Problem 4 - Precalculus with Limits

Question

The complex number \(u=a+b i\) is an __________ ___________ of the complex number \(z\) when \(z=u^{n}=(a+b i)^{n}\)

Step-by-Step Solution

Answer

\(u=a+b i\) is the nth root of \(z\), where \(n\) is any positive integer.

Step 1: Understand the question

The problem gives us a complex number \(z\), which is equal to \(u^{n}\), implying that \(u\) is an nth root of \(z\). The task here is to understand what \(u\) referred to in this context.

Step 2: Understand complex roots

In the complex number system, if \(z = u^n\), then we say that \(u\) is an nth root of \(z\). This is similar to square roots in real numbers, but here \(n\) can be any positive integer.

Step 3: Identify the root

Given that \(z = u^n\), from the definition of nth roots of a complex number, we can see that \(u=a+b i\) is the nth root of the complex number \(z\).

Key Concepts

Complex Numbers

Complex numbers are a fundamental component of mathematics. They extend the idea of the one-dimensional number line to the two-dimensional complex plane. A complex number is often represented in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by \(i^2 = -1\). Here, \(a\) is called the real part and \(b\) is called the imaginary part.

• Complex numbers allow for solutions to equations that do not have solutions in the set of real numbers, such as \(x^2 + 1 = 0\).
• They are represented in the complex plane by a point \((a, b)\).
• Complex numbers can also be expressed in polar form, where they are represented by a magnitude and an angle.

Understanding complex numbers is essential for many fields, including physics, engineering, and applied mathematics. They provide a way to describe oscillations, waves, and other phenomena that involve periodic motion.

nth Roots

Finding the nth roots of a complex number involves extending the concept of roots to the complex plane. If \(z = u^n\), then \(u\) is said to be an nth root of \(z\). Here, \(n\) is any positive integer, and the nth roots can be quite numerous.

• Unlike real numbers, a complex number has \(n\) distinct nth roots for a given \(n\).
• These roots are represented as points that are equally spaced along a circle in the complex plane.
• The general form for nth roots incorporates Euler's formula: \(u = |z|^{1/n} \cdot (\cos((\theta + 2k\pi)/n) + i \sin((\theta + 2k\pi)/n))\), where \(k = 0, 1, 2, ..., n-1\).

Understanding nth roots of complex numbers helps in solving polynomial equations and in performing operations like frequency domain analysis in signals.

Imaginary Numbers

Imaginary numbers form the building blocks of complex numbers. They arise from the need to provide solutions to equations like \(x^2 + 1 = 0\), which have no real solutions since no real number squared gives a negative number. The imaginary unit \(i\) is defined by \(i^2 = -1\).

• Imaginary numbers take the form \(bi\), where \(b\) is a real number.
• When combined with real numbers, they form complex numbers, which can be used to perform algebraic operations.
• They provide a necessary extension to the real number system, allowing for the operation of square roots on negative numbers.

Imaginary numbers are not purely theoretical; they have practical applications in engineering and physics, particularly in electrical engineering for analyzing AC circuits.