not true for all negative real numbers?

[joshuahillerup]

If everything you're working with is a set, then whatever set to the power of 0 is 1, because there is only one function that goes from whatever set to the empty set

[navsha]

1:37 the answer to ∞ to the power of 0 is actually true for that statement because 1/x=1/∞ and then you cross the symbol out and in the brackets and then you will get 1 +1 which equals 2

[Смещной_Кощка]

Hey, any number you can write like:
N = 1 × M×M×...×M
And the count of the "M" is the power.
Thats reason why any number in 0 power is 1:
N=1×M repeted 0 times=> N=1

What is inf×1? Thats inf. Thats means that you con write that:
N¹=1×inf; N²=1×inf×inf ect. =>
Inf⁰=1×inf 0 times = 1