1: Sets and Numbers
- Page ID
- 173400
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: The Language of Algebra
- This page provides an overview of fundamental mathematical concepts such as definitions, axioms, theorems, and proofs. It underscores the necessity of clear definitions to ensure understanding and communication within mathematics, as well as the critical role of logical reasoning in evaluating theorems versus conjectures.
- 1.2: Sets and Set Notation
- This page covers fundamental concepts of sets in mathematics, defining sets as collections of elements and discussing key topics such as the empty set, finite and infinite sets, and set notations. It explores operations like intersection (common elements between sets) and union (combination of all elements), with illustrative examples. The page also addresses subsets, proper subsets, and properties of set equality and containment, providing a comprehensive overview of set theory basics.
- 1.3: Sets of Numbers
- This page provides an overview of different sets of numbers crucial for mathematics, including natural numbers, whole numbers, integers, rational, irrational, real, and complex numbers. It explains their definitions, relationships, and classifications, emphasizing their properties such as positivity and negativity. Examples and checkpoints facilitate reader engagement in identifying and classifying numbers, reinforcing key concepts essential for mathematical understanding.
- 1.4: Properties of Real Numbers
- This page covers the fundamental axioms of arithmetic for addition and multiplication in real numbers, detailing properties such as commutative, associative, identity, and inverse properties, as well as the distributive property, while noting that subtraction and division do not share these properties. It provides unique existence theorems and examples to illustrate these concepts.
- 1.5: Scientific Notation
- This page explains scientific notation as a method for expressing large or small numbers as \( m \times 10^p \) for easier calculation and comparison. It provides an example of converting \(0.5\) into scientific notation as \(5 \times 10^{-1}\) and introduces a theorem for determining the order of magnitude using logarithms. The content is adapted from a Wikipedia article on scientific notation, which is shared under a Creative Commons license.
- 1.6: Inequalities and Inequality Notation
- This page covers number lines as visual tools for understanding numbers and their relationships, focusing on inequalities, which express comparisons between values. It differentiates between strict and non-strict inequalities and explains their graphing on number lines. The page also highlights key properties of inequalities, such as converse and transitivity, and discusses how to manipulate and interpret them.
- 1.7: Intervals and Interval Notation
- This page provides a comprehensive overview of mathematical intervals, defining them as subsets of real numbers between specific endpoints. It outlines open, closed, and half-open intervals with examples, discusses supremum and infimum as endpoints, and explains unions and intersections with interval notation. It includes additional examples and practice checkpoints to enhance understanding of writing intervals under various conditions.
- 1.8: Set-Builder Notation
- This page explains set-builder notation, which defines sets based on element properties. It covers the notation format \(\{ x \mid \text{condition} \}\), explains the significance of the domain, and includes examples of converting to roster notation and solving predicates. The page also highlights equivalent predicates that can define the same sets.
- 1.9: Complex Numbers
- This page introduces complex numbers and the imaginary unit \(i\), detailing their definition, representation in the complex plane, and key operations such as addition, subtraction, multiplication, and division. It covers the powers of \(i\), modulus of complex numbers, and solutions to quadratic equations resulting in complex numbers. Real solutions are deemed nonexistent for negative squares.