1.1 Real Numbers and Their Properties

Notes

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

• Natural numbers: 1, 2, 3, ... Whole numbers: 0, 1, 2, 3, ...
• Integers: whole numbers and their opposites: ..., $-2$, $-1$, 0, 1, 2, ...
• Rational numbers: any number you can write as a fraction $\displaystyle \frac{a}{b}$ with integers $a$ and $b \ne 0$. Terminating and repeating decimals are rational.
• Irrational numbers: decimals that never end or repeat, like $\pi$ and $\displaystyle \sqrt{2}$.

Key properties:

• Commutative: $a + b = b + a$ and $ab = ba$ (order doesn't matter).
• Associative: $(a + b) + c = a + (b + c)$ (grouping doesn't matter).
• Identity: $a + 0 = a$ and $a \cdot 1 = a$. Inverse: $a + (-a) = 0$ and $\displaystyle a \cdot \frac{1}{a} = 1$.