Problem 56 - Precalculus with Calculus Previews

Question

A square root of a complex number \(w=c+d i\) is defined to be a complex number \(z=a+b i\) that satisfies the polynomial equation \(z^{2}=c+d i\). Find two square roots of \(i\)

Step-by-Step Solution

Answer

The two square roots of \(i\) are \(z = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i\) and \(z = -\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}i\).

Step 1: Set up the equation

We start by setting up the equation for the square root, where the complex number we want to find the square roots for is \( w = i \). Assume \( z = a + b i \) is a square root of \( i \). Then, \( z^2 = i \) translates to: \[(a + b i)^2 = i.\] This equation implies \( a^2 - b^2 + 2ab i = 0 + 1i \).

Step 2: Equate real and imaginary parts

From \((a^2 - b^2) + 2ab i = 0 + 1i\), equate the real and the imaginary components separately. For the real parts: \[a^2 - b^2 = 0,\] For the imaginary parts: \[2ab = 1.\]

Step 3: Solve the real part equation

The equation \(a^2 - b^2 = 0\) implies \(a^2 = b^2\). Therefore, \(a = b\) or \(a = -b\).

Step 4: Solve the imaginary part equation

Substitute \(a = b\) into \(2ab = 1\): \[2a^2 = 1 \Rightarrow a^2 = \frac{1}{2} \Rightarrow a = \pm \frac{1}{\sqrt{2}}.\] Therefore, the solutions are \(a = \frac{1}{\sqrt{2}}, b = \frac{1}{\sqrt{2}}\) and \(a = -\frac{1}{\sqrt{2}}, b = -\frac{1}{\sqrt{2}}\).

Step 5: Determine both solutions

Now, substitute \(a = -b\) back into \(2ab = 1\) if the prior solutions did not cover all. It turns out the equations still hold for both solutions found. The two square roots are: \( z = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i \) and \( z = -\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}i \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They are written in the form \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

These numbers are essential in mathematics because they allow for the extension of real numbers into the complex plane, which can be particularly useful when solving polynomial equations that have no real solutions.

• Real Part: This is the \(a\) in \(a + bi\), which can be any real number.
• Imaginary Part: This is the \(b\), multiplied by \(i\), the imaginary unit.
• Imaginary Unit: \(i\) is defined as \(\sqrt{-1}\), which allows us to handle the square roots of negative numbers.

Understanding complex numbers is crucial when trying to find roots or solutions in equations where real numbers are not sufficient.

Imaginary Numbers

Imaginary numbers, as part of the complex number system, add an interesting dimension to mathematics. They are characterized by the inclusion of the imaginary unit \(i\), where \(i = \sqrt{-1}\). Imaginary numbers allow mathematicians to perform operations on numbers that would otherwise be undefined, such as the square root of a negative number.

The consideration of imaginary numbers is fundamental when working with equations like \(z^2 = c + di\) because they expand the realm of solutions. In the case where many equations have no solutions among real numbers, imaginary numbers enable the discovery of complex solutions.

• Imaginary Unit \(i\): It is used to express the square root of negative numbers.
• Complex Solutions: Imaginary numbers help in finding solutions to equations that cannot be solved using only real numbers.

This concept is not only theoretical but also practical, finding applications in engineering and physics.

Solving Polynomial Equations

Solving polynomial equations involving complex numbers requires equating and working with both real and imaginary parts. These equations often have the form \(z^2 = c + di\), which can be rewritten using a complex number \(z = a + bi\). Solving these equations involves multiple steps to find the values of \(a\) and \(b\) that satisfy the original equation.

The solution process includes:

• Expanding \((a + bi)^2\) to separate real and imaginary components.
• Setting real parts and imaginary parts equal to each other.
• Solving these two resulting equations separately.

This approach is systematic and ensures that both components of the complex number are considered in finding potential solutions.

Equating Real and Imaginary Parts

A crucial step when working with equations involving complex numbers is to equate the real and imaginary parts separately. This means breaking down the expression \((a + bi)^2 = c + di\) into two distinct equations:

• Real Part: \(a^2 - b^2 = c\)
• Imaginary Part: \(2ab = d\)

By equating these parts, you isolate two separate equations that can be solved independently.
Once the equations for the real and imaginary parts are written out, you can solve for the individual values of \(a\) and \(b\). This methodical approach often involves algebraic techniques such as substitution and factorization, leading to the discovery of both known and complex solutions. This ensures that every possible solution is accounted for in the final answer.