"Review of number systems (Real, Complex)",

TL;DR

You'll refresh your understanding of real numbers, including their various subsets like natural, integer, and rational numbers. You'll also learn about complex numbers, which extend real numbers to include imaginary components. This review provides the foundational number systems essential for later math topics.

1. The Mental Model

Think of number systems as ever-expanding containers. We start with simple counting numbers, then add zero and negatives, then fractions, then all possible points on a line, and finally, numbers that let us solve even more equations.

2. The Core Material

You're probably very familiar with real numbers. These are all the numbers you can find on a continuous number line. They include a few important subsets:

2.1 Real Numbers ($\mathbb{R}$) and Their Subsets

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• Natural Numbers ($\mathbb{N}$): These are the counting numbers: {1, 2, 3, ...}. Some definitions include 0, but we'll stick to positive integers.
• Whole Numbers ($\mathbb{W}$): These are natural numbers plus zero: {0, 1, 2, 3, ...}.
• Integers ($\mathbb{Z}$): These include all whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
• Rational Numbers ($\mathbb{Q}$): Any number that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers, and $q \neq 0$. This includes all integers (e.g., $5 = \frac{5}{1}$), and terminating or repeating decimals (e.g., $0.5 = \frac{1}{2}$, $0.333... = \frac{1}{3}$).
• Irrational Numbers: These are real numbers that cannot be expressed as a simple fraction. Their decimal representation goes on forever without repeating. Famous examples include $\pi \approx 3.14159...$ and $\sqrt{2} \approx 1.41421...$.

Real numbers are the union of rational and irrational numbers.

2.2 Complex Numbers ($\mathbb{C}$)

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Sometimes, you need to find the square root of a negative number, like $\sqrt{-1}$. Real numbers can't do this. That's where complex numbers come in.

• Imaginary Unit ($\boldsymbol{i}$): We define $i = \sqrt{-1}$. This means $i^2 = -1$.
• Structure of a Complex Number: A complex number is generally written in the form $a + bi$, where:
  • $a$ is the real part (a real number).
  • $b$ is the imaginary part (a real number).
  • $i$ is the imaginary unit.
• Examples: $3 + 2i$, $-5i$ (here, $a=0$), $7$ (here, $b=0$, so all real numbers are also complex numbers).

2.3 Operations with Complex Numbers

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• Addition/Subtraction: Just combine the real parts and the imaginary parts separately.
  • $(a + bi) + (c + di) = (a+c) + (b+d)i$
  • $(a + bi) - (c + di) = (a-c) + (b-d)i$
• Multiplication: Use the distributive property (like FOIL) and remember $i^2 = -1$.
  • $(a + bi)(c + di) = ac + adi + bci + bdi^2 = ac + (ad+bc)i - bd = (ac-bd) + (ad+bc)i$

Here's how these number systems are related:

graph TD
    C["Complex Numbers (a + bi)"] --> R["Real Numbers (a + 0i)"]
    R --> Q["Rational Numbers (p/q)"]
    R --> Ir["Irrational Numbers (e.g., π, √2)"]
    Q --> Z["Integers (...-1, 0, 1...)"]
    Z --> W["Whole Numbers (0, 1, 2...)"]
    W --> N["Natural Numbers (1, 2, 3...)"]

3. Worked Example

Let's work with some complex numbers.

Problem: Given $z_1 = 3 + 4i$ and $z_2 = 1 - 2i$, calculate $z_1 + z_2$ and $z_1 \times z_2$.

Solution:

Addition:
$z_1 + z_2 = (3 + 4i) + (1 - 2i)$
Group real and imaginary parts:
$z_1 + z_2 = (3 + 1) + (4 - 2)i$
$z_1 + z_2 = 4 + 2i$

Multiplication:
$z_1 \times z_2 = (3 + 4i)(1 - 2i)$
Use the distributive property (FOIL):
$z_1 \times z_2 = 3(1) + 3(-2i) + 4i(1) + 4i(-2i)$
$z_1 \times z_2 = 3 - 6i + 4i - 8i^2$
Remember $i^2 = -1$:
$z_1 \times z_2 = 3 - 6i + 4i - 8(-1)$
$z_1 \times z_2 = 3 - 6i + 4i + 8$
Group real and imaginary parts:
$z_1 \times z_2 = (3 + 8) + (-6 + 4)i$
$z_1 \times z_2 = 11 - 2i$

4. Key Takeaways

• Natural, whole, integers, rational, and irrational numbers are all subsets of real numbers.
• Real numbers represent all points on a continuous number line.
• Complex numbers extend real numbers by including an imaginary part, allowing for square roots of negative numbers.
• The imaginary unit $i$ is defined as $\sqrt{-1}$, meaning $i^2 = -1$.
• A complex number has the form $a+bi$, where $a$ is the real part and $b$ is the imaginary part.
• You perform arithmetic with complex numbers by treating real and imaginary parts separately (for addition/subtraction) or by using distributive property and $i^2 = -1$ (for multiplication).

Common Mistakes to Avoid:
* Forgetting that $i^2 = -1$ during complex number multiplication.
* Confusing rational numbers with irrational numbers; rational numbers can be written as $\frac{p}{q}$.
* Assuming $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ when both $a$ and $b$ are negative (e.g., $\sqrt{-1} \times \sqrt{-1} \neq \sqrt{1}$, it's $i \times i = i^2 = -1$).
* Incorrectly combining real and imaginary parts during addition or subtraction of complex numbers.

5. Now Try It

Given $z_1 = -2 + 5i$ and $z_2 = 4 + i$:
1. Identify the real part and imaginary part of $z_1$.
2. Calculate $z_1 - z_2$.
3. Calculate $z_1 \times z_2$.

Success looks like:
You correctly identify the real and imaginary parts. For the subtraction, you should get a result in the form $a+bi$. For the multiplication, you should also get a result in the form $a+bi$, ensuring you correctly handled the $i^2$ term.