Problem 3 - Precalculus: Mathematics for Calculus

Question

(a) The complex conjugate of \(3+4 i\) is \(3+4 i=\) ______. (b) \((3+4 i)(\overline{3+4 i})=\) ______.

Step-by-Step Solution

Answer

(a) Complex conjugate is 3 - 4i. (b) Product is 25.

Step 1: Find the Complex Conjugate

To find the complex conjugate of a complex number, change the sign of the imaginary part. For the complex number \(3 + 4i\), its conjugate is \(3 - 4i\).

Step 2: Write Down the Complex Conjugate

From Step 1, we found that the complex conjugate of \(3 + 4i\) is \(3 - 4i\).

Step 3: Multiply the Complex Number and its Conjugate

To calculate \((3 + 4i)(\overline{3+4i})\), multiply \(3 + 4i\) by its conjugate \(3 - 4i\). Use the formula for multiplying binomials: \[(a + bi)(a - bi) = a^2 + b^2\].Here \(a = 3\) and \(b = 4\).

Step 4: Apply the Formula

Substitute \(a = 3\) and \(b = 4\) into the formula: \[a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25\].Thus, \((3 + 4i)(3 - 4i) = 25\).

Key Concepts

Complex Conjugate

Complex conjugates are a crucial concept in the realm of complex numbers. When dealing with a complex number, such as \(3 + 4i\), the complex conjugate is obtained by changing the sign of its imaginary component. Therefore, the complex conjugate of \(3 + 4i\) becomes \(3 - 4i\).
This idea is useful in simplifying expressions and solving equations involving complex numbers.

• The real part remains the same.
• The imaginary part changes its sign (from positive to negative or negative to positive).

Understanding how to find and use complex conjugates can be very helpful when you're working on problems that require eliminating imaginary parts, such as dividing complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers involves applying the distributive property similar to multiplying polynomials. Consider two complex numbers \( (a + bi) \) and \( (c + di) \). When multiplying these, you use the formula:
\[(a + bi)(c + di) = ac + adi + bci + bdi^2\]
Remember that \(i^2 = -1\). It simplifies into \((ac - bd) + (ad + bc)i\).
For the specific case of a complex number and its conjugate, such as \((3 + 4i)(3 - 4i)\), the multiplication results in a real number because the imaginary parts cancel out:

• Multiply as per the pattern \((a + bi)(a - bi)\).
• This results in \(a^2 + b^2\), eliminating the imaginary unit \(i\).
• In our example, it simplifies to \(9 + 16 = 25\).

Multiplication by the conjugate is a method to achieve real results from complex expressions.

Imaginary Numbers

Imaginary numbers form the backbone of complex numbers. The unit of imaginary numbers is represented as \(i\), where \(i^2 = -1\). Essentially, \(i\) is defined as the square root of negative one, which does not yield a real number. Instead, it represents the idea that there are solutions beyond the real number line for certain equations.

• Imaginary numbers are paired with real numbers to form complex numbers (e.g., \(3 + 4i\)).
• They are used to represent numbers that do not exist on the traditional real number line.
• The development of \(i\) allowed for a broader understanding of equations, particularly those that require solutions involving square roots of negative numbers.

Imaginary numbers open up a new dimension in mathematics which is incredibly valuable in fields like electrical engineering and physics.