Problem 41 - Algebra 2 and Trigonometry

Question

In \(38-45,\) find the multiplicative inverse of each of the following in \(a+b i\) form. $$ 3-3 i $$

Step-by-Step Solution

Answer

The multiplicative inverse of \(3 - 3i\) is \(\frac{1}{3} + \frac{1}{3}i\).

Step 1: Identify the conjugate

The multiplicative inverse of a complex number is found by multiplying by its conjugate both in the numerator and denominator. The conjugate of the complex number \(3 - 3i\) is \(3 + 3i\).

Step 2: Multiply numerator and denominator

Multiply \(3 - 3i\) by its conjugate \(3 + 3i\), both in the numerator and the denominator:\[\frac{3 - 3i}{1} \cdot \frac{3 + 3i}{3 + 3i} = \frac{(3-3i)(3+3i)}{(3+3i)(3-3i)}\]

Step 3: Simplify the denominator

The denominator \((3 - 3i)(3 + 3i)\) is in the form of \((a - bi)(a + bi)\), which can be simplified to \(a^2 + b^2\):\[(3)^2 + (3)^2 = 9 + 9 = 18\]

Step 4: Expand the numerator

The numerator \((3 - 3i)(3 + 3i)\) is expanded as follows:\[3 \cdot 3 + 3 \cdot 3i - 3i \cdot 3 - 3i \cdot 3i = 9 + 9i - 9i + 9 = 18\]

Step 5: Combine and simplify

Combine the results of the previous steps to find the inverse:\[\frac{18}{18} = 1\]

Key Concepts

Multiplicative Inverse

In mathematics, the multiplicative inverse of a number is what you multiply that number by to get 1. For complex numbers, things look slightly different, but the concept is the same. If you have a complex number in the form \(a + bi\), its multiplicative inverse is also a complex number. Here’s how you find it:

• Determine the conjugate of the complex number.
• Multiply both the numerator and denominator by this conjugate to eliminate the imaginary part in the denominator.
• Simplify the resulting expression.

With the complex number \((3 - 3i)\), its conjugate is \((3 + 3i)\). Multiplying \((3 - 3i)\) by \((3 + 3i)\), and dividing by their product, transforms the original number into its multiplicative inverse, all while normalizing the denominator through minimizing complexity.

Conjugate of a Complex Number

The conjugate of a complex number is an essential concept. It’s about making the imaginary part disappear when multiplying complex numbers. You start with a complex number \(a + bi\), and the conjugate is \(a - bi\). Essentially, you switch the sign of the imaginary part.

• This process is vital for finding a multiplicative inverse, as it helps in getting rid of the imaginary part in the denominator.
• The formula \((a + bi)(a - bi) = a^2 + b^2\) confirms that multiplying a complex number by its conjugate results in a real number.

When we took the complex number \(3 - 3i\), its conjugate \(3 + 3i\) canceled out the imaginary parts when multiplied, leaving us with a real number in the denominator.

Simplifying Complex Fractions

Simplifying complex fractions involves making the denominator a real number. This procedure is simpler than it sounds.

• Multiply the complex fraction by the conjugate of the denominator.
• Expand the numerator and denominator using distribution.
• Apply the formula to the denominator: \((a - bi)(a + bi) = a^2 + b^2\), so it becomes a real number.
• Then, simplify the fraction by dividing the real components.

In the exercise involving \(3 - 3i\), after multiplying by the conjugate, we expanded both the numerator and the denominator. The denominator lead to \(9 + 9 = 18\). This allowed the fraction to simplify neatly to 1, showcasing the power of conjugates in handling complex numbers.