Chapter 1: Operations on Real Numbers and Algebraic Expressions

Section 1: Success in Mathematics

This section provides strategies for achieving success in a mathematics course, focusing on preparation, participation, and effective study habits.

• First Week Tips: Choose a good seat, read the syllabus, learn classmates' names, budget your time, and use a dedicated math notebook.

• Before Class: Read upcoming sections, prepare questions, and be mentally ready.

• During Class: Arrive early, avoid distractions, stay alert, take thorough notes, and ask questions.

• After Class: Reread notes, redo homework, analyze mistakes, and seek help if needed.

• Using the Text: Attend lectures, do homework, check answers, and review similar problems if errors occur.

• Preparing for Exams: Review homework and chapter tests, simulate test conditions, and use available resources like videos.

Section 2: Fractions, Decimals, and Percents

This section introduces foundational concepts of fractions, decimals, and percents, including their properties and conversions.

• Prime and Composite Numbers: A prime number has only two factors: 1 and itself. A composite number has more than two factors. 1 is neither prime nor composite.

• Prime Factorization: Express a number as a product of prime numbers. Example: 24 = 2 × 2 × 2 × 3

• Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers. Steps: Write each number as a product of primes, align common factors, multiply the highest powers of all primes present.

• Equivalent Fractions: Fractions that represent the same value. Example: \( \frac{5}{8} = \frac{15}{24} \)

• Lowest Terms: A fraction is in lowest terms if numerator and denominator have no common factors other than 1.

• Decimal Places: Tenths, hundredths, thousandths, etc., indicate the position of digits after the decimal point.

• Rounding Decimals: Identify the place value, check the digit to the right, and round accordingly.

• Converting Fractions and Decimals: Divide numerator by denominator to convert to decimal; write decimal as a fraction using place value and reduce.

• Terminating and Repeating Decimals: Terminating decimals end; repeating decimals have a digit or group of digits that repeat indefinitely.

• Percents: "Percent" means per hundred. Convert between decimals and percents by multiplying or dividing by 100.

Section 3: The Number Systems and the Real Number Line

This section explores the classification of numbers, set notation, and the real number line.

• Set Notation: A set is a collection of objects, written with braces. The empty set is denoted by { } or \( \emptyset \).

• Classification of Numbers:

  • Natural Numbers: {1, 2, 3, ...}

  • Whole Numbers: {0, 1, 2, 3, ...}

  • Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}

  • Rational Numbers: Can be written as \( \frac{p}{q} \), where p and q are integers and q ≠ 0.

  • Irrational Numbers: Cannot be written as a fraction of integers; non-repeating, non-terminating decimals.

  • Real Numbers: All rational and irrational numbers.

• The Real Number Line: A visual representation of real numbers as points on a line. The origin is 0; numbers increase to the right and decrease to the left.

• Inequalities: Use symbols <, >, = to compare numbers based on their position on the number line.

• Absolute Value: The distance from 0 to a number on the number line, always non-negative. \( |a| \geq 0 \).

Section 4: Adding, Subtracting, Multiplying, and Dividing Integers

This section covers the basic operations with integers, including rules for signs and properties of operations.

• Operations Symbols: Addition (+), Subtraction (–), Multiplication (· or ×), Division (÷ or /).

• Adding Integers: Use the number line; adding a positive moves right, adding a negative moves left.

• Additive Inverse: For any real number a, its additive inverse is –a. \( a + (–a) = 0 \).

• Subtracting Integers: \( a – b = a + (–b) \). Subtract by adding the opposite.

• Multiplying Integers: Follow the rules of signs:

  • Product of two positives: positive

  • Product of one positive and one negative: negative

  • Product of two negatives: positive

• Reciprocals: The multiplicative inverse of a nonzero number a is \( \frac{1}{a} \). \( a \cdot \frac{1}{a} = 1 \).

• Dividing Integers: The quotient of two numbers follows the same sign rules as multiplication.

Section 5: Adding, Subtracting, Multiplying, and Dividing Rational Numbers

This section extends operations to rational numbers, including fractions and decimals.

• Multiplying Fractions: Multiply numerators and denominators directly.

• Dividing Fractions: Multiply by the reciprocal of the divisor.

• Adding/Subtracting Fractions: With same denominators, add/subtract numerators. With different denominators, find the least common denominator (LCD) first.

• Adding/Subtracting Decimals: Align decimal points and add/subtract as with whole numbers.

• Multiplying Decimals: Multiply as whole numbers, then place the decimal point so the number of decimal places equals the sum of the decimal places in the factors.

• Dividing Decimals: Make the divisor a whole number by multiplying both numbers by a power of 10, then divide as with whole numbers.

Section 6: Properties of Real Numbers

This section introduces the fundamental properties that govern arithmetic operations with real numbers.

• Identity Properties:

  • Additive Identity: \( a + 0 = 0 + a = a \)

  • Multiplicative Identity: \( a \cdot 1 = 1 \cdot a = a \)

• Commutative Properties: Order does not affect the result.

  • Addition: \( a + b = b + a \)

  • Multiplication: \( a \cdot b = b \cdot a \)

• Associative Properties: Grouping does not affect the result.

  • Addition: \( a + (b + c) = (a + b) + c \)

  • Multiplication: \( a \cdot (b \cdot c) = (a \cdot b) \cdot c \)

• Properties of Zero:

  • Multiplication: \( a \cdot 0 = 0 \cdot a = 0 \)

  • Division: \( \frac{0}{a} = 0 \) for \( a \neq 0 \); \( \frac{a}{0} \) is undefined.

Section 7: Exponents and the Order of Operations

This section explains how to evaluate exponential expressions and the correct sequence for performing operations in mathematical expressions.

• Exponents: An exponent indicates how many times a base is multiplied by itself. \( a^n = a \cdot a \cdot \ldots \cdot a \) (n factors)

• Order of Operations:

  1. Evaluate expressions inside grouping symbols (parentheses, brackets).

  2. Evaluate exponents.

  3. Multiply and divide from left to right.

  4. Add and subtract from left to right.

Section 8: Simplifying Algebraic Expressions

This section introduces algebraic expressions, variables, constants, and the process of simplifying expressions using properties and combining like terms.

• Algebraic Expressions: Combinations of variables, constants, and operations.

• Variables: Letters representing numbers; constants are fixed values.

• Evaluating Expressions: Substitute values for variables and simplify.

• Like Terms: Terms with the same variable(s) and exponent(s); can be combined by adding/subtracting coefficients.

• Distributive Property: \( a(b + c) = ab + ac \); multiply each term inside parentheses by the factor outside.

• Simplifying: Combine like terms and use the distributive property to write expressions in simplest form.