Skip to main content
Mathematics LibreTexts

15.4: Chapter 4

  1. Last updated
  2. Save as PDF
  • Page ID
    129933

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Your Turn

    4.1
    1. 416
    2. 1,851
    3. 17,488
    4.2
    1. /**/9 \times {10^2} + 2 \times {10^1} + 4 \times {10^0}/**/
    2. /**/1 \times {10^3} + 2 \times {10^2} + 7 \times {10^1} + 9 \times {10^0}/**/
    3. /**/4 \times {10^6} + 1 \times {10^5} + 3 \times {10^4} + 0 \times {10^3} + 0 \times {10^2} + 4 \times {10^1} + 5 \times {10^0}/**/
    4.3
    1. 621
    2. 3,203
    3. 40,630,891
    4.4
    1. 1269
    4.5
    1. 42,136
    4.6
    1. 6,105,643
    4.7
    1. 257
    4.8
    1. 6,054
    4.9
    1. 1,248,073
    4.10
    1. 77
    2. 240
    3. 3,447
    4.11
    1. XXVII
    2. XLIX
    3. DCCXXXIX
    4. MMMDCXLVII
    4.12
    1. 0, 1, 2, 3
    4.13
    1. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B
    4.14
    1. 157
    4.15
    1. 2,014
    4.16
    1. 851
    4.17
    1. 27
    4.18
    1. 0, 1, 2, 3
    10, 11, 12, 13
    20, 21, 22, 23
    30, 31, 32, 33
    100
    4.19
    1.

    0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B

    10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B

    20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B

    30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3A, 3B

    40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4A, 4B

    50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5A, 5B

    60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6A, 6B

    70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7A, 7B

    80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8A, 8B

    90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9A, 9B

    A0, A1, A2, A3, A4, A5, A6, A7, A8, A9, AA, AB

    B0, B1, B2, B3, B4, B5, B6, B7, B8, B9, BA, BB

    100

    4.20
    1. 0, 1, 2, 10, 11, 12, 20, 21, 22, 100
    4.21
    1. 20107
    4.22
    1. 554B12
    4.23
    1. 100010012
    4.24
    1. Mayan numeral 17 is displayed.
    Mayan numeral 6 is displayed.
    2. Mayan numeral 11 is displayed.
    Mayan numeral 8 is displayed.
    Mayan numeral 5 is displayed.
    4.25
    1. The result has the digit 7 in it. In base 4, the 7 is an illegal symbol.
    4.26
    1. The remainders include 10, which in base 6 is an illegal symbol.
    4.27
    1. Since 12 is larger than 10, the base 10 number cannot have less digits than the base 12 number. Since it did, we know an error has been made.
    4.28
    1. 12426
    4.29
    1.
    + 0 1 2 3
    0 0 1 2 3
    1 1 2 3 10
    2 2 3 10 11
    3 3 10 11 12
    Base 4 Addition Table
    4.30
    1. 6337
    4.31
    1.
    + 0 1 2 3 4 5 6 7 8 9 A B C D
    0 0 1 2 3 4 5 6 7 8 9 A B C D
    1 1 2 3 4 5 6 7 8 9 A B C D 10
    2 2 3 4 5 6 7 8 9 A B C D 10 11
    3 3 4 5 6 7 8 9 A B C D 10 11 12
    4 4 5 6 7 8 9 A B C D 10 11 12 13
    5 5 6 7 8 9 A B C D 10 11 12 13 14
    6 6 7 8 9 A B C D 10 11 12 13 14 15
    7 7 8 9 A B C D 10 11 12 13 14 15 16
    8 8 9 A B C D 10 11 12 13 14 15 16 17
    9 9 A B C D 10 11 12 13 14 15 16 17 18
    A A B C D 10 11 12 13 14 15 16 17 18 19
    B B C D 10 11 12 13 14 15 16 17 18 19 1A
    C C D 10 11 12 13 14 15 16 17 18 19 1A 1B
    D D 10 11 12 13 14 15 16 17 18 19 1A 1B 1C
    Base 14 Addition Table
    4.32
    1. 13B912
    4.33
    1. 100100102
    4.34
    1. 326
    4.35
    1. 22712
    4.36
    1. The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 addition table would correct this error.
    1 3 3
    + 1 1 2
    3 1 1
    The correct answer is 3114.
    4.37
    1. This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, 9 + 9 is not 18, but instead is 13. Careful use of the base 14 addition table generates the correct answer, /**/{163_{14}}/**/.
    4.38
    1.
    * 0 1 2 3
    0 0 0 0 0
    1 0 1 2 3
    2 0 2 10 12
    3 0 3 12 21
    Base 4 Multiplication Table
    4.39
    1.
    * 0 1 2 3 4 5 6 7 8 9 A B C D
    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
    1 0 1 2 3 4 5 6 7 8 9 A B C D
    2 0 2 4 6 8 A C 10 12 14 16 18 1A 1C
    3 0 3 6 9 C 11 14 17 1A 1D 22 25 28 2B
    4 0 4 8 C 12 16 1A 20 24 28 2C 32 36 3A
    5 0 5 A 11 16 1B 22 27 2C 33 38 3D 44 49
    6 0 6 C 14 1A 22 28 30 36 3C 44 4A 52 58
    7 0 7 10 17 20 27 30 37 40 47 50 57 60 67
    8 0 8 12 1A 24 2C 36 40 48 52 5A 64 6C 76
    9 0 9 14 1D 28 33 3C 47 52 5B 66 71 7A 85
    A 0 A 16 22 2C 38 44 50 5A 66 72 7C 88 94
    B 0 B 18 25 32 3D 4A 57 64 71 7C 89 96 A3
    C 0 C 1A 28 36 44 52 60 6C 7A 88 96 A4 B2
    D 0 D 1C 2B 3A 49 58 67 76 85 94 A3 B2 C1
    4.40
    1. 40006
    2. 10101112
    4.41
    1. 436912
    4.42
    1. /**/{10_6} \div {3_6} = {2_6}/**/
    2. /**/{50_{12}} \div
    ParseError: invalid DekiScript (click for details)
    Callstack:
    at (Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/15:_Answer_Key/15.04:_Chapter_4), /content/body/div/div/div/div[42]/div[2]/span/span, line 1, column 1
    = {6_{12}}/**/
    4.43
    1. The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 multiplication table would correct this error. The correct answer is 3334.
    4.44
    1. This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, /**/{49_{14}} \times {9_{14}} = 2{\text{D}}
    ParseError: invalid DekiScript (click for details)
    Callstack:
    at (Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/15:_Answer_Key/15.04:_Chapter_4), /content/body/div/div/div/div[44]/div/span[1]/span, line 1, column 1
    /**/     is not 81, but instead is 5B. Careful use of the base 14 addition table (Table 4.9) generates the correct answer, /**/{49_{14}} \times {9_{14}} = 2{\text{D}}
    ParseError: invalid DekiScript (click for details)
    Callstack:
    at (Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/15:_Answer_Key/15.04:_Chapter_4), /content/body/div/div/div/div[44]/div/span[2]/span, line 1, column 1
    /**/    .

