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Chapter 1: Problem 48

Find the product of the given complex member and its conjugate. $$4+3 i$$

Short Answer

Expert verified
25

Step by step solution

01

Identify the given complex number

Given complex number is: \(4 + 3i\)
02

Find the conjugate of the complex number

The conjugate of a complex number \(a + bi\) is \(a - bi\). Hence, the conjugate of \(4 + 3i\) is \(4 - 3i\).
03

Set up the product of the complex number and its conjugate

To find the product, multiply the complex number by its conjugate: \((4 + 3i)(4 - 3i)\).
04

Apply the difference of squares formula

Use the identity \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 4\) and \(b = 3i\) , so it becomes \((4)^2 - (3i)^2\).
05

Simplify the expression

Calculate \((4)^2 - (3i)^2\). This results in \(16 - 9i^2\). Since \(i^2 = -1\), it simplifies to \(16 - 9(-1)\).
06

Calculate the final result

Simplify \(16 + 9\) to get \(25\).

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Key Concepts

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

Complex Conjugate
A complex conjugate is very useful in various calculations involving complex numbers. The complex conjugate of a number in the form of \(a + bi\) is \(a - bi\). Here, \(a\) and \(b\) are real numbers, while \(i\) is the imaginary unit. The conjugate essentially flips the sign of the imaginary part.
Why do this? The main reason is because multiplying a complex number by its conjugate eliminates the imaginary part. For example, if we have \(4 + 3i\) and multiply it by its conjugate \(4 - 3i\), the imaginary section disappears:
  • Given complex number: \(4+3i\)
  • Conjugate: \(4-3i\)
  • Product: \((4+3i)(4-3i)\)
Understanding conjugates is a key step in simplifying and solving more complex problems involving complex numbers.
Difference of Squares
The difference of squares formula is a useful algebraic identity applied often in mathematics. It states that \( (a + b)(a - b) = a^2 - b^2 \). When dealing with complex numbers, this identity is especially handy when you multiply a number by its conjugate. For the complex number \(4+3i\):
  • First term, \(a = 4\)
  • Second term, \(b = 3i\)
  • Applying the identity: \( (4+3i)(4-3i) = 4^2 - (3i)^2 \)
This simplifies to summarise the complex and imaginary parts, leading to a real number final result.
Imaginary Unit
The imaginary unit, represented by \(i\), is defined as the square root of -1. It plays a crucial role in expressing complex numbers. The powers of \(i\) repeat in a cycle:
  • \(i^1 = i \)
  • \(i^2 = -1 \)
  • \(i^3 = -i \)
  • \(i^4 = 1 \)
Understanding how \(i\) behaves is essential when simplifying expressions involving complex numbers. For instance, in the expression \((4+3i)(4-3i) = 4^2 - (3i)^2\), knowing that \((3i)^2 = 9i^2 = 9(-1) = -9\) allows us to simplify it further, resulting in a real number as the original imaginary part cancels out.

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