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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 0 is a lot cleaner than saying 1−1 1 1 for example The complex numbers are a field This means that every non- 0 0 element has a multiplicative inverse, and that inverse is unique While 1 i = i−1 1 i = i 1 is true (pretty much
What is the value of $1^i$? - Mathematics Stack Exchange First, a concrete example of things that can happen with complex exponentiation if you aren't careful: $1 = e^ {2\pi i}$, so we can naively try to compute $1^i = (e^ {2\pi i})^i = e^ { (2\pi i)i} = e^ {-2\pi}$ The formal moral of that example is that the value of $1^i$ depends on the branch of the complex logarithm that you use to compute the power You may already know that $1=e^ {0+2ki\pi
What is $\sqrt {i}$? - Mathematics Stack Exchange The square root of i is (1 + i) sqrt (2) [Try it out my multiplying it by itself ] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers
complex analysis - What is $0^ {i}$? - Mathematics Stack Exchange It is possible to interpret such expressions in many ways that can make sense The question is, what properties do we want such an interpretation to have? $0^i = 0$ is a good choice, and maybe the only choice that makes concrete sense, since it follows the convention $0^x = 0$ On the other hand, $0^ {-1} = 0$ is clearly false (well, almost —see the discussion on goblin's answer), and $0^0=0
Is i lt;0? | Brilliant Math Science Wiki i i is a value on the complex plane It can be thought of as either a counterclockwise or a clockwise rotation of 90 degrees ((or π 2 2π radians)) from the real number line Note that performing the same rotation twice will get 1 −1 While we can thus think of i i defined as the value such that i 2 = 1, i2 = −1, we have to be careful: while the counterclockwise and clockwise directions
Does $i^4$ equal $1?$ - Mathematics Stack Exchange I can't seem to find a solution to this for the life of me My mathematics teacher didn't know either Edit: I asked the teacher that usually teaches my course today, and she said it was incredibl
Sum of n, n², or n³ | Brilliant Math Science Wiki The series ∑ k = 1 n k a = 1 a + 2 a + 3 a + + n a k=1∑n ka = 1a +2a +3a +⋯+na gives the sum of the a th ath powers of the first n n positive numbers, where a a and n n are positive integers Each of these series can be calculated through a closed-form formula The case a = 1, n = 100 a = 1,n = 100 is famously said to have been solved by Gauss as a young schoolboy: given the tedious task