Ch. 8.3 Polar Form of Complex Numbers and DeMoivre's Theorem

This paragraph introduces the concept of polar form for complex numbers and Mobius's theorem. It explains that complex numbers can be represented in a polar coordinate system, where every complex number 'z' can be expressed as 'a + bi', with 'a' and 'b' being real numbers. The real part 'Re(z)' is 'a', and the imaginary part 'Im(z)' is 'b'. The modulus of 'z', which is the distance from the origin to the point 'a, b', and the argument of 'z', which is the angle formed with the positive real axis, are key to plotting complex numbers in the complex plane. The argument can be found using the arctan function of the ratio of 'b' to 'a'. The paragraph sets the stage for further exploration of complex numbers in polar form and their relationship with trigonometry.

The second paragraph delves into the transformation of complex numbers from their standard form to polar form. It outlines the process of converting 'a + bi' to 'r * (cos(θ) + i * sin(θ))', where 'r' is the modulus and 'θ' is the argument. The modulus is calculated as the square root of 'a^2 + b^2', and 'θ' is found using the arctan(b/a). The paragraph provides an example of converting the complex number '1 - √3i' into its polar form, demonstrating the calculation of the modulus and the argument, and then expressing the number in terms of cosine and sine of the argument. It also explains how to convert from polar form back to the standard complex form, illustrating the process with an example.

This paragraph discusses the multiplication and division of complex numbers using their polar forms. It explains that when multiplying two complex numbers in polar form, the result is obtained by multiplying their moduli and adding their arguments. For division, the modulus of the divisor is divided into the modulus of the dividend, and the argument of the dividend is subtracted from the argument of the divisor. The paragraph emphasizes the simplicity of these operations in polar form compared to the standard form, especially when dealing with powers and roots of complex numbers, as introduced by De Moivre's theorem.

The fourth paragraph introduces De Moivre's theorem, which provides a formula for finding the powers of a complex number in polar form. It states that raising a complex number to a power involves raising its modulus to that power and multiplying its argument by the exponent. The paragraph illustrates how this theorem simplifies the process of finding powers of complex numbers, as opposed to the more tedious method of repeated multiplication in standard form. It also touches on the concept of roots of complex numbers, which are related to De Moivre's theorem and involve finding values that, when raised to the nth power, yield the original complex number.

The final paragraph focuses on the process of finding the roots of complex numbers, specifically cube roots in this context. It explains that for a given complex number in polar form, there are 'n' distinct roots, each with a modulus of the original number's modulus to the 1/n power and an argument that is the original argument plus a multiple of 2π/n. The paragraph provides a step-by-step guide to finding the cube roots of the complex number '4 + 4√3i', calculating the modulus, determining the original argument, and then finding the three distinct roots by varying the multiple of 2π/n. It concludes by noting that the roots will lie on a circle with a radius equal to the nth root of the original number's modulus, and they will be evenly spaced around this circle.