Number and Algebra

TL;DR

Number and Algebra helps us understand how numbers work and how to use symbols (like letters) to represent unknown values or relationships. It's all about solving problems, from simple calculations to more complex equations, by working with patterns and rules. This topic builds the foundation for understanding almost everything else in maths, so it's super important for real-world applications.

1. The Mental Model

Think of Number and Algebra as your mathematical toolkit. Numbers are your basic tools, and algebra gives you the special wrenches and levers to handle more complex situations where you don't know all the values upfront. You're learning to translate real-world problems into mathematical statements you can solve.

2. The Core Material

This topic covers a lot of ground, but it mostly boils down to understanding different types of numbers and using algebraic expressions and equations.

2.1 Number Types

You'll work with several kinds of numbers:

• Whole Numbers: These are the counting numbers (0, 1, 2, 3...).
• Integers: These include whole numbers and their negative counterparts (...-2, -1, 0, 1, 2...).
• Rational Numbers: Any number that can be written as a fraction (like 1/2, 3, -0.75). This includes integers and terminating/repeating decimals.
• Irrational Numbers: Numbers that can't be written as a simple fraction (like $\pi$ or $\sqrt{2}$). Their decimal representations go on forever without repeating.
• Real Numbers: This is the big group that includes ALL rational and irrational numbers. Most of what you'll deal with in maths applications will be real numbers.

Understanding these categories helps you know what kind of answers to expect and how to handle them. For example, if you're counting people, you'll use whole numbers, not fractions.

2.2 Operations and Properties

You're already familiar with the four basic operations: addition, subtraction, multiplication, and division. But it's also key to remember their properties:

• Commutative Property: The order doesn't matter for addition and multiplication (e.g., $2 + 3 = 3 + 2$).
• Associative Property: How you group numbers doesn't matter for addition and multiplication (e.g., $(2 + 3) + 4 = 2 + (3 + 4)$).
• Distributive Property: Multiplication distributes over addition (e.g., $2 \times (3 + 4) = (2 \times 3) + (2 \times 4)$). This one is super important in algebra!

2.3 Algebraic Expressions

An expression is a combination of numbers, variables (letters like $x$ or $y$), and operation symbols (like +, -, *, /). It doesn't have an equals sign.

• Example: $3x + 5$ or $y^2 - 7$

You often need to simplify expressions by combining like terms. Like terms have the same variables raised to the same power.

• Example: In $3x + 2y + 5x$, the $3x$ and $5x$ are like terms. You can combine them to get $8x + 2y$.

2.4 Equations

An equation is a statement that two expressions are equal, always containing an equals sign (=).

• Example: $2x + 1 = 7$

The goal with equations is often to solve for the unknown variable. You do this by performing the same operation to both sides of the equation to keep it balanced.

Here's a simple process for solving linear equations:

graph TD
    A["Start: Equation with one variable"] --> B["Simplify both sides (distribute, combine like terms)"];
    B --> C["Get all variable terms on one side"];
    C --> D["Get all constant terms on the other side"];
    D --> E["Isolate the variable (usually by dividing)"];
    E --> F["Check your solution by plugging it back in"];
    F --> G["End: Variable's value"];

2.5 Ratios, Rates, and Percentages

These are ways to compare quantities and are constantly used in applications:

• Ratio: Compares two quantities of the same unit (e.g., 1:2 or 1/2).
• Rate: Compares two quantities of different units (e.g., km/hour or $/kg).
• Percentage: A special ratio out of 100 (e.g., 25% means 25/100).

You'll often convert between these forms and use them to solve problems involving proportions.

3. Worked Example

Let's solve a simple algebraic equation that might come from a real-world scenario.

Problem: A plumber charges a $50 call-out fee plus $30 per hour for their work. If the total bill was $170, how many hours did the plumber work?

Step 1: Define your variable.
Let $h$ be the number of hours the plumber worked.

Step 2: Set up the equation.
The call-out fee is $50.
The hourly charge is $30 times the number of hours ($30h$).
The total bill is $170.
So, the equation is:
$50 + 30h = 170$

Step 3: Solve the equation.
First, we want to get the $30h$ term by itself. Subtract 50 from both sides:
$50 + 30h - 50 = 170 - 50$
$30h = 120$

Next, we want to isolate $h$. Divide both sides by 30:
$30h / 30 = 120 / 30$
$h = 4$

Step 4: Check your answer.
If the plumber worked 4 hours:
Call-out fee + (hourly rate * hours) = $50 + (30 * 4) = $50 + $120 = $170.
This matches the total bill given in the problem, so our answer is correct.

Answer: The plumber worked 4 hours.

4. Key Takeaways

• Numbers come in different types (whole, integer, rational, irrational, real) and understanding them helps with problem-solving.
• Basic operations have properties (commutative, associative, distributive) that simplify calculations and algebraic manipulations.
• An expression is a mathematical phrase without an equals sign, while an equation states that two expressions are equal.
• When solving equations, whatever you do to one side, you must do to the other to keep it balanced.
• Simplifying expressions often involves combining "like terms" that have the same variable and exponent.
• Ratios, rates, and percentages are powerful tools for comparing quantities and solving real-world proportional problems.
• Always define your variables clearly when setting up algebraic problems.
• Don't forget to check your solution by plugging it back into the original equation.

Common Mistakes to Avoid:
- Forgetting the order of operations (BIDMAS/PEMDAS).
- Trying to combine unlike terms (e.g., $3x + 2y$ doesn't simplify further).
- Only performing an operation on one side of an equation instead of both.
- Misinterpreting negative signs, especially when distributing or subtracting.
- Not defining your variables, which can lead to confusion in complex problems.

5. Now Try It

A local bakery sells cupcakes for $3.50 each and a special birthday cake for $25. If a customer bought the birthday cake and some cupcakes, and their total bill was $56.50, how many cupcakes did they buy?

Work through the steps: define a variable, set up an equation, solve it, and then check your answer. You should be able to get a whole number of cupcakes.