Number and Algebra Fundamentals

TL;DR

This topic covers the basic building blocks of numbers and how we use letters (variables) to represent them in algebra. You'll learn about different types of numbers, how to perform basic operations, and the rules that govern algebraic expressions. Mastering these fundamentals is crucial for tackling more complex math problems.

1. The Mental Model

Think of numbers as tools for counting and measuring, and algebra as a way to solve puzzles when you don't know all the numbers yet. We use specific rules to handle these tools and solve these puzzles efficiently.

2. The Core Material

2.1 Types of Numbers

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Numbers aren't all the same! Here are the main types you'll work with:

• Natural Numbers (N): These are your counting numbers: 1, 2, 3, ... (some include 0, but usually not).
• Whole Numbers (W): Natural numbers, plus zero: 0, 1, 2, 3, ...
• Integers (Z): Whole numbers and their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational Numbers (Q): Any number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b$ is not zero. This includes all integers, decimals that terminate (like 0.5) or repeat (like 0.333...).
• Irrational Numbers: Numbers that cannot be written as a simple fraction. They have decimal representations that go on forever without repeating (e.g., $\pi$, $\sqrt{2}$).
• Real Numbers (R): All rational and irrational numbers combined. This is most of the numbers you'll encounter day-to-day.

2.2 Basic Operations

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You know these well, but it's good to review their order:

• Addition (+)
• Subtraction (-)
• Multiplication (× or *)
• Division (÷ or /)

The order in which you perform these operations is vital. Remember BODMAS/PEMDAS:

• Brackets / Parentheses
• Orders (powers and square roots) / Exponents
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

2.3 Algebraic Expressions

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In algebra, we use letters (called variables) to represent unknown numbers. An expression is a combination of numbers, variables, and operation signs (e.g., $3x + 5$).

• Terms: Parts of an expression separated by + or - signs. In $3x + 5$, $3x$ is a term and $5$ is a term.
• Coefficients: The number multiplied by a variable in a term. In $3x$, 3 is the coefficient.
• Constants: Terms that are just numbers (no variables). In $3x + 5$, 5 is the constant.
• Like Terms: Terms that have the exact same variables raised to the same powers (e.g., $3x$ and $-7x$ are like terms; $2y^2$ and $5y^2$ are like terms; $4x$ and $4x^2$ are NOT like terms). We can only add or subtract like terms.

2.4 Laws of Algebra

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These rules help you simplify expressions:

• Commutative Law:
  • Addition: $a + b = b + a$ (e.g., $2 + 3 = 3 + 2$)
  • Multiplication: $a \times b = b \times a$ (e.g., $2 \times 3 = 3 \times 2$)
• Associative Law:
  • Addition: $(a + b) + c = a + (b + c)$ (e.g., $(1 + 2) + 3 = 1 + (2 + 3)$)
  • Multiplication: $(a \times b) \times c = a \times (b \times c)$ (e.g., $(1 \times 2) \times 3 = 1 \times (2 \times 3)$)
• Distributive Law: $a \times (b + c) = (a \times b) + (a \times c)$ (e.g., $2 \times (3 + 4) = (2 \times 3) + (2 \times 4)$)

Here's how number types fit together:

graph TD
    R["Real Numbers"] --> Q["Rational Numbers"]
    R --> I["Irrational Numbers"]
    Q --> Z["Integers"]
    Z --> W["Whole Numbers"]
    W --> N["Natural Numbers"]

3. Worked Example

Let's simplify the expression: $5 + 2(3x - 1) - 4x$

1. Deal with the brackets first (Distributive Law):
   $5 + (2 \times 3x) - (2 \times 1) - 4x$
   $5 + 6x - 2 - 4x$

2. Group like terms together:
   $(6x - 4x) + (5 - 2)$

3. Combine like terms:
   $2x + 3$

So, $5 + 2(3x - 1) - 4x$ simplifies to $2x + 3$.

4. Key Takeaways

• Numbers have different categories (Natural, Whole, Integer, Rational, Irrational, Real) each with specific properties.
• Always follow the order of operations (BODMAS/PEMDAS) to get the correct result in calculations.
• Variables are placeholders for unknown numbers, allowing us to write general rules and solve equations.
• You can only add or subtract terms that are "like terms" (same variable, same power).
• The Commutative, Associative, and Distributive laws are fundamental for rearranging and simplifying expressions.

Common Mistakes to Avoid:
- Not following BODMAS/PEMDAS, especially confusing the order of multiplication/division or addition/subtraction.
- Trying to add or subtract unlike terms (e.g., $3x + 2y$ cannot be simplified further).
- Incorrectly applying the distributive law (e.g., forgetting to multiply all terms inside the bracket by the outside number).
- Mixing up negative signs when combining terms.

5. Now Try It

Simplify the following expression: $7y - 3(2y + 4) + 10$.

What to do: Apply the distributive law first, then identify and combine like terms, paying close attention to negative signs.

What success looks like: Your final simplified expression will have at most two terms (one with 'y' and one constant term). You should get $-y - 2$.