Remember when you thought numbers couldn't get any more complicated than fractions? Well, buckle up, because we're about to dive into the fascinating world of complex numbers!
Complex numbers are like the superheroes of the number world. They swoop in to save the day when we're dealing with equations that seem impossible to solve using just real numbers.
Note
A complex number is any number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $i^2 = -1$.
Let's break this down:
Example
Some complex numbers:
Now that we've met these quirky numbers, let's see how they play with others!
It's like adding or subtracting vectors - you deal with the real and imaginary parts separately.
$$(a + bi) + (c + di) = (a + c) + (b + d)i$$
Example
Let's add $(3 + 2i)$ and $(1 - 4i)$: $$(3 + 2i) + (1 - 4i) = (3 + 1) + (2 - 4)i = 4 - 2i$$
This is where things get interesting! Remember, $i^2 = -1$.
$$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$$
Example
Multiply $(2 + 3i)$ by $(1 - i)$: $$(2 + 3i)(1 - i) = 2(1) + 2(-i) + 3i(1) + 3i(-i)$$ $$= 2 - 2i + 3i + 3(-1) = (2 - 3) + (3 - 2)i = -1 + i$$
Common Mistake
Don't forget that $i^2 = -1$! It's easy to slip up and treat $i$ like a variable.
Vectors are like arrows in space. They have both magnitude (length) and direction.
Note
In Algebra II, we typically work with vectors in 2D or 3D space, represented as ordered pairs or triples.
For example, $\vec{v} = \langle 3, -2 \rangle$ is a 2D vector, and $\vec{w} = \langle 1, 4, 2 \rangle$ is a 3D vector.
Example
Let $\vec{v} = \langle 2, 3 \rangle$ and $\vec{w} = \langle -1, 4 \rangle$
Matrices are rectangular arrays of numbers. They're incredibly useful for organizing and manipulating data.
Note
A matrix is denoted by capital letters and its elements by lowercase letters with subscripts indicating their position.
For example:
$$A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$$
This is a 2x3 matrix (2 rows, 3 columns).
Example
Let's multiply two matrices:
$$\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \times \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} (1\times5 + 2\times7) & (1\times6 + 2\times8) \ (3\times5 + 4\times7) & (3\times6 + 4\times8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}$$
Tip
Matrix multiplication is not commutative! $AB \neq BA$ in general.
Number and Quantity in Algebra II takes us beyond the familiar territory of real numbers and into the exciting realms of complex numbers, vectors, and matrices. These concepts might seem abstract at first, but they have countless applications in physics, engineering, computer graphics, and more. As you continue your journey in mathematics, you'll see how these powerful tools can help solve complex problems and model real-world phenomena. Keep exploring, and don't be afraid to dive deep into these fascinating topics!