Complex Numbers

Lesson

Complex numbers can be added and subtracted very easily, following the normal laws of algebra.

We must only add and subtract like terms.

And in complex numbers, like terms are the real parts and the imaginary parts.

Let's look a an example with two complex numbers $z_1=3+2i$`z`1=3+2`i` and $z_2=6-4i$`z`2=6−4`i`

To add these two complex numbers together we must first identify the real and imaginary components of each.

Then we add the real components, and imaginary components respectfully

$\left(3+7i\right)+\left(2+i\right)$(3+7i)+(2+i) |
$=$= | $\left(2+3\right)+\left(7+1\right)i$(2+3)+(7+1)i |

$=$= | $5+8i$5+8i |

See how in this example I grouped the real and imaginary parts and then added. Sometimes this helps keep track of all the components. Sometimes you can jump straight to the answer.

$\left(9-2i\right)-\left(-2+6i\right)$(9−2i)−(−2+6i) |
$=$= | $9-2i+2-6i$9−2i+2−6i |

$=$= | $11-8i$11−8i |

In this example, I expanded the brackets observing the change of sign and then collected like terms.

Evaluate $\left(3+6i\right)+\left(7+3i\right)$(3+6`i`)+(7+3`i`).

Evaluate $\left(6+9i\right)-\left(-3-4i\right)$(6+9`i`)−(−3−4`i`).

Evaluate $\left(-6\sqrt{7}-2i\right)+\left(3\sqrt{7}+7i\right)$(−6√7−2`i`)+(3√7+7`i`).

Manipulate complex numbers and present them graphically

Apply the algebra of complex numbers in solving problems