Problem 1 - Algebra and Trigonometry

Question

Answers are given at the end of these exercises. The conjugate of \(-4-3 i\) is _______.

Step-by-Step Solution

Answer

-4 + 3i

Step 1: Understand the Concept of a Conjugate

The conjugate of a complex number is obtained by changing the sign of the imaginary part. For a complex number of the form \(a + bi\), the conjugate is \(a - bi\).

Step 2: Identify the Given Complex Number

The given complex number is \(-4 - 3i\). Here, \(-4\) is the real part (\(a\)) and \(-3i\) is the imaginary part (\(bi\)).

Step 3: Change the Sign of the Imaginary Part

To find the conjugate, change the sign of the imaginary part \(-3i\) to \(+3i\). Therefore, the conjugate of \(-4 - 3i\) is \(-4 + 3i\).

Key Concepts

complex numbers

Complex numbers are numbers that consist of a real part and an imaginary part. The standard form of a complex number is written as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Here are a few important points about complex numbers:

• The real part \(a\) is a regular number, like 3, -2, or 0.
• The imaginary part \(bi\) includes the imaginary unit \(i\), which is defined as the square root of \(-1\).
• A complex number becomes purely imaginary if \(a = 0\), for example, \3i\.
• Similarly, it becomes purely real when \(b = 0\), like 4 or -2.

Complex numbers are crucial in many fields, including engineering, physics, and mathematics. They allow us to solve equations that have no real solutions.

imaginary part

The imaginary part of a complex number is the component that includes the imaginary unit \(i\). It is written as \(bi\) where \('b'\) is a real number, and \(i\) is \(\text{{√}}{-1}\). Here are some more details:

• \textbf{Invented by Mathematicians}: Imaginary numbers were developed by mathematicians to solve problems involving the square roots of negative numbers.
• \textbf{Notation}: In \(a + bi\), the imaginary part is \(bi\). For example, in \(-4 - 3i\), it is \(-3i\).
• \textbf{Operations}: You can add, subtract, multiply, and divide imaginary numbers just like you would with real numbers, but with a key rule: \(i^2 = -1\).

When working with complex numbers, it's important to treat the imaginary part separately from the real part, especially when performing arithmetic operations.

conjugate

The conjugate of a complex number is essential in various mathematical operations, such as simplifying fractions with complex numbers. To form the conjugate, you change the sign of the imaginary part of the complex number. Here’s how you do it:

• For a complex number \(a + bi\), the conjugate is \(a - bi\).
• In the original exercise, the given complex number is \(-4 - 3i\). Its conjugate is \(-4 + 3i\).
• This operation doesn't change the real part of the complex number, only the sign of the imaginary part.

Conjugates are crucial in simplifying expressions where complex numbers appear in the denominator. By multiplying by the conjugate, you eliminate the imaginary part from the denominator, making the fraction easier to work with.