Question

1. $(5 + 6i) - (-5 + 6i) = $
2. $(3 + i) + (3 + 7i) = $
3. $(-2 + 6i) - (3 + 4i) = $

find the product of the given complex numbers.

1. $(3 + 7i)(3 + i) = $
2. $(5 + 6i)(5 - 6i) = $
3. $(3 + 4i)(-2 + 6i) = $
4. $(3 + 4i)(3 - 4i) = $
5. $(3 + 2i)(7 + 4i) = $

our expanding number system
both
eal\ and \imaginary\ were first given mathematical definitions in 1637, when french mathematician and philosopher rené descartes (rə·nā’ dě·kàrt’) used them to refer to number systems. imaginary was first used of numbers when what we now know as imaginary numbers were discovered.
imaginary numbers were first believed to be of little importance. it took the work of the outstandingly brilliant german mathematician karl friedrich gauss (kärl frē’ drĭkh gous) in the nineteenth century to show how imaginary numbers could be used in the development of mathematical theory. he used the term \complex number\ in 1832.
the enormous practicality of complex numbers in electrical research was demonstrated by german physicist charles steinmetz (stīn’ měts) early in this century. each of these men expanded our number system to include new discoveries.
complex conjugates and division
objective
to divide complex numbers
complex numbers such as $2 + 5i$ and $2 - 5i$ are called complex conjugates. thus, the complex conjugate of $a + bi$ is $a - bi$. likewise, the complex conjugate of $a - bi$ is $a + bi$.
to divide complex numbers, we use the conjugate of the divisor to find a symbol for 1. $2 - 6i$ is the conjugate for $2 + 6i$.
$\frac{2 + 3i}{2 + 6i} = \frac{(2 + 3i)(2 - 6i)}{(2 + 6i)(2 - 6i)} = \frac{4 - 12i + 6i - 18i^2}{4 - 12i + 12i - 36i^2}$
$= \frac{4 - 6i - 18(-1)}{4 - 36(-1)} = \frac{4 - 6i + 18}{4 + 36} = \frac{22 - 6i}{40}$
$= \frac{22}{40} - \frac{6i}{40} = \frac{11}{20} - \frac{3i}{20}$
the above procedure is very similar to rationalizing a denominator containing radicals.

Explanation:

Problem 8:

Step1: Remove parentheses

Using the distributive property, \((5 + 6i)-(-5 + 6i)=5 + 6i + 5 - 6i\)

Step2: Combine like terms

Combine the real parts: \(5 + 5 = 10\), and the imaginary parts: \(6i-6i = 0\). So the result is \(10+0i = 10\)

Step1: Remove parentheses

\((3 + i)+(3 + 7i)=3 + i+3 + 7i\)

Step2: Combine like terms

Real parts: \(3 + 3=6\), imaginary parts: \(i + 7i = 8i\). So the result is \(6 + 8i\)

Step1: Remove parentheses

Using the distributive property, \((-2 + 6i)-(3 + 4i)=-2 + 6i-3 - 4i\)

Step2: Combine like terms

Real parts: \(-2-3=-5\), imaginary parts: \(6i - 4i=2i\). So the result is \(-5 + 2i\)

Answer:

\(10\)

Problem 9: