1. Let's start by understanding the **Real-Number System**. It includes all rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot. 2. **Categories of Real Numbers:** - Natural numbers: 1, 2, 3, ... - Whole numbers: 0, 1, 2, 3, ... - Integers: ..., -2, -1, 0, 1, 2, ... - Rational numbers: numbers that can be written as $\frac{a}{b}$ where $a,b$ are integers and $b \neq 0$ - Irrational numbers: numbers that cannot be written as fractions, like $\sqrt{2}$ or $\pi$ 3. **Inequalities and Interval Notation:** - Inequalities compare two values using $<$, $>$, $\leq$, $\geq$. - Interval notation expresses ranges, e.g., $(a,b)$ means all numbers between $a$ and $b$ but not including $a$ and $b$. 4. **Operations with Real Numbers:** - Addition, subtraction, multiplication, and division follow standard arithmetic rules. - Properties include commutative, associative, distributive laws. 5. **Simplifying Fractions:** - Find the Greatest Common Factor (GCF) of numerator and denominator. - Divide numerator and denominator by the GCF. - Example: Simplify $\frac{12}{18}$ $$\text{GCF}(12,18) = 6$$ $$\frac{12}{18} = \frac{\cancel{6} \times 2}{\cancel{6} \times 3} = \frac{2}{3}$$ 6. **Multiplication of Fractions:** - Multiply numerators and denominators directly. - Example: $\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$ 7. **Division of Fractions:** - Multiply by the reciprocal of the divisor. - Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$ 8. **Addition and Subtraction of Fractions:** - Find a common denominator. - Example: $\frac{1}{4} + \frac{1}{6}$ $$\text{LCM of }4 \text{ and } 6 = 12$$ $$\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}$$ $$\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$$ 9. **Exponents:** - Positive exponents mean repeated multiplication. - Negative exponents mean reciprocal: $a^{-n} = \frac{1}{a^n}$ - Zero exponent means 1: $a^0 = 1$ (if $a \neq 0$) 10. **Radicals:** - Square root $\sqrt{x}$ is a number which when squared gives $x$. - Cube root $\sqrt[3]{x}$ is a number which when cubed gives $x$. 11. **Rational Exponents:** - $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ - Example: $8^{\frac{2}{3}} = (\sqrt[3]{8})^2 = 2^2 = 4$ This summary covers the key concepts you need for your midterm. Focus on understanding each step and practicing problems for mastery.