Problem 40 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{25}{4-3 i}$$

Step-by-Step Solution

Answer

The expression evaluates to \(4 + 3i\).

Step 1: Identify the conjugate

The denominator of the expression is \(4 - 3i\). The conjugate of this complex number is \(4 + 3i\).

Step 2: Multiply numerator and denominator by the conjugate

To eliminate the imaginary part from the denominator, multiply both the numerator and the denominator by \(4 + 3i\). The expression becomes: \[\frac{25}{4-3i} \times \frac{4+3i}{4+3i} = \frac{25(4+3i)}{(4-3i)(4+3i)}\]

Step 3: Simplify the denominator

The product of a complex number and its conjugate is a real number. Compute \((4 - 3i)(4 + 3i) = 4^2 - (3i)^2 = 16 - 9i^2 = 16 + 9 = 25\).

Step 4: Expand and simplify the numerator

Distribute 25 across \((4 + 3i)\):\[25(4 + 3i) = 100 + 75i\]

Step 5: Divide each part of the numerator by the simplified denominator

Now divide the real and imaginary parts of the numerator by the simplified denominator (25):\[\frac{100 + 75i}{25} = \frac{100}{25} + \frac{75i}{25} = 4 + 3i\]

Step 6: Write the expression in standard form

The final result in the form \(a + bi\) is \(4 + 3i\).

Key Concepts

Complex Conjugates

Complex conjugates play a crucial role in simplifying expressions involving complex numbers, especially in division. A complex conjugate of a number is a form where the sign of the imaginary part is changed. For instance, if you have a complex number like \(4 - 3i\), its conjugate would be \(4 + 3i\).

• Why is it important: Multiplying a complex number by its conjugate results in a real number. This is because the product eliminates the imaginary part, a desirable outcome, especially when the denominator includes a complex number.
• In our exercise: We found the conjugate of \(4 - 3i\) to be \(4 + 3i\). Multiplying both the numerator and the denominator of our original expression by this conjugate allowed us to effectively "remove" the imaginary unit from the denominator.

This process is essential in expressing complex numbers in their standard form, \(a + bi\), where both \(a\) and \(b\) are real numbers.

Imaginary Unit

The imaginary unit \(i\) is the fundamental building block for complex numbers. It's defined such that \(i^2 = -1\).

• Origin of \(i\): The need for imaginary numbers arises from the square roots of negative numbers. Since no real number squared results in a negative number, mathematicians introduced \(i\) to address these cases.
• Usage in expressions: In expressions such as \(4 - 3i\), \(i\) represents the imaginary component. When we multiply \(i\) by itself, it behaves differently than real numbers due to the relationship \(i^2 = -1\).

In our exercise, while multiplying the conjugates, we used the property \((3i)^2 = 9i^2\). By substituting \(i^2\) with \(-1\), this simplifies the expression as \(9(-1)\), leading to a real number when combined with other terms.

Complex Division

Dividing complex numbers often involves dealing with complex conjugates and requires simplification to the standard form \(a + bi\).

• Procedure: Begin by multiplying both the numerator and denominator of the fraction by the conjugate of the denominator. This eliminates any imaginary components in the denominator.
• In the example: We began by multiplying by \(4 + 3i\), the conjugate of \(4 - 3i\). This action transformed the denominator into a real number, making it simpler to divide.
• Simplification: After multiplying, factorize and expand the terms in the numerator, distributing through all components. Finally, separate real and imaginary parts by dividing each by the new, simplified denominator. End with \(a + bi\) form.

This process ensures that the complex quotient is easy to interpret and further simplifies calculations or interpretations in further mathematical applications.