What is i? If i=−1 is i imaginary? Is it used or considered often in mathematics? How is it notated?

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What is i?  If i=−1 is i imaginary?  Is it used or considered often in mathematics? How is it notated?

nick1337 · Accepted Answer

Step 1 Let z=(a+bi) be a complex number which is a square root of i, that is i=z2−(a2−b2)+2abi Equating real and imaginary parts we have, a2−b2=0 2ab=1 The ywo real solutions to this pair of equations are a=12 b=12 and a=−12 b=−12. The two square roots of i therefore are ±12(1+i)

Vasquez · Accepted Answer

Step 1 More generally, if you want to compute all the n-th roots of a complex number z0, that is, all the complex numbers z such that 1) zn=z0 you should write this equation in exponential form: z=reiθ, z0=r0eiθ0. Then (1) becomes (reiθ)n=r0eiθ⇔rneinθ=r0eiθ0 Now, if you have two complex numbers in polar coordinates which are equal, their moduluses must be equal clearly: rn=r0⇒r=+r0n since r, r0≥0 As for the arguments, we cannot simply conclude that nθ=θ0, but just that they differ in an integer multiple of 2π nθ=θ0+2kπ⇔θ=θ0+2kπn for k=0, ±1, ±2,⋯ It would seem that we have an infinite number of n-th roots, but we have enough with k=0, 1,⋯,n−1, since for instance for k=0 and k=n we obtain the same complex numbers. Thus, finally r0eiθ0n=+r0neiθ0++2kπn k=0, 1, ⋯, n−1 are all the complex n-th roots of z0 Step 2 1) For n=2, we obtain that every complex number has exactly two square roots: z0=+r0eiθ0+2kπ2, k=0, 1 =+r0eiθ02 and +r0ei(θ02+π) For instance, since i=eiπ2, we obtain i=(eiπ4=cos⁡π4+isin⁡π4=22+i22),(ei(π4+π)=cos⁡5π4+isin⁡5π4=−22−i22): Also, if z0=−1=eiπ, −1=eiπ2=i and ei(π2+π)=ei3π2=−i 2) For z0=1=et×0 and any n, we obtain the n-th roots of unity: 1n=ei2kπn k=0, 1, ⋯, n−1 For instance, if n=2, we get 1=ei×0=1 adn eiπ=−1 And for n=4 14=ei2kπ4, k=0, 1, 2, 3, that is, 14=1, i−1, −i

alenahelenash · Accepted Answer

Step 1 i12=(eπi2)12=eπi4 eπi4=cos⁡(π4)+isin⁡(π4) or simplified, 1+i2 This is of course the principal value; the other value (thanks Matt E!) is the negative square root, −1+i2 or in exponential form, −eπi4=e−3πi4

What is \sqrt{i}? If i=\sqrt{-1} is \sqrt{i} imaginary? Is it used or

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