The Most Controversial Number in Math

0^0 is highly debated in the mathematical community. Why is 0^0 undefined? Does 0^0=0? Does 0^0=1? In this video I'll address the 0^0 meaning and give you a solid 0^0 proof.

SPOILER ALERT: 0^0=1 proof can be found just using the definition of how we define exponents.

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#math #zerotothezero #0^0

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.

[BriTheMathGuy]

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[krillinslosingstreak]

Someone: "It should be 0"
Someone else: "It should be 1"
Someone else else: "Let's split the difference and call it 1/2"

[GMPranav]

Every 'controversial' number seems to have something to do with 0 and 1 colliding in some way, it's beautiful. Both are such iconic numbers.

[RCSmiths]

0, what should be the simplest number in math, always subverting our expectations. Great work as always

[ghost_inthewire]

New drinking game: Take a sip every time he says ‘zero’.

[fancychipss]

My math teacher casually threw a 0^0 at me and got mad at me for being "wrong". Now I'm pissed that half the mathematical community would have said I was right.

[biggerdoofus]

"We can't have 2 results" square roots would like a word with you.

[TheMrcoolguy]

I generally just use the idea that an empty sum = 0, empty product = 1. The identity considering the operation.

[happygood]

The meaning of 0^0 depends entirely on the context in which it appears.

In *calculus and limits, * 0^0 is considered an *indeterminate form.* This happens when you’re dealing with functions like f(x) and g(x) where both approach 0 as x approaches some value. In such cases, the value of f(x)^g(x) depends on how the functions behave, which is why 0^0 in this context doesn’t have a fixed value—it requires more information to evaluate.

In *algebra and combinatorics*, 0^0 is typically defined as 1. This is done for practical reasons, like keeping formulas consistent. For example, in the binomial theorem or when calculating the number of ways to choose zero items from zero options, defining 0^0 as 1 avoids unnecessary exceptions and makes things work smoothly.

The confusion arises because 0^0 can mean different things depending on the situation. In some cases, it’s left undefined to avoid ambiguity, while in others, it’s explicitly set to 1 to simplify calculations.

To clarify further, sources like Wolfram state that x^0 = 1 for any x ≠ 0. For x = 0, they treat 0^0 as indeterminate unless it’s in a context, like algebra, where defining it as 1 makes sense.

*Conclusion:*
- In limits, 0^0 is indeterminate.
- In algebra or combinatorics, it’s often defined as 1 for consistency.

[Cheerwine]

Fun Fact: if you take the limit of x^x with x approaching 0, it will at first appear to be getting closer to 0, but will actually increase, and approach 1.

[Scrufus]

zero doesn't even sound like a word anymore

[degroot]

I like when he writes everytime he looks disappointed

[OrionYTP]

this was the first argument i've ever seen that actually convinced me. it finally clicked! great channel!

[henrytang]

I originally accepted 0^0 = 1 based on the Desmos plot of y = x^x.

But when you brought in the binomial theorem and taylor series expansion of e^x, it made more sense.

[anshumanagrawal]

I actually was kind of convinced 0^0 should be equal to 1 earlier, but now it makes more sense thanks to you

[sqbuilder]

after hearing "zero" so many times, it doesnt even sound like a word anymore

Knuth's opinion is that if the exponent is viewed as an integer then 0^0 = 1 (because we don't have to worry about a bunch of annoying special cases as you noted) but if the exponent is viewed as a real number 0^0 is undefined (because the function x^y has an essential discontinuity at x = y = 0).

[lptotheskull]

i've never heard of numbers being controversial. and yet, here we are.

[MedK]

I've never heard 0^0 being undefined. In school, I was taught 0^0=1 and my teacher even gave us the explanation you gave at the end. 0^0=1 easily makes the most sense to me lol.

[kev_whatev]

The more I watch, the more I’m convinced it’s 1/2