(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 220013, 6183]*) (*NotebookOutlinePosition[ 220720, 6207]*) (* CellTagsIndexPosition[ 220676, 6203]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Introduction to ", StyleBox["Mathematica", FontSlant->"Italic"], "\nBy Raymond Cheong\nSeptember 10, 2003" }], "Title"], Cell[CellGroupData[{ Cell["Layout of Mathematica", "Section"], Cell[TextData[{ "This file is a ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook, which is a text file with the extension \".nb\". Usually you \ will save your ", StyleBox["Mathematica", FontSlant->"Italic"], " session as a notebook. If you modify, run, save, then reopen this \ notebook the changes you see on the screen will be preserved. However, no \ variables and function definitions are loaded into memory when you open a \ notebook. The notebook or parts of it must be re-run to load things into \ memory.\nMathematica has a \"front end\" and a \"kernel\". When you start ", StyleBox["Mathematica", FontSlant->"Italic"], " or open a notebook you are starting the front end, which acts as an GUI \ interface to the kernel. The kernel isn't loaded until you perform your \ first command, and it is the kernel that actually stores all data and \ performs all of the calculations. Naturally, the front end and kernel must \ communicate; you give a command to the front end, this is passed as input to \ the kernel, then the output of the kernel is sent back to the front end and \ the front end displays it.\nThe practical implications of this setup are as \ follows. ", StyleBox["Mathematica", FontSlant->"Italic"], " can appear to hang if the kernel is doing a very intensive calculation, \ or if the front end is given an extremely long and complicated output to \ display. In the former case, you can interrupt or abort the calculation \ without losing your front end session, notebook changes, or data in memory. \ You can even kill the kernel and save the front end session and notebook \ changes, but your data in memory will be lost (see the commands under the \ Kernel menu). On the other hand if the front end hangs, you either must wait \ or end your session and you will lose everything.\nUsually only one kernel is \ running at a time. Since the kernel stores all data, variables and function \ definitions are global across all notebooks.\nThe menus at the top allow you \ to access all features of the front end. Besides the expected commands, we \ will focus on just a few items:\n\tUnder File: Palettes (symbols and 2D \ expression templates)\n\tUnder Edit: Copy As (use to export output or images)\ \n\tUnder Format: Style (use \"Text\" to make a commenting cell like this \ one)\n\tUnder Kernel: Evaluation|Evaluate Notebook (does what it says), \ Abort Evaluation (does what it says), Quit Kernel (does what it says)\n\t\ Under Help: Help Browser (extremely useful documentation found here)" }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " notebooks are organized into cells. The blue brackets on the right hand \ margin indicates the arrangment of cells. This (text) cell is denoted by the \ bracket that extends for exactly the length of this cell. Cells can be \ grouped heirarchically (see Format|Style), and this is denoted with brackets \ to the right that cover multiple cells. You can double-click on such \ brackets to collapse or expand groups of cells. Cell grouping is a feature \ that you will probably not use when you first learn to use ", StyleBox["Mathematica", FontSlant->"Italic"], ", but you need to understand cell grouping in order to navigate this \ notebook." }], "Text"], Cell["\<\ Let's get started. Collapse this section and open the section titled \"Using \ Mathematica as a calculator\"\ \>", "Text", FontWeight->"Bold"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Using ", StyleBox["Mathematica", FontSlant->"Italic"], " as a calculator" }], "Section"], Cell[CellGroupData[{ Cell["Simple calculations", "Subsubsection"], Cell["\<\ Click on the cell below and hit Shift-Enter to run it. You will see the \ result of the calculation as output just below the cell.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(1 + 1\)], "Input"], Cell[BoxData[ \(2\)], "Output"] }, Open ]], Cell[TextData[{ "You can put multiple calculations in one cell. You can put multiple \ calculations in one line using a list, { }. Some comments:\n\tLine 1) If \ there is nothing to evaluate, the output is the same as the input\n\tLine 2, \ 3) If there is something to evaluate, the output reflects the result of the \ evaluation\n\tLine 4) The ability to evaluate can depend on the values you \ use\n\tLine 5) ", StyleBox["Mathematica", FontSlant->"Italic"], " supports all standard functions (and more). They all begin with a \ capital letter, see Help for more. You can get the \[Pi] character by \ pressing Esc-p-Esc\n\tLine 6) You can enter expressions using the Palettes \ and ", StyleBox["Mathematica", FontSlant->"Italic"], " can evaluate them just fine" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(1\), "\[IndentingNewLine]", \(1 + 1\), "\[IndentingNewLine]", \(1 + 3^2\), "\[IndentingNewLine]", \({Sqrt[4], Sqrt[3]}\), "\[IndentingNewLine]", \(Sin[2 \[Pi]]\), "\[IndentingNewLine]", \(1\/2\^10\)}], "Input"], Cell[BoxData[ \(1\)], "Output"], Cell[BoxData[ \(2\)], "Output"], Cell[BoxData[ \(10\)], "Output"], Cell[BoxData[ \({2, \@3}\)], "Output"], Cell[BoxData[ \(0\)], "Output"], Cell[BoxData[ \(1\/1024\)], "Output"] }, Open ]], Cell["\<\ Semicolon suppresses the output; % represents the most recent output \ (likewise %% represents the next-to-last output, etc.)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(3 + 3;\)\), "\[IndentingNewLine]", \(%\)}], "Input"], Cell[BoxData[ \(6\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Basic plotting (see Help for more options)", "Subsubsection"], Cell["\<\ Here is a plot of the Sinc function. 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.38919 L s .0139 .40189 m .0176 .40054 L s .01209 .41332 m .01579 .41198 L s .00843 .43642 m .01214 .4351 L s .00658 .4481 m .01029 .44678 L s .00472 .45986 m .00843 .45855 L s .00284 .4717 m .00656 .4704 L s .25 Mabswid .03716 .25514 m 0 .48963 L s 0 .48963 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s .40529 .59895 m .03716 .25514 L s .