Real Numbers and Their Properties

📋 Course Outline

1. Real Number Operations
2. Powers and Algebraic Identities
3. Methods of Mathematical Proof
4. Order and Inequalities
5. Intervals of Real Numbers
6. Absolute Value and Distance
7. Roots and Rational Powers

📖 1. Real Number Operations

🔑 Key Concepts & Definitions

• Rational number: can be written as $\frac{a}{b}$, where a and b are integers and $b\ne0$
• Irrational number: An irrational number cannot be written as $\frac{a}{b}$ with integers a and b and $b\ne0$, and therefore cannot be written as either a terminating or a repeating decimal.

📝 Essential Points

• Real numbers consist of rational numbers and irrational numbers and can be represented by points on the real-number axis.

• Subtraction is defined by $a-b=a+(-b)$, and division by a nonzero number is defined by $\frac{a}{b}=a\cdot\frac{1}{b}$ with $b\ne0$.

• For real numbers, $ab=0$ if and only if $a=0$ or $b=0$, and consequently $ab\ne0$ if and only if $a\ne0$ and $b\ne0$.

💡 Memory Hook

Rational numbers have fractional or repeating-decimal forms, whereas irrational numbers do not.

📖 2. Powers and Algebraic Identities

🔑 Key Concepts & Definitions