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11 there are multiple ways of writing out a given complex number, or a number in general And while $1$ to a large power is 1, a number very close to 1 to a large power can be anything. The complex numbers are a field

It's a fundamental formula not only in arithmetic but also in the whole of math The reason why $1^\infty$ is indeterminate, is because what it really means intuitively is an approximation of the type $ (\sim 1)^ {\rm large \, number}$ Is there a proof for it or is it just assumed?

How do i convince someone that $1+1=2$ may not necessarily be true

I once read that some mathematicians provided a very length proof of $1+1=2$ Can you think of some way to 知乎，中文互联网高质量的问答社区和创作者聚集的原创内容平台，于 2011 年 1 月正式上线，以「让人们更好的分享知识、经验和见解，找到自己的解答」为品牌使命。 注1：【】代表软件中的功能文字 注2：同一台电脑，只需要设置一次，以后都可以直接使用 注3：如果觉得原先设置的格式不是自己想要的，可以继续点击【多级列表】——【定义新多级列表】，找到相应的位置进行修改

There are infinitely many possible values for $1^i$, corresponding to different branches of the complex logarithm The confusing point here is that the formula $1^x = 1$ is not part of the definition of complex exponentiation, although it is an immediate consequence of the definition of natural number exponentiation. The theorem that $\binom {n} {k} = \frac {n!} {k Otherwise this would be restricted to $0 <k < n$

A reason that we do define $0!$ to be $1$ is so that we can cover those edge cases with the same formula, instead of having to treat them separately

We treat binomial coefficients like $\binom {5} {6}$ separately already 49 actually 1 was considered a prime number until the beginning of 20th century Unique factorization was a driving force beneath its changing of status, since it's formulation is quickier if 1 is not considered a prime But i think that group theory was the other force.

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