Why is $x^ {-1} = \frac {1} {x}$? - Mathematics Stack Exchange We know that Number (or Variable) * Its Inverse = 1 Similarly X* (Inverse of X) = 1 Implying Inverse of X = 1 x ; And by convention we assume Inverse of a number as $ {number}^ {-1}$ i e number raised to the power of minus 1;
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
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