Foundations of Algebra: Real Numbers and Expressions
TL;DR
Algebra 1 starts with understanding different types of numbers and how to work with them using basic operations. We also learn about variables and how to build and evaluate expressions. Mastering these fundamentals is crucial for everything else in algebra.
1. The Mental Model
Think of algebra as a language for describing relationships using numbers and symbols. You're learning the alphabet (numbers), vocabulary (variables, operations), and basic sentence structure (expressions).
2. The Core Material
What are Real Numbers?
Real numbers are essentially all the numbers you're likely to encounter in Algebra 1. They include:
- Natural Numbers: Counting numbers (1, 2, 3, ...).
- Whole Numbers: Natural numbers plus zero (0, 1, 2, 3, ...).
- Integers: Whole numbers and their opposites (... -2, -1, 0, 1, 2 ...).
- Rational Numbers: Numbers that can be written as a fraction where the numerator and denominator are integers (and the denominator isn't zero). Examples: 1/2, -3, 0.75 (which is 3/4).
- Irrational Numbers: Numbers that cannot be written as a simple fraction. Their decimal representation goes on forever without repeating. Famous examples are pi ($\pi \approx 3.14159...$) and the square root of 2 ($\sqrt{2} \approx 1.41421...$).
All these types of numbers together make up the set of Real Numbers. You'll often see them represented on a number line.
Working with Absolute Value
The absolute value of a number is its distance from zero on the number line. Because it's a distance, it's always non-negative. We use vertical bars, like |x|.
|5| = 5 (5 is 5 units away from 0)|-5| = 5 (-5 is 5 units away from 0)|0| = 0
Understanding Variables and Expressions
- A variable is a symbol (usually a letter like
x, y, a) that represents an unknown quantity or a quantity that can change.
- An algebraic expression is a combination of numbers, variables, and operation symbols (+, -, ×, ÷). It doesn't have an equals sign.
Examples of expressions:
* x + 5
* 3y - 7
* 2a^2 + b
Evaluating Expressions
To evaluate an expression, you substitute a given numerical value for each variable and then simplify the resulting numerical expression using the order of operations.
Order of Operations (PEMDAS/BODMAS)
This is crucial for getting the right answer every time. Remember the acronym:
- P