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Chapter 6: Problem 41

Find the product of the given complex number and its conjugate. $$ i $$

Short Answer

Expert verified
The product of \( i \) and its conjugate is \ 1\.

Step by step solution

01

Understand the Problem

The problem asks to find the product of a given complex number and its conjugate. The given complex number is represented by the imaginary unit, \( i \).
02

Write the Conjugate of the Complex Number

The complex number given is \( i \). The conjugate of \( i \) is \( -i \).
03

Multiply the Complex Number by Its Conjugate

To find the product, multiply \( i \) by its conjugate \( -i \): \( i \cdot \left ( -i \right )\ = \left ( i \right ) \left ( -i \) = -i^{2}.\
04

Simplify Using the Property of \( i \)

Recall that \( i \) is the imaginary unit and \( i^{2} = -1 \). Therefore, \ (-i^2 \ =\ -(-1) = 1\). So, \( i \cdot \left ( -i \right ) = 1.\
05

Conclude the Result

After simplifying, we find that the product of \( i \) and its conjugate is \ 1\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Unit
In mathematics, the imaginary unit is represented by the symbol \( i \). It is a fundamental element of complex numbers. The imaginary unit is defined as the square root of \( -1 \). This means:
\[ i = \sqrt{-1} \].
The introduction of the imaginary unit allows us to extend the real numbers to complex numbers, which have the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit. Without \( i \), we cannot solve equations like \( x^2 + 1 = 0 \). Complex numbers are essential in various fields, including engineering, physics, and applied mathematics.
Conjugate
The conjugate of a complex number is obtained by changing the sign of the imaginary part. For a complex number \( a + bi \), the conjugate is \( a - bi \).
This operation is useful in many mathematical applications, including simplifying fractions and finding magnitudes. For example, for the imaginary unit \( i \), which can be written as \( 0 + i \), its conjugate is \( 0 - i \) or simply \( -i \).
Conjugates help in various operations with complex numbers, such as multiplication, where they can cancel out the imaginary parts and result in a real number.
Multiplication of Complex Numbers
Multiplying complex numbers follows the distributive property, just like with real numbers. When you multiply two complex numbers \( (a + bi) \) and \( (c + di) \), you distribute each term:
\[ (a + bi) \cdot (c + di) = ac + adi + bci + bdi^2 \].
Remember that \( i^2 = -1 \), so this simplifies to:
\[ ac + adi + bci - bd \].
Combining like terms, we get:
\[ (ac - bd) + (ad + bc)i \].
This process is particularly useful when multiplying a complex number by its conjugate, resulting in a real number.
i squared property
One of the most critical properties of the imaginary unit \( i \) is that \( i^2 = -1 \). This fundamental property underpins many operations involving complex numbers. For instance, in our exercise, when we multiplied \( i \) by its conjugate \( -i \), we used this property to simplify the expression:
\( i \cdot (-i) = -i^2 = -(-1) = 1 \).
Understanding this property is essential for working with complex numbers, as it helps convert what seems like a complex expression involving \( i \) into a simpler real number expression. This property makes it easier to handle algebraic operations on complex numbers and keeps the calculations manageable.

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Most popular questions from this chapter

For each polar equation, write an equivalent rectangular equation. $$ r=\frac{3}{\sin \theta} $$

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