Skip to main content
Mathematics LibreTexts

5.4: Summary

  • Page ID
    62294
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    • A valid deduction (or “result”) is usually called a theorem, proposition, corollary, or lemma.
    • Divisibility and congruence
      • Important definitions:
        • divisor, multiple
        • congruent modulo \(n: a \equiv b(\bmod n)\)
        • remainder
        • irrational number
      • Congruence \((\bmod n)\) is reflexive, symmetric, and transitive
      • \(a \equiv b (\bmod n)\) iff \(a\) and \(b\) have the same remainder when divided by \(n\)
      • \(\sqrt{2}\) is irrational
      • Notation:
        • \(a \mid b, a \nmid b\)
        • \(a \equiv b (\bmod n)\)
    • Commutative groups
      • Important definitions:
        • commutative group (commutative, associative, identity element, negatives)
        • subgroup (closed under negatives and addition)
      • The identity element of a group is unique.
      • The negative of each element of a group is unique.
      • Notation:
        • 0 (identity element)
        • \(−g\) (negative)
    • Convergent sequences
      • Important definitions:
        • absolute value
        • converges
      • triangle inequality
      • Notation:
        • \(|x|\)
        • \(a_{n} \rightarrow L\)

    This page titled 5.4: Summary is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Dave Witte Morris & Joy Morris.

    • Was this article helpful?