Question

lesson 5-1 - day 2: divide complex numbers (rationalize the denominator) exit ticket 1. what is the complex conjugate of the complex number expressed below? (1 + 3i) conjugate: __________ 2. write the quotient in a + bi form and with a rational denominator. \\(\frac{4 - 4i}{1 + 3i}\\)

Explanation:

Question 1

Step1: Recall complex conjugate rule

The complex conjugate of a complex number \(a + bi\) is \(a - bi\), where \(a\) and \(b\) are real numbers.

Step2: Apply to \(1 + 3i\)

For the complex number \(1 + 3i\), \(a = 1\) and \(b = 3\). So its conjugate is \(1 - 3i\).

Step1: Multiply numerator and denominator by conjugate

The denominator is \(1 + 3i\), its conjugate is \(1 - 3i\). Multiply numerator \((4 - 4i)\) and denominator \((1 + 3i)\) by \(1 - 3i\):

$$ \frac{(4 - 4i)(1 - 3i)}{(1 + 3i)(1 - 3i)} $$

Step2: Expand numerator and denominator

• Numerator: \((4 - 4i)(1 - 3i)=4\times1 - 4\times3i - 4i\times1 + 4i\times3i = 4 - 12i - 4i + 12i^{2}\). Since \(i^{2}=-1\), this becomes \(4 - 16i + 12\times(-1)=4 - 16i - 12=-8 - 16i\).
• Denominator: \((1 + 3i)(1 - 3i)=1^{2}-(3i)^{2}=1 - 9i^{2}=1 - 9\times(-1)=1 + 9 = 10\).

Step3: Simplify the fraction

$$ \frac{-8 - 16i}{10}=\frac{-8}{10}-\frac{16}{10}i=-\frac{4}{5}-\frac{8}{5}i $$

Answer:

\(1 - 3i\)

Question 2