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Firma Nyheter:
- power series - How can we know the answer to 1-1+1-1+1 . . .
Summing to infinity cannot truly be evaluated as a sum as it is a process of addition that never ends, rather than a final result What we can say fairly categorically is that the only acceptable numerical answers are $1, 0,$ and $\frac{1}{2}$
- False Proof of 1=-1 - Mathematics Stack Exchange
1 $\begingroup$ Indeed what you are proving is that in the complex numbers you don't have (in general) $$\sqrt{xy}=\sqrt{x}\sqrt{y}$$ Because you find a counterexample
- arithmetic - Formal proof for $(-1) \times (-1) = 1$ - Mathematics . . .
The Law of Signs $\rm\: (-x)(-y) = xy\:$ isn't normally assumed as an axiom Rather, it is derived as a consequence of more fundamental Ring axioms $ $ [esp the distributive law $\rm\,x(y+z) = xy + xz\,$], laws which abstract the common algebraic structure shared by familiar number systems
- What does $QAQ^{-1}$ actually mean? - Mathematics Stack Exchange
Let me provide some context I was specifically looking at the application of linear algebra to the stress tensor $\sigma$ and how we get the transformed stress state in some rotated coordinate system by the following multiplication: $\sigma'=Q\sigma Q^{-1}(Q$ is a rotation matrix defined by the angle we want to rotate the axes)
- abstract algebra - Prove that 1+1=2 - Mathematics Stack Exchange
The main reason that it takes so long to get to $1+1=2$ is that Principia Mathematica starts from almost nothing, and works its way up in very tiny, incremental steps The work of G Peano shows that it's not hard to produce a useful set of axioms that can prove 1+1=2 much more easily than Whitehead and Russell do
- sequences and series - Why does this pattern work: $1 \cdot{1} = 1, 11 . . .
well, for 1^2 it isn't very hard to figure out, but here is the basic explanation there is one of 11, 111, 1111, etc in each place value until there have been as many place values as there are digits of the original number (1,111; 11,110; 111,100; and 1,111,000 for example) the amount of digits in the product is equal to (nx2)-1, where n is the original number then, it is like a bell curve
- What is the value of $1^i$? - Mathematics Stack Exchange
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
- Double induction example: $ 1 + q + q^2 - Mathematics Stack Exchange
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- 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
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