Algebra Prerequisites and Real Number Systems Study Guide for Real Numbers, Exponents, and Polynomials

Historical Evolution and Classification of Real Numbers

• The Language of Mathematics: Mathematics is considered the language of science, and its own language is composed of numbers.

• Early Enumeration: Numbers were first used approximately $100$ centuries ago in the Middle East for counting items. Farmers, cattlemen, and tradesmen utilized tokens, stones, or markers to signify specific quantities (e.g., a sheaf of grain or a head of livestock), which enabled commerce and the spread of civilization.

• Fractions: Introduced by the Egyptians between three and four thousand years ago, fractions were initially used to represent reciprocals and later to represent the division of a quantity into equal parts.

• Zero: The concept of zero as a numeral in calculations was added to the number system in India around the fifth century A.D.

• Negative Numbers: Negative numbers appeared in India in the seventh century A.D. to represent commercial debts and as solutions to equations.

• sites of natural numbers to the whole numbers: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$. This set contains three subsets: negative integers, zero, and positive integers.

• Rational Numbers: Defined as the set $\{\frac{m}{n} \mid m, n \in \text{integers}, n \neq 0\}$.

  • Every natural number, whole number, and integer is a rational number with a denominator of $1$.

  • Terminating Decimals: Decimals that end, such as $\frac{15}{8} = 1.875$.

  • Repeating Decimals: Decimals with a repeating block of numbers, such as $\frac{4}{11} = 0.363636\dots = 0.\overline{36}$. A line (overbar) indicates the repeating block.

• Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers. Their decimal forms are non-repeating and non-terminating. Examples include $\sqrt{11}$ and $0.3033033303333\dots$.

• Real Numbers: The combined set of rational and irrational numbers.

  • The Real Number Line: A horizontal line where every real number corresponds to a unique position, and every position corresponds to exactly one real number (one-to-one correspondence).

  • Classification Example: $\sqrt{25} = 5$ (rational, positive); $\sqrt{5}$ (irrational, positive); $-6\pi$ (irrational, negative).

Order of Operations and Algebraic Essentials

• Exponential Notation: The expression $a^n$ represents the base $a$ used as a factor $n$ times. $a^n$ is the $n\text{th}$ power of $a$.

• Grouping Symbols: Include parentheses $()$, brackets $[]$, braces $\{\}$, fraction bars, radicals, and absolute value bars.

• Order of Operations (PEMDAS):

  1. Parentheses: Simplify expressions within grouping symbols.

  2. Exponents: Simplify expressions with exponents or radicals.

  3. Multiplication and Division: Perform in order from left to right.

  4. Addition and Subtraction: Perform in order from left to right.

• Properties of Real Numbers (for all real numbers $a, b, c$):

  • Commutative Property: $a+b = b+a$ and $a \times b = b \times a$. Subtraction and division are NOT commutative.

  • Associative Property: $a+(b+c) = (a+b)+c$ and $a(bc) = (ab)c$. Subtraction and division are NOT associative.

  • Distributive Property: $a \times (b+c) = a \times b + a \times c$. Multiplication distributes over addition and subtraction.

  • Identity Property: Additive identity is $0$ ($a+0=a$); Multiplicative identity is $1$ ($a \times 1=a$).

  • Inverse Property: Additive inverse of $a$ is $-a$ ($a+(-a)=0$); Multiplicative inverse (reciprocal) of nonzero $a$ is $\frac{1}{a}$ ($a \times (\frac{1}{a}) = 1$).

• Algebraic Expressions: Collections of constants (fixed values) and variables (varying quantities) joined by operations. Evaluating an expression entails substituting specific values for variables and simplifying.

• Formulas: Equations expressing relationships between quantities. Example: Surface area of a cylinder $S = 2\pi r(r + h)$.

Exponents and Scientific Notation

• Product Rule: $a^m \times a^n = a^{m+n}$.

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ (where $a \neq 0$ and m > n).