    Check Your Understanding

    1. A system in which the position of a numeral determines the value associated with that numeral.
    2. 279
    3. /**/4 \times {10^4} + 5 \times {10^3} + 2 \times {10^2} + 0 \times {10^1} + 9 \times {10^0}/**/
    4. 10
    5. A numeral is a symbol representing a number. A number is a quantity or amount.
    6. 601,947
    7. 60
    8. 20
    9. Roman numerals do not use place value.
    10. 341
    11. 209
    12. 247
    13. CDLXXIX
    14. A base 25 system would require 25 symbols.
    15. Since the 4 is the second digit, its place value is 181 times 4, or 72.
    16.

    329

    17. 409
    18. 5118
    19. 2,126
    20. In base 4, 5 is not a valid symbol. So, a mistake has been made.
    21.
    + 0 1 2 3 4 5 6 7
    0 0 1 2 3 4 5 6 7
    1 1 2 3 4 5 6 7 10
    2 2 3 4 5 6 7 10 11
    3 3 4 5 6 7 10 11 12
    4 4 5 6 7 10 11 12 13
    5 5 6 7 10 11 12 13 14
    6 6 7 10 11 12 13 14 15
    7 7 10 11 12 13 14 15 16
    22. 1216
    23. 78
    24. 8214
    25. 512
    26. In base 8, 8 is not a valid symbol. So, a mistake has been made.
    27. A common base 14 error is performing the operation in base 10.
    28. the addition table
    29. The process is the same, except the multiplication table for the base is used instead of the familiar base 10 rules.
    30. the multiplication table for the base
    31. 22406
    32. B14
    33. The 8 is not a symbol used in base 5.
    34. A symbol that is not used in that base is present, or a base 10 rule is used.

    This page titled 15.4: Chapter 4 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      Author
      OpenStax
      License
      CC BY
      License Version
      4.0
      OER program or Publisher
      OpenStax
      Show Page TOC
      no
    2. Tags
      1. source@https://openstax.org/details/books/contemporary-mathematics