Problem 42 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (2-3 i)^{-1} $$

Step-by-Step Solution

Answer

\(\frac{2}{13} + \frac{3}{13}i\)

Step 1: Understand the Expression

The exercise requires evaluating the expression \((2 - 3i)^{-1}\), which is the inverse or reciprocal of the complex number \(2 - 3i\). This means we need to find a complex number \(a + bi\) such that when multiplied by \(2 - 3i\), it gives 1.

Step 2: Multiply by the Conjugate

To evaluate \((2 - 3i)^{-1}\), we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 - 3i\) is \(2 + 3i\). So we have:\[(2 - 3i)^{-1} = \frac{1 \cdot (2 + 3i)}{(2 - 3i)(2 + 3i)}\]

Step 3: Simplify the Denominator

Next, we simplify the denominator using the formula for the product of a complex number and its conjugate: \[(a - bi)(a + bi) = a^2 + b^2\]Substitute \(a = 2\) and \(b = 3\):\[(2 - 3i)(2 + 3i) = 2^2 + 3^2 = 4 + 9 = 13\]

Step 4: Simplify the Whole Expression

Now that we know the simplified denominator is 13, we simplify the expression:\[(2 - 3i)^{-1} = \frac{2 + 3i}{13} = \frac{2}{13} + \frac{3i}{13}\]

Step 5: Write the Final Result

Rewrite the result in the form \(a + bi\):\[\frac{2}{13} + \frac{3}{13}i\]Thus, the inverse of \(2 - 3i\) is \(\frac{2}{13} + \frac{3}{13}i\).

Key Concepts

Inverse of a Complex Number

The inverse of a complex number, also known as the reciprocal, is a complex number that, when multiplied by the original number, yields 1. For a complex number of the form \(a + bi\), its inverse is generally obtained by multiplying both the numerator and the denominator by the complex conjugate of the denominator. This helps eliminate the imaginary unit \(i\) from the denominator.
To find the inverse of \(2 - 3i\), we are fundamentally looking for a number \(a + bi\) such that:

• \((2 - 3i) \cdot (a + bi) = 1\)

The process requires using the complex conjugate to simplify effectively, ensuring that the denominator becomes a real number, making division straightforward.

Complex Conjugate

A complex conjugate is a vital part of working with complex numbers, especially when trying to simplify expressions or find the inverse. For any complex number \(a + bi\), its conjugate is \(a - bi\). Multiplying a complex number by its conjugate results in a real number.
In our exercise with \(2 - 3i\), its conjugate is \(2 + 3i\). When we multiply the number by its conjugate, it helps us eliminate the imaginary part and obtain:

• \((2 - 3i)(2 + 3i) = 4 + 9 = 13\)

This product is essential because it transforms our complex fraction into one with a real denominator, aiding in further simplification.

Simplifying Complex Fractions

Simplifying complex fractions involves expressing the numerator and denominator in a form that removes the imaginary unit from the denominator. This is often done using the complex conjugate.
For our given expression \((2 - 3i)^{-1}\), we started by multiplying the entire expression by \(\frac{2 + 3i}{2 + 3i}\), where \(2 + 3i\) is the conjugate of the denominator. This gives us:

• Numerator: \(1 \cdot (2 + 3i) = 2 + 3i\)
• Denominator: \((2 - 3i)(2 + 3i) = 13\)

Thus, the complex fraction simplifies to \(\frac{2 + 3i}{13}\). This step ensures the fraction is expressed as the complex form \(a + bi\), with real and imaginary parts isolated as \(\frac{2}{13} + \frac{3}{13}i\).

Complex Number Multiplication

Multiplying complex numbers involves applying the distributive property, similar to multiplying binomials. Each part of the numbers is multiplied and then combined, accounting for the imaginary unit \(i\), where \(i^2 = -1\).
Consider multiplying \(2 - 3i\) and \(2 + 3i\), we distribute each term:

• \(2 \cdot 2 = 4\)
• \(2 \cdot 3i = 6i\)
• \(-3i \cdot 2 = -6i\)
• \(-3i \cdot 3i = -9i^2\)

Combining these, where \(-9i^2\) becomes \(+9\) since \(i^2 = -1\), we achieve a result of \(4 + 9 = 13\). By doing this multiplication, one typically reaches a real number when using conjugates, simplifying further division and interpretation.