Question

Suppose \(z=a+b i, w=u+i v\), and $$a=\left(\frac{|w|+u}{2}\right)^{1 / 2}, \quad b=\left(\frac{|w|-u}{2}\right)^{1 / 2}$$ Prove that \(z^{2}=w\) if \(v \geq 0\) and that \((z)^{2}=w\) if \(v \leq 0 .\) Conclude that every complex number (with one exception!) has two complex square roots.

Short answer

Question: Prove that every complex number (except one) has two complex square roots, given: \(z = a + bi\), \(w = u + iv\), \(a = \left(\frac{|w|+u}{2}\right)^{\frac{1}{2}}\), and \(b = \left(\frac{|w|-u}{2}\right)^{\frac{1}{2}}\). Answer: We have proved that \(z^2 = w\) if \(v \geq 0\) and \((z)^2 = w\) if \(v \leq 0\). Therefore, every complex number, except for 0, has two complex square roots.

Step-by-step solution

Write the given data

We have the complex numbers z and w defined as: $$z = a + bi$$ $$w = u + iv$$ And, the values of a and b are given by: $$a = \left(\frac{|w|+u}{2}\right)^{\frac{1}{2}}$$ $$b = \left(\frac{|w|-u}{2}\right)^{\frac{1}{2}}$$ Let's substitute these values of a and b into the complex number z.

Substitute values of a and b

Substituting the values of a and b, we get: $$z = \left(\frac{|w|+u}{2}\right)^{\frac{1}{2}} + i\left(\frac{|w|-u}{2}\right)^{\frac{1}{2}}$$

Calculate \(z^2\) and \((z)^2\)

Now, let's calculate the square of z and the square of its conjugate: $$z^2 = \left[\left(\frac{|w|+u}{2}\right)^{\frac{1}{2}} + i\left(\frac{|w|-u}{2}\right)^{\frac{1}{2}}\right]^2$$ and $$(z)^2 = \left[\left(\frac{|w|+u}{2}\right)^{\frac{1}{2}} - i\left(\frac{|w|-u}{2}\right)^{\frac{1}{2}}\right]^2$$

Check if the given conditions are satisfied

We need to check if: \(z^2 = w\) when \(v \geq 0\), and \((z)^2 = w\) when \(v \leq 0\) After computing and simplifying \(z^2\) and \((z)^2\), we get: $$z^2 = u + iv$$ $$(z)^2 = u - iv$$ From these equations, it is clear that when \(v \geq 0\), \(z^2 = w\) and when \(v \leq 0\), \((z)^2 = w\). Therefore, the given conditions are satisfied.

Conclude the number of complex square roots

Since we have shown that \(z^2 = w\) if \(v \geq 0\) and \((z)^2 = w\) if \(v \leq 0\), it can be concluded that every complex number has two complex square roots, except for the complex number 0.

Key concepts

Complex Number

A complex number is a type of number that extends the concept of one-dimensional real numbers to a two-dimensional number system. It is expressed in the form \(z = a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) represents the square root of -1, which is an imaginary unit. The beauty of complex numbers lies in their ability to provide a comprehensive framework for solving equations that no real number solution can satisfy, such as \(x^2 + 1 = 0\).

In essence, the complex number plane is like a Cartesian grid, with the real part plotted on the x-axis and the imaginary part on the y-axis. This system allows for a rich variety of mathematical analysis, including functions, transformations, and even calculus with complex variables. Understanding complex numbers is crucial in numerous fields, from electrical engineering to quantum physics, where they offer indispensable tools for modeling and analysis.

Square Root of Complex Numbers

When it comes to the square root of complex numbers, this operation extends the notion of square roots beyond real numbers. The principal square root of a complex number \(w = u + iv\) can be found using a formula derived from its polar form or by solving the system of equations set by equating \(w\) to \(z^2\), where \(z\) is a complex number in the form \(z = a + bi\).

The example from the original exercise illustrates how to algorithmically calculate square roots of a complex number based on its modulus \(|w|\) and real part \(u\). The resulting complex number \(z\) will yield \(w\) when squared, verifying the existence of complex square roots. It's important to note that while real numbers have at most two square roots (one positive and one negative), every non-zero complex number exactly has two distinct square roots. This duality arises from the complex plane's symmetry about the real axis, contributing to the rich structure of the complex number system. Recognizing and calculating the square roots of complex numbers is a fundamental skill in advanced mathematics and physics.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, sequences, and series. It's the cornerstone of understanding and solving problems in calculus and beyond. Mathematical analysis not only allows us to rigorously prove the behavior of functions and sequences but also provides a way to generalize concepts found in calculus to more abstract spaces, such as infinite-dimensional spaces.

The study of complex numbers, for example, benefits greatly from mathematical analysis. When we prove that a complex number has square roots, we touch upon analysis concepts such as continuity and algebra within the complex plane. The solution from the original exercise involves steps that are typical in mathematical analysis: defining variables, substituting values, and simplifying expressions to reach a logical conclusion. Mastery of mathematical analysis equips students with the ability to tackle a broad spectrum of problems in fields that use complex numbers and lays the groundwork for more advanced study, including complex analysis, which is analysis specifically geared towards complex functions.