Problem 3 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) \(3-4i\); (b) 25

Step 1: Understanding the Complex Conjugate

The complex conjugate of a number of the form \( a + bi \) is given by \( a - bi \). Here, \( a = 3 \) and \( b = 4 \). Therefore, the complex conjugate of \( 3 + 4i \) is \( 3 - 4i \).

Step 2: Setting Up the Multiplication

To find \((3 + 4i)(\overline{3 + 4i})\), substitute the complex conjugate we found in Step 1 into the expression. This gives us \((3 + 4i)(3 - 4i)\).

Step 3: Applying the Difference of Squares

Recognize that \((a + bi)(a - bi)\) is a difference of squares: \(a^2 - (bi)^2\). Here \(a = 3\) and \(b = 4\), so the expression becomes \(3^2 - (4i)^2\).

Step 4: Calculating Each Term

Calculate \(3^2 = 9\) and \((4i)^2 = 16i^2\). Since \(i^2 = -1\), \(16i^2 = 16(-1) = -16\).

Step 5: Completing the Multiplication

Substitute back into the equation: \((3+4i)(3-4i) = 9 - (-16) = 9 + 16 = 25\). The result of the multiplication is 25.

Key Concepts

Complex Conjugate

The complex conjugate is an important concept in dealing with complex numbers. A complex number generally takes the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The complex conjugate of this number is simply \( a - bi \).
This operation essentially flips the sign of the imaginary component. For example, the complex conjugate of \( 3 + 4i \) changes the imaginary part, resulting in \( 3 - 4i \).
Finding complex conjugates is useful in various scenarios, including simplifying complex number calculations and solving equations involving complex numbers.

Difference of Squares

The difference of squares is a mathematical identity that states \( (a+b)(a-b) = a^2 - b^2 \). When applied to complex numbers, it can simplify expressions significantly.
For example, when multiplying a complex number by its conjugate, such as \((3+4i)(3-4i)\), it forms a difference of squares pattern:
\[ (3+4i)(3-4i) = 3^2 - (4i)^2 \]
This allows us to calculate the product without performing lengthy multiplications, ultimately simplifying to a real number in cases like this.
Multiplying by conjugates is particularly useful because it helps eliminate the imaginary unit from the expression, leading to a real number result.

Imaginary Unit

The imaginary unit, denoted by \( i \), is a fundamental concept in complex numbers. It is defined by the property that \( i^2 = -1 \).
This defining equation points to the principal difference between real and complex numbers - the factor of conquering negative square roots.
When dealing with complex numbers, the imaginary unit enables us to express numbers that were otherwise not representable in the realm of real numbers. For instance, \( 4i \) is simply \( 4 \) times the imaginary unit \( i \).
In the context of multiplying conjugates, \( (4i)^2 = 16i^2 \) illustrates how \( i^2 = -1 \) alters the multiplication, converting what might seem complex at first into a simplified real number expression.