Foundations

Every number you will use in Algebra 1 is a real number. Real numbers sort into nested families:

• Natural (counting) numbers: $1, 2, 3, 4, \dots$
• Whole numbers: the naturals plus zero: $0, 1, 2, 3, \dots$
• Integers: whole numbers and their negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$
• Rational numbers: any number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \ne 0$. Their decimals either end or repeat.
• Irrational numbers: real numbers that cannot be written as a fraction. Their decimals go on forever with no repeating pattern.

Each family sits inside the next: every natural number is whole, every whole number is an integer, every integer is rational.

The number line places every real number in order from small (left) to large (right). For any two numbers, the one farther right is greater.

• $a < b$ means $a$ is to the left of $b$.
• $a > b$ means $a$ is to the right of $b$.
• $a \le b$ means $a$ is less than or equal to $b$; likewise $a \ge b$.

Negatives flip your intuition: the more negative a number looks, the smaller it is. So $-8 < -3$, because $-8$ sits farther left.

The absolute value of a number is its distance from $0$ on the number line. Distance is never negative, so absolute value is never negative.

A number and its opposite always have the same absolute value.

Adding:

• Same signs: add the values, keep the sign. $(-4) + (-6) = -10$.
• Different signs: subtract the smaller value from the larger, keep the sign of the larger. $\;(-9) + 4 = -5$.

Subtracting: “add the opposite.” Change subtraction to addition and flip the sign of the second number:

The rule is the same for both operations:

• Same signs $\rightarrow$ positive answer.
• Different signs $\rightarrow$ negative answer.

A fraction is in lowest terms when the numerator and denominator share no common factor except $1$. Divide both by their greatest common factor (GCF).

• Multiply: multiply straight across, then simplify. $\;\dfrac{2}{3}\times\dfrac{4}{5} = \dfrac{8}{15}$.
• Divide: multiply by the reciprocal (flip the second fraction). $\;\dfrac{2}{3}\div\dfrac{4}{5} = \dfrac{2}{3}\times\dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}$.
• Add / Subtract: rewrite with a common denominator first, then add or subtract the numerators.

• Fraction to decimal: divide the numerator by the denominator. $\;\tfrac{3}{8} = 3 \div 8 = 0.375$.
• Decimal to fraction: write the decimal over its place value, then simplify. $\;0.35 = \tfrac{35}{100} = \tfrac{7}{20}$.

Simplify in this order:

• P --- Parentheses / grouping (including absolute value and fraction bars)
• E --- Exponents
• MD --- Multiplication and Division, left to right
• AS --- Addition and Subtraction, left to right

Multiplication and division rank equally; do whichever comes first reading left to right. Same for addition and subtraction.

• Commutative (order): $a + b = b + a$ and $a \cdot b = b \cdot a$.
• Associative (grouping): $(a+b)+c = a+(b+c)$ and $(ab)c = a(bc)$.
• Distributive (spread over a sum): $a(b+c) = ab + ac$.
• Identity: adding $0$ or multiplying by $1$ changes nothing: $a + 0 = a$, $a \cdot 1 = a$.

To evaluate an algebraic expression, replace each variable with its given value (use parentheses!) and then follow PEMDAS.

Like terms have exactly the same variable part (same letters with the same exponents). You may add or subtract like terms by combining their coefficients; the variable part stays the same.

Constants (plain numbers) are like terms with each other.

A set of numbers is closed under an operation if combining any two members always lands you back inside that set.

• The integers are closed under $+$, $-$, and $\times$, but not under $\div$: $\;1 \div 2 = \tfrac{1}{2}$ escapes the integers.
• The real numbers are closed under $+$, $-$, $\times$, and $\div$ (dividing by $0$ excepted).

Because the reals also obey commutativity, associativity, and distributivity, carry additive and multiplicative identities ($0$ and $1$), and give every number an additive inverse $-a$ and every nonzero number a multiplicative inverse $\tfrac{1}{a}$, mathematicians call $\mathbb{R}$ a field. These field properties are the rules that justify every algebra step you take.

Beyond “distance from $0$,” absolute value measures the distance between any two numbers:

Order does not matter, because $|a - b| = |b - a|$. For example, the gap between $-3$ and $4$ is

This is why $|x - 5| = 2$ describes every point exactly $2$ units from $5$, namely $x = 3$ and $x = 7$.

The number families nest as a chain of subsets, each written with its own symbol:

that is, naturals $\subset$ integers $\subset$ rationals $\subset$ reals. The irrationals are the reals that are left over when you remove the rationals --- they fill every gap the fractions miss, which is why the number line has no holes.

When grouping symbols are nested --- parentheses inside brackets inside braces, or a fraction bar acting as grouping above and below --- always resolve the innermost group first and work outward. A fraction bar groups its entire numerator and entire denominator as if each were parenthesized.

The Greek capital sigma $\Sigma$ is shorthand for “add up a list.” The expression

reads: “start the counter $i$ at $1$, plug it into the rule after the $\Sigma$, and add the results up to $i = n$.” The letter $i$ is just an index; it disappears once the sum is computed. For instance,

You will meet $\Sigma$ again with sequences and series --- for now, just read it as a compact “add these terms.”