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:

1. Parentheses (or Brackets)
2. Exponents (or Orders/Indices)
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Let's use an example: 10 - 2 * 3 + (6 / 2)

1. Parentheses: 10 - 2 * 3 + 3
2. Exponents: (None in this example)
3. Multiplication/Division (left to right):
   • 2 * 3 = 6
   • 10 - 6 + 3
4. Addition/Subtraction (left to right):
   • 10 - 6 = 4
   • 4 + 3 = 7
     So, 10 - 2 * 3 + (6 / 2) = 7.

3. Worked Example

Let's evaluate the expression 3x^2 - |y - 4| + 5 when x = -2 and y = 1.

1. Substitute the values:
   3(-2)^2 - |1 - 4| + 5

2. Address parentheses/absolute value first:

   • (-2)^2 = (-2) * (-2) = 4
   • |1 - 4| = |-3| = 3

3. Substitute these results back into the expression:
   3(4) - 3 + 5

4. Perform multiplication:
   12 - 3 + 5

5. Perform addition and subtraction from left to right:

   • 12 - 3 = 9
   • 9 + 5 = 14

The evaluated expression is 14.

4. Key Takeaways

• Real numbers encompass all numbers you'll use in everyday algebra: natural, whole, integers, rational, and irrational.
• The absolute value of a number is its distance from zero and is always non-negative.
• Variables are symbols representing unknown or changing values.
• Algebraic expressions combine numbers, variables, and operation symbols without an equals sign.
• To evaluate an expression, substitute values for variables and follow the order of operations.
• PEMDAS/BODMAS ensures you perform operations in the correct sequence: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Common Mistakes to Avoid:
- Mixing up a negative number with its absolute value (e.g., thinking |-5| = -5).
- Incorrectly applying the order of operations, especially with multiplication/division or addition/subtraction.
- Forgetting to multiply 3 by (-2)^2 as 3 * 4 instead of (3 * -2)^2.
- Not simplifying what's inside absolute value bars before taking the absolute value.

5. Now Try It

Evaluate the expression 7 - 2|x + 5| + y^3 when x = -7 and y = -1. Show your steps clearly.

What success looks like: You should arrive at a single numerical answer, demonstrating correct substitution, absolute value calculation, exponent handling, and the order of operations.