Problem 38

Question

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$8 i$$

Step-by-Step Solution

Verified

Answer

The complex conjugate of $8i$ is $-8i$ and the product of the number $8i$ and its conjugate $-8i$ is $64$.

1Step 1: Find the complex conjugate

The complex number given is exclusively imaginary, $8i$, where the real part is zero. So, its complex conjugate will be the same number with the sign of imaginary part flipped. Hence the complex conjugate of $8i$ is $-8i$.

2Step 2: Multiply the complex number by its complex conjugate

To find the product of the complex number $8i$ and its conjugate $-8i$, simply multiply the two numbers together. Thus, $8i * -8i = -64i^2$. Considering that $i^2 = -1$, the result simplifies to $64$.

Key Concepts

Complex ConjugateMultiplication of Complex NumbersImaginary Unit

Complex Conjugate

A complex conjugate is conceptually like a mirror image in mathematics. It reflects a given complex number on the imaginary axis.
This is achieved by changing the sign of the imaginary part, resulting in a number that is geometrically opposite on the complex plane.

For example, the complex number $c = a + bi$ has a complex conjugate of $a - bi$.
In this exercise, we have the number $8i$, which is purely imaginary.

Since the real part is zero, we simply change $8i$ into $-8i$, the complex conjugate.
This property helps in various calculations, including multiplication and division of complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers might seem tricky at first. However, it's quite simple once you know the method.
You multiply complex numbers just like binomials by using the distributive property. Consider each component and carefully carry out the multiplication across terms:

When multiplying two complex numbers $ (a + bi)$ and $ (c + di)$, you calculate as follows:
\[ (a+bi)(c+di) = ac + adi + bci + bdi^2\]
The term $bdi^2$ simplifies since $i^2 = -1$, affecting the final real and imaginary components.

In the solution given: $8i \ imes -8i$ becomes $-64i^2$.
And since $i^2 = -1$, this important rule helps simplify the product to a real number, \64\. Multiplication of a complex number by its conjugate always yields a real number as a result.

Imaginary Unit

The imaginary unit $i$ is a fundamental concept in complex numbers.
It's defined as the square root of -1 and serves as the basis for constructing imaginary and hence complex numbers.

This property is expressed as $i^2 = -1$, which often helps in simplifications during calculations.
The imaginary unit allows representation of numbers that are not on the traditional number line.

A complex number is typically represented in the form $a + bi$, where $a$ and $b$ are real numbers, signifying real and imaginary components.
In the given problem, $8i$ is purely imaginary. The manipulations such as multiplying by conjugate or simplification through $i^2 = -1$ consistently hinge on the properties of this fascinating mathematical entity.

Problem 38

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