Lesson 1.1: Real Numbers: Algebra Essentials

New Concept

Welcome to the foundation of algebra!
We'll explore the real number system, learning to classify numbers, perform calculations using the order of operations, and use key properties to simplify and evaluate algebraic expressions.

What’s next

You'll start by classifying numbers, then dive into interactive examples on the order of operations. Get ready for practice cards to master simplifying expressions!

Property

The set of natural numbers includes the numbers used for counting: {1, 2, 3, …}.
The set of whole numbers is the set of natural numbers plus zero: {0, 1, 2, 3, …}.
The set of integers adds the negative natural numbers to the set of whole numbers: {…, -3, -2, -1, 0, 1, 2, 3, …}.
The set of rational numbers includes fractions written as $\{\frac{m}{n} | m \text{ and } n \text{ are integers and } n \neq 0\}$.
The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: {$h$ | $h$ is not a rational number}.

Examples

• Classify $\sqrt{64}$: This simplifies to $8$. So, it is a natural number (N), whole number (W), integer (I), and rational number (Q).
• Classify $\frac{14}{3}$: As a fraction of integers, it is a rational number (Q). As a decimal, it is $4.666...$, which is a repeating decimal.
• Classify $\sqrt{13}$: This cannot be simplified to a whole number or a fraction of integers, so it is an irrational number (Q').

Explanation

Think of number sets like nesting dolls. Naturals fit inside wholes, which fit inside integers, which fit inside rationals. Irrationals are a separate group, and all of them together form the real numbers.

Property

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS:

• P (arentheses)
• E (xponents)
• M (ultiplication) and D (ivision)
• A (ddition) and S (ubtraction)

To simplify an expression using the order of operations:

1. Simplify any expressions within grouping symbols.
2. Simplify any expressions containing exponents or radicals.
3. Perform any multiplication and division in order, from left to right.
4. Perform any addition and subtraction in order, from left to right.

Examples

• Evaluate $(4 \cdot 3)^2 - 5(4 + 2)$:

First, solve inside parentheses to get $(12)^2 - 5(6)$. Then, handle the exponent: $144 - 5(6)$.
Next, multiply: $144 - 30$.
Finally, subtract to get $114$.

• Evaluate $\frac{4^2 - 6}{5} + \sqrt{21 - 5}$:

Simplify the numerator and under the radical first: $\frac{16 - 6}{5} + \sqrt{16} = \frac{10}{5} + 4$.
Then divide: $2 + 4 = 6$.

• Evaluate $15 - |6 - 10| + 4(5 - 2)$:

Handle absolute value and parentheses: $15 - |-4| + 4(3) = 15 - 4 + 12$.
Add and subtract from left to right: $11 + 12 = 23$.

Property

Commutative Property: $a + b = b + a$ (Addition); $a \cdot b = b \cdot a$ (Multiplication)
Associative Property: $a + (b + c) = (a + b) + c$ (Addition); $a(bc) = (ab)c$ (Multiplication)
Distributive Property: $a \cdot (b + c) = a \cdot b + a \cdot c$
Identity Property: $a + 0 = a$ (Additive Identity); $a \cdot 1 = a$ (Multiplicative Identity)
Inverse Property: $a + (-a) = 0$ (Additive Inverse); $a \cdot (\frac{1}{a}) = 1$ for $a \neq 0$ (Multiplicative Inverse)

Examples

• Simplify $4 \cdot 9 + 4 \cdot 1$: Using the distributive property, this becomes $4 \cdot (9 + 1) = 4 \cdot 10 = 40$.
• Simplify $(15 + 7) + (-7)$: Using the associative property of addition, this is $15 + [7 + (-7)] = 15 + 0 = 15$.
• Simplify $\frac{3}{8} \cdot (\frac{5}{6} \cdot \frac{8}{3})$: Use the commutative and associative properties to regroup: $(\frac{3}{8} \cdot \frac{8}{3}) \cdot \frac{5}{6} = 1 \cdot \frac{5}{6} = \frac{5}{6}$.

Explanation

These properties are the fundamental rules that let you legally rearrange, regroup, and rewrite expressions. They are the 'why' behind the algebraic steps you take to simplify problems and solve equations.

Property

An algebraic expression is a collection of constants and variables joined by algebraic operations.
To evaluate an algebraic expression, replace each variable in the expression with its given value, then simplify the resulting expression using the order of operations.

Examples

• Evaluate $4y - 9$ for $y = 5$: Substitute $5$ for $y$ to get $4(5) - 9 = 20 - 9 = 11$.
• Evaluate $\frac{x+7}{x-2}$ for $x=5$: Substitute $5$ for $x$ to get $\frac{5+7}{5-2} = \frac{12}{3} = 4$.
• Evaluate $\sqrt{a^2 + 2b^2}$ for $a = 4$ and $b = 3$: Substitute the values to get $\sqrt{4^2 + 2(3^2)} = \sqrt{16 + 2(9)} = \sqrt{16 + 18} = \sqrt{34}$.

Explanation

Evaluating an expression is like using a recipe. The expression is the formula, the variables are the ingredients, and the given values are the specific amounts you must plug in to find the final result.

Property

To simplify an algebraic expression, use the properties of real numbers to make it easier to evaluate.
The goal is to combine like terms.
Use the commutative property to reorder terms and the distributive property to remove parentheses.
The simplified expression is equivalent to the original.

Examples

• Simplify $7x - 4y + 2x + 6y$: Reorder to get $7x + 2x - 4y + 6y$. Combine like terms to get $9x + 2y$.
• Simplify $5(a - 3) + 2a$: Use the distributive property to get $5a - 15 + 2a$. Combine like terms to get $7a - 15$.
• Simplify $(6m - \frac{1}{2}n) - (2m + \frac{3}{2}n)$: Distribute the negative to get $6m - \frac{1}{2}n - 2m - \frac{3}{2}n$. Combine like terms to get $4m - 2n$.

Explanation

Simplifying an expression is like tidying up a messy room. You group similar items (like terms) together to make the expression cleaner and easier to understand, without changing its actual value.