How do you solve a complex number to the power of a complex number?

Now to get to the power of a complex number: z^n = (x + iy)^n in mod-arg form you just apply deMoivre’s theorem: The modulus is raised to the power and the argument is multiplied by the power. Now this only works for real powers. z = r [ cos(o) + i sin(o) ] where r is the modulus and o is the argument.

How does de moivre’s theorem work?

De Moivre’s theorem gives a formula for computing powers of complex numbers. We first gain some intuition for de Moivre’s theorem by considering what happens when we multiply a complex number by itself. This shows that by squaring a complex number, the absolute value is squared and the argument is multiplied by 2.

What is de Moivre’s theorem?

De Moivre’s Theorem. De Moivre’s theorem gives a formula for computing powers of complex numbers. We first gain some intuition for de Moivre’s theorem by considering what happens when we multiply a complex number by itself. Recall that using the polar form, any complex number \\(z=a+ib\\) can be represented as \\(z = r…

What did Abraham de Moivre do for math?

Abraham De Moivre (1667 – 1754) further extended the study of such numbers when he published Miscellanea Analytica in 1730, utilizing trigonometry to represent powers of complex numbers. His work is the subject of the mathematical portion of this paper, and his life is described in more detail in the next section.

How did Gauss prove the fundamental theorem of algebra?

In 1799, as part of his dissertation, Gauss relied on this knowledge to prove that any polynomial with real coefficients could be written as the product of linear or quadratic factors. Any such polynomial would then have a solution contained in the set of complex numbers (Mazur, 207). This we now know as the Fundamental Theorem of Algebra.

What is Wessel’s visual interpretation of complex numbers?

4 The Norwegian surveyor Caspar Wessel presented his visual interpretation of complex numbers to the Royal Danish Academy of Science in 1797. Wessel described a complex number a + bi,as point (a, b) on a plane consisting of a real axis and an imaginary axis (Nahin, 1998).