1. **Definition of Term: Rational Numbers** Rational numbers are numbers that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ and $q$ are integers and $q \neq 0$. 2. **Keywords and Explanations:** - Quotient: Result of division. - Integers: Whole numbers including negative, zero, and positive. - Denominator $q \neq 0$: Division by zero is undefined. 3. **Examples:** - $\frac{1}{2}$, $-\frac{3}{4}$, $5$ (can be written as $\frac{5}{1}$) 1. **Definition of Term: Irrational Numbers** Irrational numbers are numbers that cannot be expressed as a simple fraction; their decimal expansions are non-terminating and non-repeating. 2. **Keywords and Explanations:** - Non-terminating: Decimal goes on forever. - Non-repeating: No repeating pattern in decimal digits. 3. **Examples:** - $\sqrt{2}$, $\pi$, $e$ 1. **Definition of Term: Real Numbers** Real numbers include all rational and irrational numbers; they represent all points on the number line. 2. **Keywords and Explanations:** - Number line: Visual representation of real numbers. - Includes both rational and irrational numbers. 3. **Examples:** - $-3$, $0$, $\frac{7}{8}$, $\sqrt{5}$ 1. **Definition of Term: Decimal Expansions of Real Numbers** Decimal expansion is the representation of a real number in decimal form, which can be terminating, non-terminating repeating, or non-terminating non-repeating. 2. **Keywords and Explanations:** - Terminating decimal: Ends after finite digits. - Non-terminating repeating decimal: Infinite digits with a repeating pattern. - Non-terminating non-repeating decimal: Infinite digits without repetition. 3. **Examples:** - Terminating: $0.75$ - Non-terminating repeating: $0.333...$ - Non-terminating non-repeating: $\pi = 3.14159...$ 1. **Definition of Term: Operations on Real Numbers** Operations include addition, subtraction, multiplication, and division performed on real numbers. 2. **Keywords and Explanations:** - Closure: Real numbers are closed under these operations. - Commutative, associative, distributive laws apply. 3. **Examples:** - $2 + 3 = 5$ - $5 - 7 = -2$ - $4 \times 0.5 = 2$ - $\frac{6}{2} = 3$ 1. **Definition of Term: Laws of Exponents for Real Numbers** Rules that govern the operations involving powers of real numbers. 2. **Keywords and Explanations:** - Product of powers: $a^m \times a^n = a^{m+n}$ - Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$ - Power of a power: $(a^m)^n = a^{mn}$ - Power of a product: $(ab)^m = a^m b^m$ - Power of a quotient: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$ - Zero exponent: $a^0 = 1$ (for $a \neq 0$) - Negative exponent: $a^{-m} = \frac{1}{a^m}$ 3. **Examples:** - $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$ - $\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$ - $(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$