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .40529 .59895 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .39787 .81114 L s .39787 .81114 m .40529 .59895 L s 0 0 m 1 0 L 1 .81114 L 0 .81114 L closepath clip newpath .5 Mabswid 0 .191 .72 r .38681 .73482 .3999 .75306 .42145 .77321 .40833 .75488 Metetra .156 .413 .855 r .40833 .75488 .42145 .77321 .44366 .78665 .43054 .76822 Metetra .534 .617 .881 r .43054 .76822 .44366 .78665 .4665 .78779 .45344 .76925 Metetra .723 .69 .818 r .45344 .76925 .4665 .78779 .48978 .77447 .47682 .75581 Metetra .797 .708 .767 r .47682 .75581 .48978 .77447 .5132 .74865 .50039 .72986 Metetra .824 .713 .743 r .50039 .72986 .5132 .74865 .53654 .71583 .52391 .69691 Metetra .824 .713 .743 r .52391 .69691 .53654 .71583 .55973 .68321 .54728 .66416 Metetra .797 .708 .766 r .54728 .66416 .55973 .68321 .58289 .6577 .57058 .63853 Metetra .723 .686 .814 r .57058 .63853 .58289 .6577 .60627 .64417 .59407 .62488 Metetra .555 .617 .869 r .59407 .62488 .60627 .64417 .63017 .64443 .61804 .62501 Metetra .279 .468 .866 r .61804 .62501 .63017 .64443 .65483 .65684 .64273 .6373 Metetra .055 .326 .808 r .64273 .6373 .65483 .65684 .6803 .67665 .66821 .657 Metetra .055 .326 .808 r .66821 .657 .6803 .67665 .70639 .69694 .69431 .67717 Metetra .273 .465 .866 r .69431 .67717 .70639 .69694 .73268 .71032 .72064 .69043 Metetra .547 .617 .874 r .72064 .69043 .73268 .71032 .75866 .71102 .74669 .691 Metetra .723 .694 .823 r .74669 .691 .75866 .71102 .78388 .6968 .77204 .67664 Metetra .806 .722 .772 r .77204 .67664 .78388 .6968 .80818 .66969 .79651 .6494 Metetra .837 .73 .745 r .79651 .6494 .80818 .66969 .83177 .63537 .82028 .61494 Metetra .837 .73 .745 r .82028 .61494 .83177 .63537 .8552 .60129 .8439 .58071 Metetra .805 .721 .772 r .8439 .58071 .8552 .60129 .87922 .57452 .86808 .55381 Metetra .723 .691 .819 r .86808 .55381 .87922 .57452 .90454 .56008 .89353 .53922 Metetra .566 .616 .862 r .89353 .53922 .90454 .56008 .9317 .55979 .92079 .5388 Metetra .359 .499 .863 r .92079 .5388 .9317 .55979 .96083 .572 .94998 .55086 Metetra .214 .407 .837 r .94998 .55086 .96083 .572 .99157 .59183 .98078 .57055 Metetra .219 .325 .759 r .37408 .70435 .38681 .73482 .40833 .75488 .3955 .7241 Metetra .381 .425 .782 r .3955 .7241 .40833 .75488 .43054 .76822 .41764 .73717 Metetra .566 .532 .776 r .41764 .73717 .43054 .76822 .45344 .76925 .44051 .73801 Metetra .686 .595 .748 r .44051 .73801 .45344 .76925 .47682 .75581 .46392 .72446 Metetra .748 .625 .723 r .46392 .72446 .47682 .75581 .50039 .72986 .48757 .6985 Metetra .774 .637 .709 r .48757 .6985 .50039 .72986 .52391 .69691 .51118 .66556 Metetra .774 .637 .709 r .51118 .66556 .52391 .69691 .54728 .66416 .53464 .63284 Metetra .748 .625 .723 r .53464 .63284 .54728 .66416 .57058 .63853 .55802 .60718 Metetra .687 .595 .747 r .55802 .60718 .57058 .63853 .59407 .62488 .58154 .59343 Metetra .576 .536 .773 r .58154 .59343 .59407 .62488 .61804 .62501 .60549 .59336 Metetra .427 .449 .78 r .60549 .59336 .61804 .62501 .64273 .6373 .63011 .60537 Metetra .311 .378 .77 r .63011 .60537 .64273 .6373 .66821 .657 .6555 .62473 Metetra .311 .378 .77 r .6555 .62473 .66821 .657 .69431 .67717 .68151 .64456 Metetra .424 .448 .781 r .68151 .64456 .69431 .67717 .72064 .69043 .70777 .65751 Metetra .572 .534 .774 r .70777 .65751 .72064 .69043 .74669 .691 .73381 .65787 Metetra .686 .596 .75 r .73381 .65787 .74669 .691 .77204 .67664 .75919 .6434 Metetra .751 .629 .724 r .75919 .6434 .77204 .67664 .79651 .6494 .78373 .61613 Metetra .78 .642 .709 r .78373 .61613 .79651 .6494 .82028 .61494 .80761 .58169 Metetra .78 .642 .709 r .80761 .58169 .82028 .61494 .8439 .58071 .83133 .54749 Metetra .751 .628 .724 r .83133 .54749 .8439 .58071 .86808 .55381 .85558 .52057 Metetra .686 .595 .749 r .85558 .52057 .86808 .55381 .89353 .53922 .88108 .50587 Metetra .581 .538 .771 r .88108 .50587 .89353 .53922 .92079 .5388 .90832 .50522 Metetra .46 .466 .778 r .90832 .50522 .92079 .5388 .94998 .55086 .93746 .51696 Metetra .377 .415 .773 r .93746 .51696 .94998 .55086 .98078 .57055 .96817 .53626 Metetra .314 .344 .733 r .36161 .66687 .37408 .70435 .3955 .7241 .38289 .68619 Metetra .431 .418 .744 r .38289 .68619 .3955 .7241 .41764 .73717 .40492 .6989 Metetra .571 .503 .74 r .40492 .6989 .41764 .73717 .44051 .73801 .42772 .6995 Metetra .671 .561 .722 r .42772 .6995 .44051 .73801 .46392 .72446 .45111 .68586 Metetra .728 .592 .704 r .45111 .68586 .46392 .72446 .48757 .6985 .47478 .65993 Metetra .752 .605 .694 r .47478 .65993 .48757 .6985 .51118 .66556 .49845 .6271 Metetra .752 .605 .694 r .49845 .6271 .51118 .66556 .53464 .63284 .52196 .59449 Metetra .728 .592 .704 r .52196 .59449 .53464 .63284 .55802 .60718 .54536 .56887 Metetra .672 .562 .722 r .54536 .56887 .55802 .60718 .58154 .59343 .56886 .55503 Metetra .579 .508 .739 r .56886 .55503 .58154 .59343 .60549 .59336 .59274 .55472 Metetra .462 .437 .745 r .59274 .55472 .60549 .59336 .63011 .60537 .61724 .56635 Metetra .375 .382 .74 r .61724 .56635 .63011 .60537 .6555 .62473 .64248 .58525 Metetra .375 .382 .74 r .64248 .58525 .6555 .62473 .68151 .64456 .66834 .6046 Metetra .46 .436 .745 r .66834 .6046 .68151 .64456 .70777 .65751 .69448 .61715 Metetra .575 .506 .74 r .69448 .61715 .70777 .65751 .73381 .65787 .72044 .61724 Metetra .67 .561 .722 r .72044 .61724 .73381 .65787 .75919 .6434 .7458 .60266 Metetra .729 .593 .704 r .7458 .60266 .75919 .6434 .78373 .61613 .77037 .57543 Metetra .756 .608 .693 r .77037 .57543 .78373 .61613 .80761 .58169 .79431 .54112 Metetra .756 .608 .693 r .79431 .54112 .80761 .58169 .83133 .54749 .81808 .50705 Metetra .729 .593 .704 r .81808 .50705 .83133 .54749 .85558 .52057 .84236 .48016 Metetra .671 .561 .722 r .84236 .48016 .85558 .52057 .88108 .50587 .86783 .46536 Metetra .583 .51 .738 r .86783 .46536 .88108 .50587 .90832 .50522 .89499 .46444 Metetra .485 .451 .744 r .89499 .46444 .90832 .50522 .93746 .51696 .924 .47577 Metetra .42 .41 .743 r .924 .47577 .93746 .51696 .96817 .53626 .95456 .49455 Metetra .314 .344 .733 r .34913 .62936 .36161 .66687 .38289 .68619 .37028 .64826 Metetra .431 .418 .744 r .37028 .64826 .38289 .68619 .40492 .6989 .39219 .6606 Metetra .571 .503 .74 r .39219 .6606 .40492 .6989 .42772 .6995 .41492 .66096 Metetra .671 .561 .722 r .41492 .66096 .42772 .6995 .45111 .68586 .43829 .64722 Metetra .728 .592 .704 r .43829 .64722 .45111 .68586 .47478 .65993 .46198 .62133 Metetra .752 .605 .694 r .46198 .62133 .47478 .65993 .49845 .6271 .4857 .58862 Metetra .752 .605 .694 r .4857 .58862 .49845 .6271 .52196 .59449 .50926 .55612 Metetra .728 .592 .704 r .50926 .55612 .52196 .59449 .54536 .56887 .53269 .53054 Metetra .672 .562 .722 r .53269 .53054 .54536 .56887 .56886 .55503 .55617 .5166 Metetra .579 .508 .739 r .55617 .5166 .56886 .55503 .59274 .55472 .57997 .51605 Metetra .462 .437 .745 r .57997 .51605 .59274 .55472 .61724 .56635 .60436 .52731 Metetra .375 .382 .74 r .60436 .52731 .61724 .56635 .64248 .58525 .62946 .54573 Metetra .375 .382 .74 r .62946 .54573 .64248 .58525 .66834 .6046 .65516 .56461 Metetra .46 .436 .745 r .65516 .56461 .66834 .6046 .69448 .61715 .68118 .57675 Metetra .575 .506 .74 r .68118 .57675 .69448 .61715 .72044 .61724 .70706 .57658 Metetra .67 .561 .722 r .70706 .57658 .72044 .61724 .7458 .60266 .7324 .56189 Metetra .729 .593 .704 r .7324 .56189 .7458 .60266 .77037 .57543 .757 .53471 Metetra .756 .608 .693 r .757 .53471 .77037 .57543 .79431 .54112 .78099 .50053 Metetra .756 .608 .693 r .78099 .50053 .79431 .54112 .81808 .50705 .80481 .46657 Metetra .729 .593 .704 r .80481 .46657 .81808 .50705 .84236 .48016 .82912 .43973 Metetra .671 .561 .722 r .82912 .43973 .84236 .48016 .86783 .46536 .85457 .42483 Metetra .583 .51 .738 r .85457 .42483 .86783 .46536 .89499 .46444 .88166 .42364 Metetra .485 .451 .744 r .88166 .42364 .89499 .46444 .924 .47577 .91054 .43454 Metetra .42 .41 .743 r .91054 .43454 .924 .47577 .95456 .49455 .94094 .45281 Metetra .224 .327 .76 r .33628 .59855 .34913 .62936 .37028 .64826 .35732 .61713 Metetra .384 .427 .782 r .35732 .61713 .37028 .64826 .39219 .6606 .37916 .62919 Metetra .567 .532 .776 r .37916 .62919 .39219 .6606 .41492 .66096 .40186 .62935 Metetra .686 .595 .748 r .40186 .62935 .41492 .66096 .43829 .64722 .42526 .61552 Metetra .748 .625 .723 r .42526 .61552 .43829 .64722 .46198 .62133 .44902 .5896 Metetra .774 .636 .709 r .44902 .5896 .46198 .62133 .4857 .58862 .47284 .55692 Metetra .774 .636 .709 r .47284 .55692 .4857 .58862 .50926 .55612 .4965 .52444 Metetra .748 .624 .723 r .4965 .52444 .50926 .55612 .53269 .53054 .52 .49884 Metetra .687 .595 .747 r .52 .49884 .53269 .53054 .55617 .5166 .5435 .4848 Metetra .577 .536 .772 r .5435 .4848 .55617 .5166 .57997 .51605 .56729 .48405 Metetra .429 .45 .78 r .56729 .48405 .57997 .51605 .60436 .52731 .59161 .49501 Metetra .314 .38 .771 r .59161 .49501 .60436 .52731 .62946 .54573 .61661 .51309 Metetra .314 .38 .771 r .61661 .51309 .62946 .54573 .65516 .56461 .64222 .53161 Metetra .427 .449 .78 r .64222 .53161 .65516 .56461 .68118 .57675 .66817 .54345 Metetra .573 .535 .774 r .66817 .54345 .68118 .57675 .70706 .57658 .69403 .54304 Metetra .686 .596 .749 r .69403 .54304 .70706 .57658 .7324 .56189 .7194 .52825 Metetra .751 .628 .724 r .7194 .52825 .7324 .56189 .757 .53471 .74408 .50104 Metetra .78 .642 .709 r .74408 .50104 .757 .53471 .78099 .50053 .76818 .46688 Metetra .78 .642 .709 r .76818 .46688 .78099 .50053 .80481 .46657 .79211 .43296 Metetra .751 .628 .724 r .79211 .43296 .80481 .46657 .82912 .43973 .81649 .40609 Metetra .686 .595 .748 r .81649 .40609 .82912 .43973 .85457 .42483 .84198 .39107 Metetra .582 .538 .771 r .84198 .39107 .85457 .42483 .88166 .42364 .86905 .38965 Metetra .462 .467 .778 r .86905 .38965 .88166 .42364 .91054 .43454 .89787 .40023 Metetra .379 .416 .773 r .89787 .40023 .91054 .43454 .94094 .45281 .92819 .4181 Metetra 0 .223 .744 r .32267 .57922 .33628 .59855 .35732 .61713 .34368 .59768 Metetra .192 .43 .86 r .34368 .59768 .35732 .61713 .37916 .62919 .36551 .60962 Metetra .544 .617 .876 r .36551 .60962 .37916 .62919 .40186 .62935 .38826 .60966 Metetra .723 .686 .814 r .38826 .60966 .40186 .62935 .42526 .61552 .41176 .59569 Metetra .795 .705 .765 r .41176 .59569 .42526 .61552 .44902 .5896 .43569 .56965 Metetra .821 .71 .742 r .43569 .56965 .44902 .5896 .47284 .55692 .45969 .53684 Metetra .821 .71 .742 r .45969 .53684 .47284 .55692 .4965 .52444 .48353 .50423 Metetra .795 .704 .765 r .48353 .50423 .4965 .52444 .52 .49884 .50719 .47851 Metetra .722 .683 .811 r .50719 .47851 .52 .49884 .5435 .4848 .5308 .46434 Metetra .563 .617 .864 r .5308 .46434 .5435 .4848 .56729 .48405 .55464 .46345 Metetra .303 .478 .866 r .55464 .46345 .56729 .48405 .59161 .49501 .57899 .47427 Metetra .092 .346 .817 r .57899 .47427 .59161 .49501 .61661 .51309 .60399 .49221 Metetra .092 .346 .817 r .60399 .49221 .61661 .51309 .64222 .53161 .6296 .51059 Metetra .298 .475 .866 r .6296 .51059 .64222 .53161 .66817 .54345 .65558 .52228 Metetra .555 .617 .869 r .65558 .52228 .66817 .54345 .69403 .54304 .68152 .52173 Metetra .723 .691 .819 r .68152 .52173 .69403 .54304 .7194 .52825 .70702 .50679 Metetra .803 .718 .77 r .70702 .50679 .7194 .52825 .74408 .50104 .73187 .47944 Metetra .834 .726 .745 r .73187 .47944 .74408 .50104 .76818 .46688 .75615 .44514 Metetra .834 .726 .745 r .75615 .44514 .76818 .46688 .79211 .43296 .78028 .41108 Metetra .803 .717 .77 r .78028 .41108 .79211 .43296 .81649 .40609 .80483 .38408 Metetra .723 .687 .816 r .80483 .38408 .81649 .40609 .84198 .39107 .83044 .36891 Metetra .572 .616 .858 r .83044 .36891 .84198 .39107 .86905 .38965 .85761 .36733 Metetra .376 .505 .861 r .85761 .36733 .86905 .38965 .89787 .40023 .88649 .37774 Metetra .238 .419 .839 r .88649 .37774 .89787 .40023 .92819 .4181 .91686 .39544 Metetra .667 .115 0 r .30804 .57318 .32267 .57922 .34368 .59768 .32908 .59175 Metetra .429 0 0 r .32908 .59175 .34368 .59768 .36551 .60962 .35099 .60376 Metetra .311 .737 .972 r .35099 .60376 .36551 .60962 .38826 .60966 .37388 .60376 Metetra .764 .883 .931 r .37388 .60376 .38826 .60966 .41176 .59569 .39758 .58964 Metetra .868 .858 .83 r .39758 .58964 .41176 .59569 .43569 .56965 .42177 .56334 Metetra .894 .841 .786 r .42177 .56334 .43569 .56965 .45969 .53684 .44606 .53022 Metetra .894 .841 .786 r .44606 .53022 .45969 .53684 .48353 .50423 .47019 .49731 Metetra .867 .856 .829 r .47019 .49731 .48353 .50423 .50719 .47851 .49409 .47133 Metetra .766 .872 .925 r .49409 .47133 .50719 .47851 .5308 .46434 .51792 .457 Metetra .409 .762 .995 r .51792 .457 .5308 .46434 .55464 .46345 .54192 .45606 Metetra 0 .35 .736 r .54192 .45606 .55464 .46345 .57899 .47427 .56638 .46691 Metetra .481 0 0 r .56638 .46691 .57899 .47427 .60399 .49221 .59149 .48495 Metetra .481 0 0 r .59149 .48495 .60399 .49221 .6296 .51059 .61721 .50343 Metetra 0 .34 .725 r .61721 .50343 .6296 .51059 .65558 .52228 .64334 .51517 Metetra .371 .754 .989 r .64334 .51517 .65558 .52228 .68152 .52173 .66947 .51457 Metetra .759 .898 .939 r .66947 .51457 .68152 .52173 .70702 .50679 .69521 .49945 Metetra .88 .892 .836 r .69521 .49945 .70702 .50679 .73187 .47944 .72034 .47181 Metetra .914 .878 .786 r .72034 .47181 .73187 .47944 .75615 .44514 .74493 .43718 Metetra .914 .878 .786 r .74493 .43718 .75615 .44514 .78028 .41108 .76935 .40279 Metetra .879 .89 .836 r .76935 .40279 .78028 .41108 .80483 .38408 .79417 .37551 Metetra .763 .886 .933 r .79417 .37551 .80483 .38408 .83044 .36891 .82003 .36016 Metetra .451 .767 .997 r .82003 .36016 .83044 .36891 .85761 .36733 .84742 .3585 Metetra .018 .484 .875 r .84742 .3585 .85761 .36733 .88649 .37774 .87651 .36893 Metetra 0 .289 .736 r .87651 .36893 .88649 .37774 .91686 .39544 .90707 .38671 Metetra .715 .254 .154 r .2923 .57892 .30804 .57318 .32908 .59175 .31345 .59782 Metetra .624 .125 .045 r .31345 .59782 .32908 .59175 .35099 .60376 .3355 .61005 Metetra .271 0 0 r .3355 .61005 .35099 .60376 .37388 .60376 .3586 .61009 Metetra .508 .901 .759 r .3586 .61009 .37388 .60376 .39758 .58964 .3826 .59579 Metetra .852 .998 .805 r .3826 .59579 .39758 .58964 .42177 .56334 .40714 .56912 Metetra .918 .974 .774 r .40714 .56912 .42177 .56334 .44606 .53022 .43182 .53551 Metetra .918 .974 .774 r .43182 .53551 .44606 .53022 .47019 .49731 .45633 .50213 Metetra .855 .997 .808 r .45633 .50213 .47019 .49731 .49409 .47133 .48058 .47579 Metetra .554 .928 .8 r .48058 .47579 .49409 .47133 .51792 .457 .5047 .46127 Metetra .157 0 0 r .5047 .46127 .51792 .457 .54192 .45606 .52895 .46035 Metetra .558 .01 0 r .52895 .46035 .54192 .45606 .56638 .46691 .55362 .4714 Metetra .672 .157 .007 r .55362 .4714 .56638 .46691 .59149 .48495 .57892 .48974 Metetra .672 .157 .007 r .57892 .48974 .59149 .48495 .61721 .50343 .60486 .50855 Metetra .562 .018 0 r .60486 .50855 .61721 .50343 .64334 .51517 .63123 .52052 Metetra .208 0 0 r .63123 .52052 .64334 .51517 .66947 .51457 .65766 .51995 Metetra .438 .854 .694 r .65766 .51995 .66947 .51457 .69521 .49945 .68376 .50462 Metetra .791 .988 .738 r .68376 .50462 .69521 .49945 .72034 .47181 .70927 .47657 Metetra .882 .985 .712 r .70927 .47657 .72034 .47181 .74493 .43718 .73426 .44141 Metetra .882 .985 .712 r .73426 .44141 .74493 .43718 .76935 .40279 .75907 .40651 Metetra .796 .99 .743 r .75907 .40651 .76935 .40279 .79417 .37551 .78427 .37882 Metetra .491 .89 .744 r .78427 .37882 .79417 .37551 .82003 .36016 .81048 .36326 Metetra .082 0 0 r .81048 .36326 .82003 .36016 .84742 .3585 .8382 .3616 Metetra .457 0 0 r .8382 .3616 .84742 .3585 .87651 .36893 .86762 .37222 Metetra .585 .017 0 r .86762 .37222 .87651 .36893 .90707 .38671 .89853 .3903 Metetra .699 .282 .251 r .27563 .59183 .2923 .57892 .31345 .59782 .29691 .61118 Metetra .63 .19 .186 r .29691 .61118 .31345 .59782 .3355 .61005 .31914 .62375 Metetra .405 0 .002 r .31914 .62375 .3355 .61005 .3586 .61009 .34251 .62388 Metetra .34251 .62388 .3586 .61009 .3826 .59579 .36686 .60938 Metetra .668 .924 .622 r .36686 .60938 .3826 .59579 .40714 .56912 .39182 .58226 Metetra .826 .973 .666 r .39182 .58226 .40714 .56912 .43182 .53551 .41695 .54807 Metetra .826 .973 .666 r .41695 .54807 .43182 .53551 .45633 .50213 .4419 .51411 Metetra .676 .93 .63 r .4419 .51411 .45633 .50213 .48058 .47579 .46657 .48733 Metetra .46657 .48733 .48058 .47579 .5047 .46127 .49104 .4726 Metetra .346 0 0 r .49104 .4726 .5047 .46127 .52895 .46035 .5156 .47173 Metetra .594 .116 .09 r .5156 .47173 .52895 .46035 .55362 .4714 .54053 .48309 Metetra .675 .216 .155 r .54053 .48309 .55362 .4714 .57892 .48974 .56609 .50188 Metetra .675 .216 .155 r .56609 .50188 .57892 .48974 .60486 .50855 .5923 .52117 Metetra .597 .121 .096 r .5923 .52117 .60486 .50855 .63123 .52052 .61898 .53347 Metetra .373 0 0 r .61898 .53347 .63123 .52052 .65766 .51995 .64578 .53298 Metetra .64578 .53298 .65766 .51995 .68376 .50462 .67229 .51742 Metetra .67229 .51742 .68376 .50462 .70927 .47657 .69826 .48888 Metetra .689 .888 .515 r .69826 .48888 .70927 .47657 .73426 .44141 .72371 .45308 Metetra .689 .888 .515 r .72371 .45308 .73426 .44141 .75907 .40651 .74898 .41753 Metetra .525 .817 .478 r .74898 .41753 .75907 .40651 .78427 .37882 .77462 .38936 Metetra .77462 .38936 .78427 .37882 .81048 .36326 .80126 .37356 Metetra .305 0 0 r .80126 .37356 .81048 .36326 .8382 .3616 .82939 .37195 Metetra .543 .025 0 r .82939 .37195 .8382 .3616 .86762 .37222 .85922 .38286 Metetra .631 .123 .022 r .85922 .38286 .86762 .37222 .89853 .3903 .89056 .40141 Metetra .699 .282 .251 r .25842 .60515 .27563 .59183 .29691 .61118 .27983 .62498 Metetra .63 .19 .186 r .27983 .62498 .29691 .61118 .31914 .62375 .30225 .6379 Metetra .405 0 .002 r .30225 .6379 .31914 .62375 .34251 .62388 .32589 .63813 Metetra .32589 .63813 .34251 .62388 .36686 .60938 .35061 .62342 Metetra .668 .924 .622 r .35061 .62342 .36686 .60938 .39182 .58226 .376 .59583 Metetra .826 .973 .666 r .376 .59583 .39182 .58226 .41695 .54807 .40159 .56103 Metetra .826 .973 .666 r .40159 .56103 .41695 .54807 .4419 .51411 .42701 .52647 Metetra .676 .93 .63 r .42701 .52647 .4419 .51411 .46657 .48733 .45209 .49924 Metetra .45209 .49924 .46657 .48733 .49104 .4726 .47694 .4843 Metetra .346 0 0 r .47694 .4843 .49104 .4726 .5156 .47173 .5018 .4835 Metetra .594 .116 .09 r .5018 .4835 .5156 .47173 .54053 .48309 .52701 .49517 Metetra .675 .216 .155 r .52701 .49517 .54053 .48309 .56609 .50188 .55282 .51444 Metetra .675 .216 .155 r .55282 .51444 .56609 .50188 .5923 .52117 .5793 .53421 Metetra .597 .121 .096 r .5793 .53421 .5923 .52117 .61898 .53347 .60631 .54687 Metetra .373 0 0 r .60631 .54687 .61898 .53347 .64578 .53298 .63348 .54647 Metetra .63348 .54647 .64578 .53298 .67229 .51742 .66042 .53067 Metetra .66042 .53067 .67229 .51742 .69826 .48888 .68686 .50161 Metetra .689 .888 .515 r .68686 .50161 .69826 .48888 .72371 .45308 .7128 .46514 Metetra .689 .888 .515 r .7128 .46514 .72371 .45308 .74898 .41753 .73854 .42894 Metetra .525 .817 .478 r .73854 .42894 .74898 .41753 .77462 .38936 .76465 .40027 Metetra .76465 .40027 .77462 .38936 .80126 .37356 .79172 .38422 Metetra .305 0 0 r .79172 .38422 .80126 .37356 .82939 .37195 .82027 .38265 Metetra .543 .025 0 r .82027 .38265 .82939 .37195 .85922 .38286 .85053 .39388 Metetra .631 .123 .022 r .85053 .39388 .85922 .38286 .89056 .40141 .88231 .41291 Metetra .717 .257 .159 r .24125 .61161 .25842 .60515 .27983 .62498 .26275 .63181 Metetra .627 .13 .053 r .26275 .63181 .27983 .62498 .30225 .6379 .28532 .64498 Metetra .281 0 0 r .28532 .64498 .30225 .6379 .32589 .63813 .30919 .64526 Metetra .497 .894 .749 r .30919 .64526 .32589 .63813 .35061 .62342 .33422 .63036 Metetra .85 .998 .802 r .33422 .63036 .35061 .62342 .376 .59583 .36001 .60236 Metetra .917 .975 .772 r .36001 .60236 .376 .59583 .40159 .56103 .38602 .56703 Metetra .917 .975 .772 r .38602 .56703 .40159 .56103 .42701 .52647 .41186 .53195 Metetra .852 .998 .805 r .41186 .53195 .42701 .52647 .45209 .49924 .43733 .50432 Metetra .545 .923 .792 r .43733 .50432 .45209 .49924 .47694 .4843 .46249 .48918 Metetra .169 0 0 r .46249 .48918 .47694 .4843 .5018 .4835 .48762 .48839 Metetra .562 .018 0 r .48762 .48839 .5018 .4835 .52701 .49517 .51305 .50029 Metetra .674 .162 .015 r .51305 .50029 .52701 .49517 .55282 .51444 .53906 .51991 Metetra .674 .162 .015 r .53906 .51991 .55282 .51444 .5793 .53421 .56576 .54006 Metetra .567 .025 0 r .56576 .54006 .5793 .53421 .60631 .54687 .59303 .55298 Metetra .219 0 0 r .59303 .55298 .60631 .54687 .63348 .54647 .62053 .55261 Metetra .425 .844 .682 r .62053 .55261 .63348 .54647 .66042 .53067 .64785 .53659 Metetra .786 .986 .732 r .64785 .53659 .66042 .53067 .68686 .50161 .67471 .50707 Metetra .879 .985 .708 r .67471 .50707 .68686 .50161 .7128 .46514 .70109 .47002 Metetra .879 .985 .708 r .70109 .47002 .7128 .46514 .73854 .42894 .72727 .43324 Metetra .791 .988 .738 r .72727 .43324 .73854 .42894 .76465 .40027 .75379 .40413 Metetra .48 .883 .733 r .75379 .40413 .76465 .40027 .79172 .38422 .78125 .38784 Metetra .095 0 0 r .78125 .38784 .79172 .38422 .82027 .38265 .81017 .38628 Metetra .464 0 0 r .81017 .38628 .82027 .38265 .85053 .39388 .84078 .39772 Metetra .59 .024 0 r .84078 .39772 .85053 .39388 .88231 .41291 .87293 .41711 Metetra .692 .148 0 r .22471 .60548 .24125 .61161 .26275 .63181 .24623 .62582 Metetra .48 0 0 r .24623 .62582 .26275 .63181 .28532 .64498 .26887 .63907 Metetra .258 .718 .952 r .26887 .63907 .28532 .64498 .30919 .64526 .29288 .63931 Metetra .761 .894 .937 r .29288 .63931 .30919 .64526 .33422 .63036 .31815 .62423 Metetra .871 .867 .832 r .31815 .62423 .33422 .63036 .36001 .60236 .34423 .59594 Metetra .898 .848 .787 r .34423 .59594 .36001 .60236 .38602 .56703 .37059 .56025 Metetra .898 .848 .787 r .37059 .56025 .38602 .56703 .41186 .53195 .39676 .52481 Metetra .87 .865 .831 r .39676 .52481 .41186 .53195 .43733 .50432 .42252 .49689 Metetra .764 .883 .931 r .42252 .49689 .43733 .50432 .46249 .48918 .44792 .48157 Metetra .371 .754 .989 r .44792 .48157 .46249 .48918 .48762 .48839 .47322 .48072 Metetra 0 .304 .683 r .47322 .48072 .48762 .48839 .51305 .50029 .49877 .49266 Metetra .526 0 0 r .49877 .49266 .51305 .50029 .53906 .51991 .52489 .51241 Metetra .526 0 0 r .52489 .51241 .53906 .51991 .56576 .54006 .5517 .53268 Metetra 0 .293 .671 r .5517 .53268 .56576 .54006 .59303 .55298 .57912 .54567 Metetra .327 .743 .978 r .57912 .54567 .59303 .55298 .62053 .55261 .60683 .54524 Metetra .755 .909 .944 r .60683 .54524 .62053 .55261 .64785 .53659 .63443 .52901 Metetra .883 .902 .837 r .63443 .52901 .64785 .53659 .67471 .50707 .66162 .49917 Metetra .918 .887 .786 r .66162 .49917 .67471 .50707 .70109 .47002 .68836 .46172 Metetra .918 .887 .786 r .68836 .46172 .70109 .47002 .72727 .43324 .71489 .42456 Metetra .883 .899 .837 r .71489 .42456 .72727 .43324 .75379 .40413 .74172 .39512 Metetra .759 .898 .939 r .74172 .39512 .75379 .40413 .78125 .38784 .76946 .37863 Metetra .42 .764 .996 r .76946 .37863 .78125 .38784 .81017 .38628 .79863 .37698 Metetra 0 .453 .846 r .79863 .37698 .81017 .38628 .84078 .39772 .82947 .38844 Metetra 0 .252 .696 r .82947 .38844 .84078 .39772 .87293 .41711 .86184 .40793 Metetra 0 .144 .68 r .20919 .58447 .22471 .60548 .24623 .62582 .23065 .60466 Metetra .101 .386 .845 r .23065 .60466 .24623 .62582 .26887 .63907 .25326 .61777 Metetra .52 .617 .889 r .25326 .61777 .26887 .63907 .29288 .63931 .27732 .61787 Metetra .723 .694 .823 r .27732 .61787 .29288 .63931 .31815 .62423 .30271 .60264 Metetra .8 .713 .769 r .30271 .60264 .31815 .62423 .34423 .59594 .32899 .57419 Metetra .827 .717 .743 r .32899 .57419 .34423 .59594 .37059 .56025 .35557 .53835 Metetra .827 .717 .743 r .35557 .53835 .37059 .56025 .39676 .52481 .38196 .50276 Metetra .8 .712 .768 r .38196 .50276 .39676 .52481 .42252 .49689 .40791 .47469 Metetra .723 .691 .819 r .40791 .47469 .42252 .49689 .44792 .48157 .43344 .45921 Metetra .544 .617 .876 r .43344 .45921 .44792 .48157 .47322 .48072 .45881 .45821 Metetra .241 .452 .865 r .45881 .45821 .47322 .48072 .49877 .49266 .48437 .46998 Metetra .001 .296 .792 r .48437 .46998 .49877 .49266 .52489 .51241 .51046 .48955 Metetra .001 .296 .792 r .51046 .48955 .52489 .51241 .5517 .53268 .53726 .50965 Metetra .235 .449 .864 r .53726 .50965 .5517 .53268 .57912 .54567 .5647 .52246 Metetra .534 .617 .881 r .5647 .52246 .57912 .54567 .60683 .54524 .5925 .52187 Metetra .723 .7 .828 r .5925 .52187 .60683 .54524 .63443 .52901 .62025 .50547 Metetra .809 .727 .774 r .62025 .50547 .63443 .52901 .66162 .49917 .64765 .47545 Metetra .842 .735 .746 r .64765 .47545 .66162 .49917 .68836 .46172 .67462 .43783 Metetra .842 .735 .746 r .67462 .43783 .68836 .46172 .71489 .42456 .70138 .40051 Metetra .809 .726 .774 r .70138 .40051 .71489 .42456 .74172 .39512 .72842 .3709 Metetra .723 .696 .824 r .72842 .3709 .74172 .39512 .76946 .37863 .75632 .35423 Metetra .555 .617 .869 r .75632 .35423 .76946 .37863 .79863 .37698 .78559 .35239 Metetra .332 .489 .865 r .78559 .35239 .79863 .37698 .82947 .38844 .81649 .36365 Metetra .178 .389 .832 r .81649 .36365 .82947 .38844 .86184 .40793 .8489 .38293 Metetra .126 .269 .741 r .19471 .55051 .20919 .58447 .23065 .60466 .21604 .57032 Metetra .324 .395 .781 r .21604 .57032 .23065 .60466 .25326 .61777 .23856 .5831 Metetra .547 .525 .782 r .23856 .5831 .25326 .61777 .27732 .61787 .26258 .58295 Metetra .684 .597 .752 r .26258 .58295 .27732 .61787 .30271 .60264 .288 .56759 Metetra .751 .629 .724 r .288 .56759 .30271 .60264 .32899 .57419 .31436 .53912 Metetra .778 .641 .709 r .31436 .53912 .32899 .57419 .35557 .53835 .34107 .50331 Metetra .778 .641 .709 r .34107 .50331 .35557 .53835 .38196 .50276 .36758 .46775 Metetra .751 .628 .724 r .36758 .46775 .38196 .50276 .40791 .47469 .39362 .43965 Metetra .685 .596 .751 r .39362 .43965 .40791 .47469 .43344 .45921 .41918 .42405 Metetra .56 .53 .778 r .41918 .42405 .43344 .45921 .45881 .45821 .44451 .4228 Metetra .384 .427 .782 r .44451 .4228 .45881 .45821 .48437 .46998 .46998 .43421 Metetra .247 .341 .764 r .46998 .43421 .48437 .46998 .51046 .48955 .49595 .45336 Metetra .247 .341 .764 r .49595 .45336 .51046 .48955 .53726 .50965 .52263 .47302 Metetra .381 .425 .782 r .52263 .47302 .53726 .50965 .5647 .52246 .54998 .48545 Metetra .554 .528 .78 r .54998 .48545 .5647 .52246 .5925 .52187 .57775 .48458 Metetra .683 .597 .754 r .57775 .48458 .5925 .52187 .62025 .50547 .60554 .46803 Metetra .755 .633 .725 r .60554 .46803 .62025 .50547 .64765 .47545 .63304 .43798 Metetra .786 .647 .709 r .63304 .43798 .64765 .47545 .67462 .43783 .66014 .40041 Metetra .786 .647 .709 r .66014 .40041 .67462 .43783 .70138 .40051 .68703 .36312 Metetra .755 .633 .725 r .68703 .36312 .70138 .40051 .72842 .3709 .71416 .33349 Metetra .684 .597 .752 r .71416 .33349 .72842 .3709 .75632 .35423 .7421 .31667 Metetra .566 .532 .776 r .7421 .31667 .75632 .35423 .78559 .35239 .77135 .31455 Metetra .427 .449 .78 r .77135 .31455 .78559 .35239 .81649 .36365 .80217 .3254 Metetra .331 .389 .772 r .80217 .3254 .81649 .36365 .8489 .38293 .83448 .34418 Metetra .232 .292 .719 r .1809 .50916 .19471 .55051 .21604 .57032 .20206 .52845 Metetra .378 .386 .741 r .20206 .52845 .21604 .57032 .23856 .5831 .22444 .54078 Metetra .55 .491 .743 r .22444 .54078 .23856 .5831 .26258 .58295 .24837 .54034 Metetra .667 .559 .724 r .24837 .54034 .26258 .58295 .288 .56759 .27377 .52487 Metetra .73 .594 .704 r .27377 .52487 .288 .56759 .31436 .53912 .30016 .49644 Metetra .756 .608 .693 r .30016 .49644 .31436 .53912 .34107 .50331 .32693 .46076 Metetra .756 .608 .693 r .32693 .46076 .34107 .50331 .36758 .46775 .35351 .42535 Metetra .73 .593 .704 r .35351 .42535 .36758 .46775 .39362 .43965 .37957 .39729 Metetra .668 .56 .723 r .37957 .39729 .39362 .43965 .41918 .42405 .40511 .38158 Metetra .561 .497 .742 r .40511 .38158 .41918 .42405 .44451 .4228 .43035 .38003 Metetra .42 .412 .744 r .43035 .38003 .44451 .4228 .46998 .43421 .45568 .39098 Metetra .314 .344 .733 r .45568 .39098 .46998 .43421 .49595 .45336 .48147 .40955 Metetra .314 .344 .733 r .48147 .40955 .49595 .45336 .52263 .47302 .50796 .42863 Metetra .418 .41 .744 r .50796 .42863 .52263 .47302 .54998 .48545 .53516 .44056 Metetra .556 .495 .742 r .53516 .44056 .54998 .48545 .57775 .48458 .56283 .43936 Metetra .666 .559 .724 r .56283 .43936 .57775 .48458 .60554 .46803 .59059 .42268 Metetra .731 .595 .704 r .59059 .42268 .60554 .46803 .63304 .43798 .61812 .39269 Metetra .761 .61 .692 r .61812 .39269 .63304 .43798 .66014 .40041 .64529 .35527 Metetra .761 .61 .692 r .64529 .35527 .66014 .40041 .68703 .36312 .67225 .31813 Metetra .731 .595 .704 r .67225 .31813 .68703 .36312 .71416 .33349 .69942 .28855 Metetra .667 .559 .724 r .69942 .28855 .71416 .33349 .7421 .31667 .72733 .27161 Metetra .566 .5 .741 r .72733 .27161 .7421 .31667 .77135 .31455 .75648 .26915 Metetra .452 .431 .745 r .75648 .26915 .77135 .31455 .80217 .3254 .78715 .27948 Metetra .375 .382 .74 r .78715 .27948 .80217 .3254 .83448 .34418 .81925 .29762 Metetra .232 .292 .719 r .16708 .46777 .1809 .50916 .20206 .52845 .18807 .48654 Metetra .378 .386 .741 r .18807 .48654 .20206 .52845 .22444 .54078 .21031 .49842 Metetra .55 .491 .743 r .21031 .49842 .22444 .54078 .24837 .54034 .23415 .49769 Metetra .667 .559 .724 r .23415 .49769 .24837 .54034 .27377 .52487 .25952 .4821 Metetra .73 .594 .704 r .25952 .4821 .27377 .52487 .30016 .49644 .28595 .45372 Metetra .756 .608 .693 r .28595 .45372 .30016 .49644 .32693 .46076 .31278 .41819 Metetra .756 .608 .693 r .31278 .41819 .32693 .46076 .35351 .42535 .33942 .38291 Metetra .73 .593 .704 r .33942 .38291 .35351 .42535 .37957 .39729 .36551 .35489 Metetra .668 .56 .723 r .36551 .35489 .37957 .39729 .40511 .38158 .39102 .33907 Metetra .561 .497 .742 r .39102 .33907 .40511 .38158 .43035 .38003 .41618 .33722 Metetra .42 .412 .744 r .41618 .33722 .43035 .38003 .45568 .39098 .44136 .34771 Metetra .314 .344 .733 r .44136 .34771 .45568 .39098 .48147 .40955 .46697 .36571 Metetra .314 .344 .733 r .46697 .36571 .48147 .40955 .50796 .42863 .49328 .3842 Metetra .418 .41 .744 r .49328 .3842 .50796 .42863 .53516 .44056 .52032 .39563 Metetra .556 .495 .742 r .52032 .39563 .53516 .44056 .56283 .43936 .54789 .39409 Metetra .666 .559 .724 r .54789 .39409 .56283 .43936 .59059 .42268 .57563 .37729 Metetra .731 .595 .704 r .57563 .37729 .59059 .42268 .61812 .39269 .60319 .34735 Metetra .761 .61 .692 r .60319 .34735 .61812 .39269 .64529 .35527 .63043 .31008 Metetra .761 .61 .692 r .63043 .31008 .64529 .35527 .67225 .31813 .65746 .2731 Metetra .731 .595 .704 r .65746 .2731 .67225 .31813 .69942 .28855 .68466 .24357 Metetra .667 .559 .724 r .68466 .24357 .69942 .28855 .72733 .27161 .71254 .2265 Metetra .566 .5 .741 r .71254 .2265 .72733 .27161 .75648 .26915 .74159 .22371 Metetra .452 .431 .745 r .74159 .22371 .75648 .26915 .78715 .27948 .7721 .23352 Metetra .375 .382 .74 r .7721 .23352 .78715 .27948 .81925 .29762 .80401 .25101 Metetra .133 .273 .743 r .15245 .4334 .16708 .46777 .18807 .48654 .1733 .45177 Metetra .328 .397 .781 r .1733 .45177 .18807 .48654 .21031 .49842 .19544 .46331 Metetra .548 .525 .782 r .19544 .46331 .21031 .49842 .23415 .49769 .21924 .46233 Metetra .684 .597 .752 r .21924 .46233 .23415 .49769 .25952 .4821 .24464 .44662 Metetra .751 .628 .724 r .24464 .44662 .25952 .4821 .28595 .45372 .27116 .4182 Metetra .778 .64 .709 r .27116 .4182 .28595 .45372 .31278 .41819 .29811 .3827 Metetra .778 .64 .709 r .29811 .3827 .31278 .41819 .33942 .38291 .32487 .34746 Metetra .751 .628 .724 r .32487 .34746 .33942 .38291 .36551 .35489 .35106 .31942 Metetra .685 .596 .751 r .35106 .31942 .36551 .35489 .39102 .33907 .3766 .30346 Metetra .561 .53 .778 r .3766 .30346 .39102 .33907 .41618 .33722 .40172 .30136 Metetra .387 .428 .782 r .40172 .30136 .41618 .33722 .44136 .34771 .42681 .31148 Metetra .251 .344 .764 r .42681 .31148 .44136 .34771 .46697 .36571 .4523 .32905 Metetra .251 .344 .764 r .4523 .32905 .46697 .36571 .49328 .3842 .47848 .34709 Metetra .384 .427 .782 r .47848 .34709 .49328 .3842 .52032 .39563 .50542 .35813 Metetra .556 .528 .78 r .50542 .35813 .52032 .39563 .54789 .39409 .53296 .35631 Metetra .683 .597 .753 r .53296 .35631 .54789 .39409 .57563 .37729 .56074 .33936 Metetra .755 .633 .725 r .56074 .33936 .57563 .37729 .60319 .34735 .5884 .30938 Metetra .785 .647 .709 r .5884 .30938 .60319 .34735 .63043 .31008 .61577 .27216 Metetra .785 .647 .709 r .61577 .27216 .63043 .31008 .65746 .2731 .64293 .23522 Metetra .755 .632 .725 r .64293 .23522 .65746 .2731 .68466 .24357 .67023 .20566 Metetra .684 .597 .752 r .67023 .20566 .68466 .24357 .71254 .2265 .69815 .18844 Metetra .567 .532 .776 r .69815 .18844 .71254 .2265 .74159 .22371 .72718 .18536 Metetra .429 .45 .78 r .72718 .18536 .74159 .22371 .7721 .23352 .75761 .19475 Metetra .334 .391 .772 r .75761 .19475 .7721 .23352 .80401 .25101 .7894 .21174 Metetra 0 .179 .71 r .13627 .41103 .15245 .4334 .1733 .45177 .15705 .42923 Metetra .142 .407 .853 r .15705 .42923 .1733 .45177 .19544 .46331 .17915 .4406 Metetra .531 .617 .883 r .17915 .4406 .19544 .46331 .21924 .46233 .203 .43946 Metetra .723 .691 .819 r .203 .43946 .21924 .46233 .24464 .44662 .22853 .42358 Metetra .798 .709 .767 r .22853 .42358 .24464 .44662 .27116 .4182 .25524 .39501 Metetra .825 .714 .743 r .25524 .39501 .27116 .4182 .29811 .3827 .28243 .35936 Metetra .825 .714 .743 r .28243 .35936 .29811 .3827 .32487 .34746 .30942 .32397 Metetra .797 .709 .767 r .30942 .32397 .32487 .34746 .35106 .31942 .33579 .29578 Metetra .723 .687 .816 r .33579 .29578 .35106 .31942 .3766 .30346 .36147 .27966 Metetra .553 .617 .871 r .36147 .27966 .3766 .30346 .40172 .30136 .38665 .27738 Metetra .27 .464 .866 r .38665 .27738 .40172 .30136 .42681 .31148 .41174 .2873 Metetra .042 .319 .804 r .41174 .2873 .42681 .31148 .4523 .32905 .43719 .30466 Metetra .042 .319 .804 r .43719 .30466 .4523 .32905 .47848 .34709 .46334 .32249 Metetra .263 .461 .866 r .46334 .32249 .47848 .34709 .50542 .35813 .4903 .33332 Metetra .544 .617 .876 r .4903 .33332 .50542 .35813 .53296 .35631 .51792 .3313 Metetra .723 .696 .824 r .51792 .3313 .53296 .35631 .56074 .33936 .54585 .31417 Metetra .807 .723 .772 r .54585 .31417 .56074 .33936 .5884 .30938 .57373 .28402 Metetra .838 .731 .746 r .57373 .28402 .5884 .30938 .61577 .27216 .60134 .24663 Metetra .838 .731 .746 r .60134 .24663 .61577 .27216 .64293 .23522 .62875 .20953 Metetra .806 .722 .772 r .62875 .20953 .64293 .23522 .67023 .20566 .65625 .17981 Metetra .723 .692 .82 r .65625 .17981 .67023 .20566 .69815 .18844 .68433 .1624 Metetra .563 .617 .864 r .68433 .1624 .69815 .18844 .72718 .18536 .71346 .15911 Metetra .352 .497 .864 r .71346 .15911 .72718 .18536 .75761 .19475 .74394 .16825 Metetra .205 .403 .836 r .74394 .16825 .75761 .19475 .7894 .21174 .77576 .18498 Metetra .627 .066 0 r .1181 .40254 .13627 .41103 .15705 .42923 .13887 .42083 Metetra .354 0 0 r .13887 .42083 .15705 .42923 .17915 .4406 .16102 .43224 Metetra .371 .754 .989 r .16102 .43224 .17915 .4406 .203 .43946 .185 .43102 Metetra .766 .868 .923 r .185 .43102 .203 .43946 .22853 .42358 .21077 .41494 Metetra .863 .847 .827 r .21077 .41494 .22853 .42358 .25524 .39501 .23781 .38607 Metetra .889 .833 .785 r .23781 .38607 .25524 .39501 .28243 .35936 .26536 .35005 Metetra .889 .833 .785 r .26536 .35005 .28243 .35936 .30942 .32397 .29272 .3143 Metetra .862 .846 .826 r .29272 .3143 .30942 .32397 .33579 .29578 .31941 .28581 Metetra .768 .858 .917 r .31941 .28581 .33579 .29578 .36147 .27966 .34532 .26949 Metetra .451 .767 .997 r .34532 .26949 .36147 .27966 .38665 .27738 .37067 .26711 Metetra 0 .405 .797 r .37067 .26711 .38665 .27738 .41174 .2873 .39587 .27703 Metetra .415 0 0 r .39587 .27703 .41174 .2873 .43719 .30466 .4214 .29444 Metetra .415 0 0 r .4214 .29444 .43719 .30466 .46334 .32249 .44763 .31234 Metetra 0 .397 .788 r .44763 .31234 .46334 .32249 .4903 .33332 .47473 .32318 Metetra .42 .764 .996 r .47473 .32318 .4903 .33332 .51792 .3313 .50256 .32106 Metetra .764 .883 .931 r .50256 .32106 .51792 .3313 .54585 .31417 .53078 .30369 Metetra .876 .879 .834 r .53078 .30369 .54585 .31417 .57373 .28402 .55901 .2732 Metetra .908 .868 .787 r .55901 .2732 .57373 .28402 .60134 .24663 .58701 .23542 Metetra .908 .868 .787 r .58701 .23542 .60134 .24663 .62875 .20953 .61479 .19792 Metetra .875 .877 .834 r .61479 .19792 .62875 .20953 .65625 .17981 .64264 .16787 Metetra .766 .872 .925 r .64264 .16787 .65625 .17981 .68433 .1624 .67102 .15022 Metetra .486 .768 .994 r .67102 .15022 .68433 .1624 .71346 .15911 .70039 .14679 Metetra .087 .519 .905 r .70039 .14679 .71346 .15911 .74394 .16825 .73109 .1559 Metetra 0 .336 .783 r .73109 .1559 .74394 .16825 .77576 .18498 .76311 .17265 Metetra .687 .187 .051 r .09788 .40634 .1181 .40254 .13887 .42083 .1187 .42496 Metetra .562 .018 0 r .1187 .42496 .13887 .42083 .16102 .43224 .14096 .43659 Metetra .083 0 0 r .14096 .43659 .16102 .43224 .185 .43102 .16517 .43538 Metetra .651 .97 .878 r .16517 .43538 .185 .43102 .21077 .41494 .19128 .41907 Metetra .877 .98 .836 r .19128 .41907 .21077 .41494 .23781 .38607 .21876 .38974 Metetra .922 .951 .794 r .21876 .38974 .23781 .38607 .26536 .35005 .24681 .35316 Metetra .922 .951 .794 r .24681 .35316 .26536 .35005 .29272 .3143 .27465 .31685 Metetra .878 .978 .837 r .27465 .31685 .29272 .3143 .31941 .28581 .30177 .28793 Metetra .678 .975 .896 r .30177 .28793 .31941 .28581 .34532 .26949 .32804 .27136 Metetra .32804 .27136 .34532 .26949 .37067 .26711 .35365 .26895 Metetra .457 0 0 r .35365 .26895 .37067 .26711 .39587 .27703 .37905 .27905 Metetra .614 .059 0 r .37905 .27905 .39587 .27703 .4214 .29444 .40475 .29677 Metetra .614 .059 0 r .40475 .29677 .4214 .29444 .44763 .31234 .43117 .31498 Metetra .464 0 0 r .43117 .31498 .44763 .31234 .47473 .32318 .45851 .32603 Metetra .004 0 0 r .45851 .32603 .47473 .32318 .50256 .32106 .48667 .3239 Metetra .607 .954 .844 r .48667 .3239 .50256 .32106 .53078 .30369 .5153 .30626 Metetra .852 .998 .805 r .5153 .30626 .53078 .30369 .55901 .2732 .54402 .27526 Metetra .913 .98 .762 r .54402 .27526 .55901 .2732 .58701 .23542 .57254 .23685 Metetra .913 .98 .762 r .57254 .23685 .58701 .23542 .61479 .19792 .60083 .19874 Metetra .855 .997 .808 r .60083 .19874 .61479 .19792 .64264 .16787 .62915 .1682 Metetra .641 .966 .87 r .62915 .1682 .64264 .16787 .67102 .15022 .65795 .15027 Metetra .137 .684 .769 r .65795 .15027 .67102 .15022 .70039 .14679 .6877 .14679 Metetra .308 0 0 r .6877 .14679 .70039 .14679 .73109 .1559 .71875 .15604 Metetra .478 0 0 r .71875 .15604 .73109 .1559 .76311 .17265 .75113 .17308 Metetra .675 .216 .155 r .07603 .41762 .09788 .40634 .1187 .42496 .09692 .43673 Metetra .581 .091 .056 r .09692 .43673 .1187 .42496 .14096 .43659 .11933 .44869 Metetra .258 0 0 r .11933 .44869 .14096 .43659 .16517 .43538 .14382 .44756 Metetra .419 .832 .647 r .14382 .44756 .16517 .43538 .19128 .41907 .17034 .43098 Metetra .802 .994 .761 r .17034 .43098 .19128 .41907 .21876 .38974 .19835 .40111 Metetra .891 .989 .752 r .19835 .40111 .21876 .38974 .24681 .35316 .22698 .36383 Metetra .891 .989 .752 r .22698 .36383 .24681 .35316 .27465 .31685 .25539 .32683 Metetra .806 .995 .765 r .25539 .32683 .27465 .31685 .30177 .28793 .28303 .29737 Metetra .467 .867 .694 r .28303 .29737 .30177 .28793 .32804 .27136 .30971 .28053 Metetra .163 0 0 r .30971 .28053 .32804 .27136 .35365 .26895 .33565 .27815 Metetra .519 0 0 r .33565 .27815 .35365 .26895 .37905 .27905 .36131 .28854 Metetra .631 .123 .022 r .36131 .28854 .37905 .27905 .40475 .29677 .38723 .30672 Metetra .631 .123 .022 r .38723 .30672 .40475 .29677 .43117 .31498 .41389 .32542 Metetra .523 0 0 r .41389 .32542 .43117 .31498 .45851 .32603 .44154 .3368 Metetra .206 0 0 r .44154 .3368 .45851 .32603 .48667 .3239 .47011 .33472 Metetra .47011 .33472 .48667 .3239 .5153 .30626 .49925 .31676 Metetra .713 .952 .668 r .49925 .31676 .5153 .30626 .54402 .27526 .52854 .28516 Metetra .826 .973 .666 r .52854 .28516 .54402 .27526 .57254 .23685 .55766 .24598 Metetra .826 .973 .666 r .55766 .24598 .57254 .23685 .60083 .19874 .58655 .20711 Metetra .72 .956 .675 r .58655 .20711 .60083 .19874 .62915 .1682 .61543 .17597 Metetra .401 .818 .63 r .61543 .17597 .62915 .1682 .65795 .15027 .64474 .15772 Metetra .101 0 0 r .64474 .15772 .65795 .15027 .6877 .14679 .67496 .15423 Metetra .43 0 0 r .67496 .15423 .6877 .14679 .71875 .15604 .70646 .16376 Metetra .552 0 0 r .70646 .16376 .71875 .15604 .75113 .17308 .73929 .18125 Metetra .675 .216 .155 r .0534 .42931 .07603 .41762 .09692 .43673 .07436 .44891 Metetra .581 .091 .056 r .07436 .44891 .09692 .43673 .11933 .44869 .09692 .46124 Metetra .258 0 0 r .09692 .46124 .11933 .44869 .14382 .44756 .12168 .46018 Metetra .419 .832 .647 r .12168 .46018 .14382 .44756 .17034 .43098 .14864 .44333 Metetra .802 .994 .761 r .14864 .44333 .17034 .43098 .19835 .40111 .17719 .41289 Metetra .891 .989 .752 r .17719 .41289 .19835 .40111 .22698 .36383 .20642 .37488 Metetra .891 .989 .752 r .20642 .37488 .22698 .36383 .25539 .32683 .23543 .33717 Metetra .806 .995 .765 r .23543 .33717 .25539 .32683 .28303 .29737 .2636 .30716 Metetra .467 .867 .694 r .2636 .30716 .28303 .29737 .30971 .28053 .29072 .29004 Metetra .163 0 0 r .29072 .29004 .30971 .28053 .33565 .27815 .31699 .28769 Metetra .519 0 0 r .31699 .28769 .33565 .27815 .36131 .28854 .3429 .29838 Metetra .631 .123 .022 r .3429 .29838 .36131 .28854 .38723 .30672 .36905 .31704 Metetra .631 .123 .022 r .36905 .31704 .38723 .30672 .41389 .32542 .39596 .33625 Metetra .523 0 0 r .39596 .33625 .41389 .32542 .44154 .3368 .42392 .34799 Metetra .206 0 0 r .42392 .34799 .44154 .3368 .47011 .33472 .45291 .34595 Metetra .45291 .34595 .47011 .33472 .49925 .31676 .48257 .32768 Metetra .713 .952 .668 r .48257 .32768 .49925 .31676 .52854 .28516 .51246 .29544 Metetra .826 .973 .666 r .51246 .29544 .52854 .28516 .55766 .24598 .54221 .25545 Metetra .826 .973 .666 r .54221 .25545 .55766 .24598 .58655 .20711 .57172 .21579 Metetra .72 .956 .675 r .57172 .21579 .58655 .20711 .61543 .17597 .60119 .18404 Metetra .401 .818 .63 r .60119 .18404 .61543 .17597 .64474 .15772 .63102 .16546 Metetra .101 0 0 r .63102 .16546 .64474 .15772 .67496 .15423 .66173 .16196 Metetra .43 0 0 r .66173 .16196 .67496 .15423 .70646 .16376 .69368 .17177 Metetra .552 0 0 r .69368 .17177 .70646 .16376 .73929 .18125 .72698 .18974 Metetra .689 .192 .058 r .03112 .43373 .0534 .42931 .07436 .44891 .0521 .45371 Metetra .567 .025 0 r .0521 .45371 .07436 .44891 .09692 .46124 .07476 .46629 Metetra .096 0 0 r .07476 .46629 .09692 .46124 .12168 .46018 .09976 .46524 Metetra .645 .968 .873 r .09976 .46524 .12168 .46018 .14864 .44333 .12709 .44814 Metetra .876 .981 .835 r .12709 .44814 .14864 .44333 .17719 .41289 .15614 .4172 Metetra .922 .952 .793 r .15614 .4172 .17719 .41289 .20642 .37488 .18593 .37856 Metetra .922 .952 .793 r .18593 .37856 .20642 .37488 .23543 .33717 .21548 .34022 Metetra .877 .98 .836 r .21548 .34022 .23543 .33717 .2636 .30716 .24413 .30973 Metetra .673 .975 .893 r .24413 .30973 .2636 .30716 .29072 .29004 .27164 .29234 Metetra .27164 .29234 .29072 .29004 .31699 .28769 .29819 .28997 Metetra .464 0 0 r .29819 .28997 .31699 .28769 .3429 .29838 .3243 .30086 Metetra .618 .066 0 r .3243 .30086 .3429 .29838 .36905 .31704 .35061 .31987 Metetra .618 .066 0 r .35061 .31987 .36905 .31704 .39596 .33625 .3777 .33945 Metetra .471 0 0 r .3777 .33945 .39596 .33625 .42392 .34799 .40591 .35143 Metetra .017 0 0 r .40591 .35143 .42392 .34799 .45291 .34595 .43525 .34939 Metetra .599 .95 .838 r .43525 .34939 .45291 .34595 .48257 .32768 .46538 .33081 Metetra .85 .998 .802 r .46538 .33081 .48257 .32768 .51246 .29544 .49581 .29801 Metetra .912 .981 .76 r .49581 .29801 .51246 .29544 .54221 .25545 .52615 .25732 Metetra .912 .981 .76 r .52615 .25732 .54221 .25545 .57172 .21579 .55623 .21697 Metetra .852 .998 .805 r .55623 .21697 .57172 .21579 .60119 .18404 .58622 .18467 Metetra .634 .964 .865 r .58622 .18467 .60119 .18404 .63102 .16546 .61653 .16577 Metetra .124 .674 .757 r .61653 .16577 .63102 .16546 .66173 .16196 .64764 .16221 Metetra .318 0 0 r .64764 .16221 .66173 .16196 .69368 .17177 .67998 .17221 Metetra .485 0 0 r .67998 .17221 .69368 .17177 .72698 .18974 .71365 .1905 Metetra .658 .103 0 r .01027 .42483 .03112 .43373 .0521 .45371 .0312 .44492 Metetra .411 0 0 r .0312 .44492 .0521 .45371 .07476 .46629 .05388 .45755 Metetra .327 .743 .978 r .05388 .45755 .07476 .46629 .09976 .46524 .07903 .45642 Metetra .764 .879 .929 r .07903 .45642 .09976 .46524 .12709 .44814 .10663 .43908 Metetra .866 .855 .829 r .10663 .43908 .12709 .44814 .15614 .4172 .13606 .40778 Metetra .892 .839 .786 r .13606 .40778 .15614 .4172 .18593 .37856 .16629 .36872 Metetra .892 .839 .786 r .16629 .36872 .18593 .37856 .21548 .34022 .19628 .32996 Metetra .866 .854 .828 r .19628 .32996 .21548 .34022 .24413 .30973 .2253 .29911 Metetra .766 .868 .923 r .2253 .29911 .24413 .30973 .27164 .29234 .25309 .28148 Metetra .42 .764 .996 r .25309 .28148 .27164 .29234 .29819 .28997 .27982 .279 Metetra 0 .365 .753 r .27982 .279 .29819 .28997 .3243 .30086 .30603 .28989 Metetra .464 0 0 r .30603 .28989 .3243 .30086 .35061 .31987 .33239 .30898 Metetra .464 0 0 r .33239 .30898 .35061 .31987 .3777 .33945 .35955 .32864 Metetra 0 .355 .742 r .35955 .32864 .3777 .33945 .40591 .35143 .3879 .34063 Metetra .384 .757 .991 r .3879 .34063 .40591 .35143 .43525 .34939 .41747 .33848 Metetra .761 .894 .937 r .41747 .33848 .43525 .34939 .46538 .33081 .44793 .31962 Metetra .879 .888 .836 r .44793 .31962 .46538 .33081 .49581 .29801 .47879 .28642 Metetra .912 .876 .787 r .47879 .28642 .49581 .29801 .52615 .25732 .50959 .24525 Metetra .912 .876 .787 r .50959 .24525 .52615 .25732 .55623 .21697 .54012 .20444 Metetra .878 .886 .835 r .54012 .20444 .55623 .21697 .58622 .18467 .57053 .17174 Metetra .764 .883 .931 r .57053 .17174 .58622 .18467 .61653 .16577 .60118 .15256 Metetra .461 .767 .996 r .60118 .15256 .61653 .16577 .64764 .16221 .63257 .14885 Metetra .036 .494 .883 r .63257 .14885 .64764 .16221 .67998 .17221 .66513 .1588 Metetra 0 .302 .749 r .66513 .1588 .67998 .17221 .71365 .1905 .69902 .17712 Metetra 0 g .25 Mabswid .68874 0 m .96935 .42924 L s .96935 .42924 m 1 .6535 L s 1 .6535 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .03716 .25514 m 0 .48963 L s 0 .48963 m .70298 .24544 L s .70298 .24544 m .68874 0 L s .68874 0 m .03716 .25514 L s .03716 .25514 m .68874 0 L s .09674 .23181 m .10144 .23638 L s [(-5)] .08733 .22266 1 .97262 Mshowa .34306 .13535 m .34735 .14032 L s [(0)] .3345 .12542 .86223 1 Mshowa .61454 .02905 m .61829 .03444 L s [(5)] .60704 .01829 .69632 1 Mshowa .125 Mabswid .14416 .21324 m .14694 .21603 L s .19248 .19432 m .19521 .19716 L s .24172 .17504 m .2444 .17792 L s .2919 .15539 m .29453 .15832 L s .39523 .11493 m .39774 .11796 L s .44843 .0941 m .45088 .09718 L s .50269 .07285 m .50508 .07598 L s .55805 .05117 m .56037 .05435 L s .05019 .25004 m .05305 .25274 L s .67219 .00648 m .67437 .00976 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 233.563}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCacheValid->False] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Force numerical output with the function N", "Subsubsection"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " treats Sqrt[2] as a symbolic expression. Use N[ ] to force ", StyleBox["Mathematica", FontSlant->"Italic"], " to evaluate it numerically.\nThe notation expression//function is \ equivalent to function[expression]. You will find it useful to use //N and \ //Simplify" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Sqrt[2]\), "\[IndentingNewLine]", \(N[Sqrt[2]]\), "\[IndentingNewLine]", \(Sqrt[2] // N\)}], "Input"], Cell[BoxData[ \(\@2\)], "Output"], Cell[BoxData[ \(1.4142135623730951`\)], "Output"], Cell[BoxData[ \(1.4142135623730951`\)], "Output"] }, Open ]], Cell["Tell N how many digits you want", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[Sqrt[2], 100]\)], "Input"], Cell[BoxData[ \(1.4142135623730950488016887242096980785696718753769480731766797379907324\ 784621070388503875343276415727350138462309123`100. \)], "Output"] }, Open ]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Working with expressions in Mathematica", "Section"], Cell[CellGroupData[{ Cell["Mathematica has a powerful symbolic engine", "Subsubsection"], Cell["\<\ The Simplify expression attempts to apply algebraic transformation rules to \ simplify your expressions. Note the use of //Simplify\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Simplify[x^3 + 3 x^2\ y + 3\ x\ y^2 + y^3]\), "\[IndentingNewLine]", \(Simplify[\((x^2 - y^2)\)/\((x + y)\)]\), "\[IndentingNewLine]", \(Sin[x]^2 + Cos[x]^2 // Simplify\)}], "Input"], Cell[BoxData[ \(\((x + y)\)\^3\)], "Output"], Cell[BoxData[ \(x - y\)], "Output"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell["\<\ Sometimes you will want to Expand polynomials. Simplify will reverse the \ process.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Expand[\((x + y)\)^5]\), "\[IndentingNewLine]", \(Simplify[Expand[\((x + y)\)^5]]\)}], "Input"], Cell[BoxData[ \(x\^5 + 5\ x\^4\ y + 10\ x\^3\ y\^2 + 10\ x\^2\ y\^3 + 5\ x\ y\^4 + y\^5\)], "Output"], Cell[BoxData[ \(\((x + y)\)\^5\)], "Output"] }, Open ]], Cell["\<\ On occassion Simplify will miss obvious simplifications. Use FullSimplify to \ try to compress further.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(Simplify[Log[x - x^2] - Log[x]]\), "\[IndentingNewLine]", \(FullSimplify[Log[x - x^2] - Log[x]]\)}], "Input"], Cell[BoxData[ \(\(-Log[x]\) + Log[\(-\((\(-1\) + x)\)\)\ x]\)], "Output"], Cell[BoxData[ \(Log[1 - x]\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Variables can take on values or expressions and ", StyleBox["Mathematica", FontSlant->"Italic"], " can handle both" }], "Subsubsection"], Cell[BoxData[ \(\(Clear[a, f];\)\)], "Input"], Cell["\<\ Use = to assign a value or expression to a variable. Note that the right \ hand side is evaluated first, then the result is applied to the left hand \ side.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(a = 3 + 4;\)\), "\[IndentingNewLine]", \(\(f = x^2 + 2 x + 1;\)\), "\[IndentingNewLine]", \(a\), "\[IndentingNewLine]", \(f\)}], "Input"], Cell[BoxData[ \(7\)], "Output"], Cell[BoxData[ \(1 + 2\ x + x\^2\)], "Output"] }, Open ]], Cell["\<\ You can use all the functions described so far on variables as well.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(N[a]\), "\[IndentingNewLine]", \(Simplify[f]\)}], "Input"], Cell[BoxData[ \(7.`\)], "Output"], Cell[BoxData[ \(\((1 + x)\)\^2\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Expressions can be tricky", "Subsubsection"], Cell[BoxData[ \(\(Clear[f, x];\)\)], "Input"], Cell["\<\ If we set f to be an expression in terms of x, then set x to a particular \ value we get the expected result for f. If we then change the value of x, we \ still get the expected result for f.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(f = x^2;\)\), "\[IndentingNewLine]", \(x = 3; \ f\), "\[IndentingNewLine]", \(x = 4; \ f\)}], "Input"], Cell[BoxData[ \(9\)], "Output"], Cell[BoxData[ \(16\)], "Output"] }, Open ]], Cell["\<\ However, if we re-run the same commands we get unexpected results. This is \ because in the definition for f, the right hand side is evaluted first. At \ this point x has a value of 4, therefore f is assigned 16. 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