Why is $1/i$ equal to $-i$? - Mathematics Stack Exchange
网页2015年5月11日 · 1 = ei0 i = eiπ 2. Now we perform the division making use of the exponent rules: 1 i = ei0 eiπ 2 = e − iπ 2. If you consult the unit circle (since the magnitude is 1), you will find that a phase of − π 2 corresponds to − i. Alternatively you can apply Euler's formula: e − iπ 2 = cos(− π 2) + isin(− π 2) = − i. Share.
arithmetic - Formal proof for $(-1) \times (-1) = 1$ - Mathematics ...
网页My point is: if one formalizes this proof, then it's really not shorter (or much different) than other methods. Continuing, you need to use the distributive law x + x(− 1) = x(1 + (− 1)) = x ⋅ 0 = 0, etc, just as in the other proofs. At a high level, all of the proofs are equivalent, if boiled down to the raw ring axioms.
abstract algebra - Prove that 1+1=2 - Mathematics Stack Exchange
网页2013年1月15日 · Here is a relevant excerpt: As you can see, it ends with "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." The theorem above, ∗54 ⋅ 43 ∗ 54 ⋅ 43, is already a couple of hundred pages into the book (Wikipedia says 370 or so) I wrote a blog article a few years ago …
Proof by induction, 1 · 1! + 2 · 2! + ... + n · n! = (n + 1)! − 1
网页2016年4月17日 · Bernard's answer highlights the key algebraic step, but I thought I might mention something that I have found useful when dealing with induction problems: whenever you have an induction problem like this that involves a …
What does $QAQ^{-1}$ actually mean? - Mathematics Stack …
网页2020年4月28日 · 1. Let's say we have the matrix acting on an arbitrary vector. w = Av w = A v. where v ↦ Av v ↦ A v is a linear transformation T: Rn → Rn T: R n → R n. If we want to "rotate" the matrix, we have to check and see what kinds of rules such a transformation to follow. The relationship we want to hold is. w′ =A′v′ w ′ = A ′ v ′.
Series expansion: $\frac{1/}{(1-x)^n}$ - Mathematics Stack Exchange
网页Here is an (intuitive) proof that explains the coeffcients in ∞ ∑ k = 0(k + n − 1 n − 1)xk = 1 (1 − x)n. Since 1 1 − x = ∞ ∑ k = 0xk, we just need to figure out how to multiply it n times. To calculate the coefficient of a specific power, say xk, you only need the first k + 1 terms in each series. So we have, (1 + x + x2 + x3 ...
General term formula of series 1/1 + 1/2 + 1/3 ... +1/n
网页$$\ln(n+1)\le\sum_{i=1}^n\frac1i\le\ln(n)+1$$ This is a rather tight upper limit and lower limit you can use to approximate your answer. One could also note that $$\sum_{i=1}^n\frac1i=\int_0^1\sum_{i=0}^{n-1}x^i\ dx=\int_0^1\frac{1-x^n}{1-x}\ dx$$ We also have the Euler-Maclaurin expansion:
factorial - Why does 0! = 1? - Mathematics Stack Exchange
网页The theorem that (n k) = n! k!(n−k)! (n k) = n! k! (n − k)! already assumes 0! 0! is defined to be 1 1. Otherwise this would be restricted to 0 <k <n 0 <k <n. A reason that we do define 0! 0! to be 1 1 is so that we can cover those edge cases with the same formula, instead of having to treat them separately.
The sequence of integers $1, 11, 111, 1111, \ldots$ have two …
网页2016年5月9日 · Prove that the sequence $\\{1, 11, 111, 1111, .\\ldots\\}$ will contain two numbers whose difference is a multiple of $2017$. I have been computing some of the immediate multiples of $2017$ to see how
Formula for $1^2+2^2+3^2+...+n^2$ - Mathematics Stack …
网页It comes from that fact that the difference operator. Δf(n) = f(n + 1) − f(n) acting on polynomials f(x) of degree d will result in polynomials in degree d − 1 (check this!) - the difference between f(n) = 12 +22 + ⋯ +n2 and f(n + 1) =12 + ⋯ + (n + 1)2 is simply (n + 1)2, which is a quadratic in n, hence we should expect f to be cubic.