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Complex Numbers

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Complex numbers, comprising a real and an imaginary part, are fundamental in mathematics and engineering. This overview covers their square roots, representation in polar form, and multiplication. It delves into De Moivre's Theorem for computing powers and roots, and practical applications, including solving complex equations and finding nth roots of unity. Examples illustrate the process of finding roots of complex numbers and their conversion between forms.

Introduction to Complex Numbers

Complex numbers are integral to various fields of mathematics and engineering, consisting of a real part and an imaginary part. They are expressed in the form a+bi, where 'a' is the real component, 'b' is the imaginary coefficient, and 'i' is the imaginary unit, satisfying the equation i² = -1. This system enables the solution of equations that have no real solutions, such as those involving the square root of a negative number. For example, the square root of -16 is represented as 4i, where 'i' is the imaginary unit.
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Square Roots of Complex Numbers

To find the square root of a complex number, one must identify a pair (a, b) such that (a+bi)² equals the original complex number. For instance, to determine the square root of 3+4i, equate (a+bi)² to 3+4i and solve for 'a' and 'b'. This involves expanding the left side to a² - b² + 2abi and matching the real and imaginary parts with those of 3+4i, resulting in a system of equations. Solving this system yields the values of 'a' and 'b' that give the square root of the complex number.

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00

In various ______ and engineering, complex numbers are used, which include both a real and an ______ part.

fields of mathematics

imaginary

01

Equation for expanding (a+bi)²

Expand to a² - b² + 2abi, where a and b are real numbers, i is the imaginary unit.

02

Solving system for square root of 3+4i

Match real parts a² - b² to 3, and imaginary parts 2ab to 4, then solve for a and b.

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