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  • Formal proof for $ (-1) \times (-1) = 1$ - Mathematics Stack Exchange
    Is there a formal proof for $(-1) \\times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math Is there a proof for it or is it just assumed?
  • Prove that $e^ {i\pi} = -1$ - Mathematics Stack Exchange
    The $\pi$ itself is defined as the total angle which connects $1$ to $-1$ along the arch Summarizing, we can say that because the circle can be defined through the action of the group of shifts which preserve the distance between a point and another point, the relation between π and e arises
  • What would base $1$ be? - Mathematics Stack Exchange
    The examples given with base 10 and 2 in the question are positional bases In a positional base 1, you only got one digit, with no value: 0 All positions will have zero value, and you can only represent one number: 0 – Bijective base 1 would be one way to make it funcitonal, but that isn't a positional base
  • Binomial expansion of $ (1-x)^n$ - Mathematics Stack Exchange
    I'm not sure how appropriate it is to answer questions this old, but compared to the methods above, I feel the easiest way to see the answer to this question is to take a = -x And substitute that into the binomial expansion: (1+a)^n This yields exactly the ordinary expansion Then, by substituting -x for a, we see that the solution is simply the ordinary binomial expansion with alternating
  • linear algebra - Product of inverse matrices $ (AB)^ {-1 . . .
    I am unsure how to go about doing this inverse product problem: The question says to find the value of each matrix expression where A and B are the invertible 3 x 3
  • Why is $1 i$ equal to $-i$? - Mathematics Stack Exchange
    There are multiple ways of writing out a given complex number, or a number in general Usually we reduce things to the "simplest" terms for display -- saying $0$ is a lot cleaner than saying $1-1$ for example The complex numbers are a field This means that every non-$0$ element has a multiplicative inverse, and that inverse is unique While $1 i = i^ {-1}$ is true (pretty much by definition
  • Taylor series of $\ln (1+x)$? - Mathematics Stack Exchange
    Note that $$\frac {1} {1+x}=\sum_ {n \ge 0} (-1)^nx^n$$ Integrating both sides gives you \begin {align} \ln (1+x) =\sum_ {n \ge 0}\frac { (-1)^nx^ {n+1}} {n+1}\\ =x
  • Why is $1$ not a prime number? - Mathematics Stack Exchange
    If 1 is defined to be a prime number, then we'd have to change that theorem to: "every positive integer can be written uniquely as a product of primes, except for infinite multiplications by 1" So we choose to go with the easier path of defining 1 to not be a prime


















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