- #1

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##f(x) = \sqrt{x}##

Refers to the principal root of any real number x.

Is it true that this is not the case when dealing with complex numbers? Does ##\sqrt{z}##, where ##z ∈ ℂ##, represent more than one value?

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- Thread starter PFuser1232
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In summary, the square root function is well-defined for real numbers, but not necessarily for complex numbers. When dealing with complex numbers, the principal square root may have more than one value depending on the definition of the nth root.

- #1

- 479

- 20

##f(x) = \sqrt{x}##

Refers to the principal root of any real number x.

Is it true that this is not the case when dealing with complex numbers? Does ##\sqrt{z}##, where ##z ∈ ℂ##, represent more than one value?

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- #2

- 149

- 15

If you define

more reading youll enjoy:

http://en.wikipedia.org/wiki/Root_of_unity

- #3

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http://en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers

Complex numbers are numbers that contain both a real and an imaginary component. They are written in the form a+bi, where a is the real part and bi is the imaginary part. Real numbers, on the other hand, only contain a single value and do not have an imaginary component.

The roots of complex numbers refer to the solutions of the equation z^n = a+bi, where n is a positive integer and a+bi is a complex number. In other words, they are the values of z that, when raised to the nth power, equal the given complex number.

To find the roots of a complex number, you can use the polar form of complex numbers. You can convert the complex number into polar form, then use the formula z^n = r^n(cos(nθ) + i sin(nθ)) to find the roots. You can also use De Moivre's theorem to find the roots.

Yes, complex numbers can have multiple roots. For example, the complex number 1+0i has two roots, 1 and -1, when raised to the power of 2. This is because the equation z^2 = 1+0i has multiple solutions.

Complex numbers have various applications in science and mathematics. They are used in electrical engineering for analyzing alternating current circuits, in quantum mechanics for describing wave functions, and in signal processing for analyzing signals with both real and imaginary components. They are also used in solving polynomial equations and in fractal geometry.

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