Number Systems and Expressions

TL;DR

Understanding number systems helps you classify numbers, while algebraic expressions let you represent unknown values and relationships concisely. You'll learn the different types of numbers and how to simplify expressions using basic operations. Master these now, they're the building blocks for all future math.

1. The Mental Model

Think of number systems as different ways to categorize things you count or measure. Expressions are like math sentences with blanks (variables) that show relationships between these numbers, even if you don't know their exact value yet.

2. The Core Material

2.1 Exploring Number Systems

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Numbers aren't just numbers; they come in different types, each with its own special properties. You'll usually start with natural numbers and expand your understanding from there.

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

• Natural Numbers (N): These are the counting numbers: 1, 2, 3, 4... Think of them as what you use to count apples.
• Whole Numbers (W): Just like natural numbers, but they include zero: 0, 1, 2, 3... So, if you have no apples, that's a whole number.
• Integers (Z): This group includes all whole numbers and their negative counterparts: ...-3, -2, -1, 0, 1, 2, 3... This lets you talk about things like temperature below zero.
• Rational Numbers (Q): Any number that can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q$ isn't zero. Examples include $\frac{1}{2}$, $-3$ (which is $\frac{-3}{1}$), $0.75$ (which is $\frac{3}{4}$). When written as decimals, they either terminate (like $0.25$) or repeat (like $0.333...$).
• Irrational Numbers (I): These are numbers that can't be written as a simple fraction. Their decimal representations go on forever without repeating. Famous examples are $\pi$ (pi) and $\sqrt{2}$.
• Real Numbers (R): This is the big group that includes all rational and irrational numbers. Most of the math you do in Math 10 will deal with real numbers.
• Complex Numbers (N): You'll see these much later, but they are numbers that include the imaginary unit $i$ (where $i^2 = -1$). They're written as $a + bi$.

2.2 Understanding Expressions

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An expression is a combination of numbers, variables, and operations (like +, -, ×, ÷). It doesn't have an equals sign, so you can't "solve" it, but you can simplify or evaluate it.

• Variables: Letters (like $x, y, a$) that represent unknown values.
• Constants: Numbers with a fixed value (like $5, -10, \pi$).
• Terms: Parts of an expression separated by addition or subtraction (e.g., in $3x + 7y - 2$, the terms are $3x$, $7y$, and $-2$).
• Coefficients: The numerical part of a term that multiplies a variable (e.g., in $3x$, $3$ is the coefficient).

2.3 Simplifying Expressions

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Simplifying means making an expression easier to read and work with, often by combining "like terms."

• Like terms have the same variables raised to the same powers. For example, $3x$ and $5x$ are like terms; $3x$ and $5x^2$ are not.
• Order of Operations (BEDMAS/PEMDAS): Always follow this order:
  1. Brackets (or Parentheses)
  2. Exponents
  3. Division and Multiplication (from left to right)
  4. Addition and Subtraction (from left to right)

2.4 Evaluating Expressions

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When you're given values for the variables, you can substitute them into an expression and calculate a single numerical answer.

3. Worked Example

Let's simplify and then evaluate the expression: $5(2x - 3) + 7x - 4$ when $x = 2$.

1. Distribute: Multiply the $5$ into the bracket.
   $5(2x) - 5(3) + 7x - 4$
   $10x - 15 + 7x - 4$

2. Combine like terms: Group the $x$ terms and the constant terms.
   $(10x + 7x) + (-15 - 4)$
   $17x - 19$

3. Evaluate: Substitute $x = 2$ into the simplified expression.
   $17(2) - 19$
   $34 - 19$
   $15$

So, the simplified expression is $17x - 19$, and when $x=2$, its value is $15$.

4. Key Takeaways

• Numbers fit into different categories like Natural, Whole, Integer, Rational, and Irrational, which all belong to Real Numbers.
• An expression combines numbers, variables, and operations but doesn't have an equals sign.
• Variables are symbols (usually letters) representing unknown values.
• Simplifying an expression involves combining like terms using the order of operations (BEDMAS/PEMDAS).
• Evaluating an expression means substituting given values for variables and calculating a single numerical answer.
• Always apply the distributive property before combining like terms.

Common Mistakes to Avoid

• Mixing up like terms; remember, $3x$ and $3x^2$ are not the same.
• Forgetting to distribute a number to all terms inside a bracket.
• Not following the correct order of operations (BEDMAS/PEMDAS).
• Confusing an expression (no equals sign) with an equation (has an equals sign).

5. Now Try It

Simplify the expression $3(4a + 2b - 5) - 2(6a - 3b + 1)$ and then evaluate it for $a = -1$ and $b = 3$. Your final success will be a single number representing the evaluated expression.