• Power Rule: $(a^m)^n = a^{m \times n}$.

• Zero Exponent Rule: $a^0 = 1$ for any nonzero real number $a$. The expression $0^0$ is undefined.

• Negative Rule: $a^{-n} = \frac{1}{a^n}$ and $a^n = \frac{1}{a^{-n}}$ (where $a \neq 0$).

• Power of a Product Rule: $(ab)^n = a^n b^n$.

• Power of a Quotient Rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$.

• Scientific Notation: A number in the form $a \times 10^n$, where 1 \leq |a| < 10 and $n$ is an integer.

  • Standard to Scientific: Move the decimal to the right of the first digit. Moving left results in a positive $n$ (large numbers); moving right results in a negative $n$ (small numbers).

  • Calculations: When performing operations with scientific notation, multiply/divide the decimals and use exponent rules for powers of $10$. Adjust the final result to proper scientific notation if necessary.

Radicals and Rational Expressions

• Principal Square Root: The nonnegative number $\sqrt{a}$ that, when multiplied by itself, equals $a$.

• Product Rule for Square Roots: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ (for $a, b \geq 0$).

• Quotient Rule for Square Roots: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ (for $b \neq 0$).

• Rationalizing Denominators:

  • Single term: For $\frac{1}{\sqrt{c}}$, multiply by $\frac{\sqrt{c}}{\sqrt{c}}$.

  • Two terms: For $\frac{1}{a + \sqrt{c}}$, multiply by the conjugate $\frac{a - \sqrt{c}}{a - \sqrt{c}}$.

• nth Roots: The principal $n\text{th}$ root $\sqrt[n]{a}$ is the number with the same sign as $a$ that, when raised to the $n\text{th}$ power, equals $a$.

• Rational Exponents: $a^{\frac{m}{n}} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$. The numerator is the power, and the denominator is the root index.

Polynomials and Operations

• Polynomial Definition: A sum or difference of terms consisting of a variable raised to a nonnegative integer power: $a_nx^n + \dots + a_1x + a_0$.

  • Monomial: One term.

  • Binomial: Two terms.

  • Trinomial: Three terms.

  • Degree: The highest power of the variable.

  • Leading Coefficient: Key coefficient attached to the term with the highest degree.

• Multiplying Polynomials:

  • Distributive Property: Multiply each term of the first polynomial by each term of the second.

  • FOIL Method: For binomials only (First, Outer, Inner, Last).

• Special Products:

  • Perfect Square Trinomial: $(x + a)^2 = x^2 + 2ax + a^2$.

  • Difference of Squares: $(a + b)(a - b) = a^2 - b^2$.

Factoring Polynomials

• Greatest Common Factor (GCF): The largest polynomial that divides evenly into all terms of an expression. It should always be the first step in factoring.

• Trinomials ($x^2 + bx + c$): Find two numbers $p, q$ such that $pq = c$ and $p + q = b$.

• Factoring by Grouping: Used for trinomials $ax^2 + bx + c$ ($a \neq 1$). Find $p, q$ such that $pq = ac$ and $p + q = b$, then split the middle term and factor portions separately.

• Factoring Sum and Difference of Cubes:

  • Sum: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

  • Difference: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

  • SOAP Memo: Signs are Same, Opposite, Always Positive.

• Fractional/Negative Exponents: Factor by pulling out the variable raised to the lowest power value.

Rational Expressions

• Rational Expression: The quotient of two polynomial expressions.

• Simplifying: Factor numerator and denominator; cancel common factors (common factor divided by itself equals $1$).

• Operators:

  • Multiplication: Multiply numerators and denominators across.

  • Division: Multiply the first expression by the reciprocal of the second.

  • Addition/Subtraction: Requires finding the Least Common Denominator (LCD). Multiply each term by the form of $1$ necessary to reach the LCD.

• Complex Rational Expressions: Contains rational expressions within the numerator or denominator. Simplified by combining the top and bottom into single fractions, then dividing.