Chapter 1: Operations on Real Numbers and Algebraic Expressions

Section 1: Success in Mathematics

Success in mathematics requires preparation, organization, and active engagement. The following strategies help students maximize their learning and performance in algebra courses.

• First Week Tips: Choose a good seat, read the syllabus, connect with classmates, budget study time, and use a dedicated math notebook.

• Before Class: Read upcoming sections, prepare questions, and mentally prepare for learning.

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

• After Class: Review notes, reread sections, complete homework, analyze mistakes, and seek help if needed.

• Using the Text Effectively: Attend lectures, practice homework, check answers, and ask about unclear problems.

• Preparing for Exams: Review homework and chapter problems, test yourself, watch prep videos, and arrive prepared.

Section 2: Fractions, Decimals, and Percents

This section covers foundational concepts for working with fractions, decimals, and percents, including prime factorization, least common multiples, 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 Example: The prime factorization of 24 is 2 × 2 × 2 × 3.

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

• Equivalent Fractions: Fractions that represent the same value. Example: is equivalent to .

• Least Common Denominator (LCD): The LCD is the LCM of the denominators of a group of fractions.

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

• Decimal Places: Decimals are named by their place value: tenths, hundredths, thousandths, etc.

• Rounding Decimals: Identify the place value, check the digit to the right, round up if 5 or more, otherwise leave as is.

• Converting Fractions to Decimals: Divide the numerator by the denominator.

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

• Converting Decimals to Fractions: Use the place value as the denominator and reduce to lowest terms.

• Percent: Percent means parts per hundred. To convert percent to decimal, divide by 100.

Section 3: The Number Systems and the Real Number Line

Understanding the classification of numbers and their representation on the real number line is fundamental in algebra.

• Set Notation: A set is a collection of objects, written with braces. The empty set is denoted by { } or a special symbol.

• Classification of Numbers:

  • Natural Numbers: {1, 2, 3, ...}

  • Whole Numbers: {0, 1, 2, 3, ...}

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

  • Rational Numbers: Numbers that can be written as , where p and q are integers.

  • Irrational Numbers: Numbers that cannot be written as a fraction of two integers.

  • Real Numbers: All rational and irrational numbers.

• The Real Number Line: Real numbers are represented as points on a line. The origin is 0, negative numbers are left, positive numbers are right.

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

• Absolute Value: The absolute value is the distance from 0 to a on the number line.

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

Operations on signed numbers are essential for algebraic calculations. This section explains the rules and properties for these operations.

• Operations Symbols: Addition (+), Subtraction (–), Multiplication (ā), Division (/).

• Additive Inverse: The opposite of a number a is –a. .

• Subtracting Integers: ; subtract by adding the opposite.

• Multiplying Integers: Rules:

  • Product of two positives is positive.

  • Product of one positive and one negative is negative.

  • Product of two negatives is positive.

• Reciprocals: The reciprocal of a nonzero number a is .

• Dividing Integers: Rules:

  • Quotient of two positives is positive.

  • Quotient of one positive and one negative is negative.

  • Quotient of two negatives is positive.

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

Rational numbers include fractions and decimals. This section covers the arithmetic operations for these numbers.

• Multiplying Fractions: Multiply numerators and denominators.

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

• Adding/Subtracting Fractions: With same denominators, add/subtract numerators. With different denominators, find the LCD first.

• Adding/Subtracting Decimals: Align decimal points and add/subtract digits in like place values.

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

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

Section 6: Properties of Real Numbers

Understanding the properties of real numbers is crucial for simplifying and solving algebraic expressions.

• Identity Properties:

  • Additive Identity:

  • Multiplicative Identity:

• Commutative Properties: and

• Associative Properties: and

• Multiplication and Division Properties of Zero:

  • Multiplication:

  • Division: for ; is undefined.

Section 7: Exponents and the Order of Operations

Exponents and the order of operations are essential for evaluating mathematical expressions correctly.

• Exponents: means multiply a by itself n times. Example: .

• Order of Operations:

  1. Perform operations within grouping symbols first.

  2. Evaluate exponents.

  3. Perform multiplication and division from left to right.

  4. Perform addition and subtraction from left to right.

Section 8: Simplifying Algebraic Expressions

Algebraic expressions are simplified by evaluating, identifying like terms, using the distributive property, and combining like terms.

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

• Evaluating Expressions: Substitute values for variables and simplify.

• Like Terms: Terms with the same variable factors and exponents.

• Combining Like Terms: Add or subtract coefficients of like terms.

• Distributive Property: