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Mathematics LibreTexts

2: Arithmetic

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    176506

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    • 2.1: Factors, Prime Factorizations, and Least Common Multiples
      This page explains prime and composite numbers, highlighting the Fundamental Theorem of Arithmetic which asserts that every integer greater than 1 is either prime or can be uniquely factored into primes. It delves into least common multiples (LCM) and greatest common factors (GCF), illustrating their determination through prime factorization.
    • 2.2: Mathematical Operations - Addition, Subtraction, Multiplication, and Division
      This page covers basic mathematical operations: addition, subtraction, multiplication, and division. It defines these operations, detailing properties such as commutativity and associativity, particularly for addition and multiplication. Subtraction is noted as anti-commutative. The page further explains multiplication as repeated addition and presents the process of division with examples.
    • 2.3: Factorials
      This page defines factorials (\( n! \)), providing both non-recursive and recursive definitions, including the special case \( 0! = 1 \). It introduces the recurrence relation \( n! = n(n-1)! \) and offers examples for evaluating factorials. Common mistakes regarding factorial distribution over addition are also noted. Overall, the page underscores the importance of factorials in mathematics.
    • 2.4: Mathematical Operations - Powers and Square Roots
      This page covers exponentiation as repeated multiplication of a base \(b\) raised to a natural number power \(n\), explaining the notation \(b^n\) and geometric interpretations of squares and cubes. It defines square roots with a focus on the principal square root, provides examples for evaluating powers and square roots, clarifies the distinction between \((-b)^n\) and \(-b^n\), and introduces estimation techniques for square roots.
    • 2.5: The Order of Operations
      This page outlines the order of operations in mathematics, which dictates the sequence for evaluating arithmetic expressions. It emphasizes precedence levels, with multiplication and exponentiation prioritized over addition and subtraction, while parentheses can alter this order. The hierarchy is defined as: Parentheses, Exponentiation, Multiplication/Division (left to right), and Addition/Subtraction (left to right).
    • 2.6: Simplifying Fractions and Mixed Numbers
      This page covers key definitions and theorems about fractions, including proportions, equivalent fractions, and types of fractions like proper, improper, and mixed numbers. It stresses the importance of simplifying fractions and offers warnings regarding factor cancellation. Theorems on creating equivalent fractions and converting between improper fractions and mixed numbers are discussed, with practical examples that guide readers on finding equivalents and reducing to simplest form.
    • 2.7: Multiplying and Dividing Fractions
      This page explains multiplication and division of fractions, detailing how to multiply numerators and denominators and simplify through cross-cancellation. It discusses the scaling effect and the "keep-change-flip" rule for division by the reciprocal. The concept of reciprocals is defined, accompanied by examples, while highlighting that a common denominator is not required for these operations.
    • 2.8: Adding and Subtracting Fractions
      This page explains the fundamental concepts of adding and subtracting fractions, highlighting the need for a common denominator and defining essential terms such as equivalent fractions and least common denominator (LCD). It describes theorems for handling fractions with both like and unlike denominators, and includes practical examples to illustrate these procedures, stressing the importance of reducing fractions to their lowest terms.
    • 2.9: Mathematical Operations - Higher-Index Roots
      This page explores nth roots, defined as numbers fulfilling the equation \(y^n = x\) with a positive integer \(n\). It introduces radical notation (\(\sqrt[n]{x}\)) and highlights the principal nth root's dependence on the parity of \(n\). The content covers properties and theorems about nth roots, including examples and manipulation through radical expressions and rational exponents, while warning against potential misunderstandings with negative radicands.
    • 2.10: The Absolute Value of a Number
      This page explains absolute value (|x|) as the non-negative magnitude of a real number, highlighting its significance as the distance from zero on the number line. It covers essential properties like non-negativity and the triangle inequality, stresses the importance of order of operations in evaluating expressions, and provides examples for calculating absolute values and applying them in distance problems.
    • 2.11: Ratios, Proportions, and Proportional Reasoning
      This page covers essential concepts of ratios and proportions, explaining that a ratio compares two quantities while a proportion equates two ratios. It emphasizes simplifying ratios and maintaining order for meaningful comparisons. The page details directly and inversely proportional relationships, introduces the Cross-Products Property, and outlines related theorems, supported by examples that demonstrate practical applications of these ideas.
    • 2.12: The Arithmetic of Complex Numbers
      This page introduces complex numbers, defined as \(a + bi\) where \(i\) represents the imaginary unit. It covers their properties, equality, and operations like addition, subtraction, and multiplication, paralleling real-number rules. The text also emphasizes caution when handling square roots of negative numbers and provides examples to illustrate operations and simplification techniques involving \(i\).
    • 2.13: Percents
      This page explains the concept of percentage as a ratio out of 100 and details the conversion process between percentages, decimals, and fractions. It highlights the significance of accurate conversions and common mistakes. Theorems on percent proportion clarify the meanings of "is" and "of" in percentage problems, aiding in solving for unknowns. Practical examples illustrate these concepts effectively.
    • 2.14: Unit Analysis and Unit Conversion
      This page explains unit analysis fundamentals, defining key concepts and theorems for unit conversion. It clarifies dimensions and emphasizes that only comparable quantities can be combined. A conversion factor, treated as one, is introduced. Theorems ensure dimensional consistency in equations, while cautions about squared units and temperature conversions are noted. Various examples demonstrate practical applications of unit conversions.
    • 2.15: Measurement, Approximation, Rounding, and Significant Figures
      This page explains key concepts in measurements, including approximation, accuracy, precision, significant figures, and rounding. It clarifies that approximation is a close but not exact value, differentiates between accuracy (closeness to true value) and precision (consistency), and outlines rules for identifying significant figures.

    This page titled 2: Arithmetic is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Roy Simpson, Cosumnes River College.