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<H2><A ID="SECTION00013000000000000000"> Multi-Line Displays</A> </H2>

<P> <I>Scientific Notebook</I> provides a range of alignment options for multiline mathematical displays. Here is a series of multiline displays. <BR> <DIV ALIGN="CENTER"> <!– MATH

x = 17y$\displaystyle \tag$2 (1)
y > a + b + c + d + e + f + g + h + i + j +  
    k + l + m + n + o + p$\displaystyle \tag$3 (2)

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>x</I></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP>17<I>y</I><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">2</TD> <TD WIDTH=10 ALIGN="RIGHT"> (1)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>y</I></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>&gt;</TD> <TD ALIGN="LEFT" NOWRAP><I>a</I> + <I>b</I> + <I>c</I> + <I>d</I> + <I>e</I> + <I>f</I> + <I>g</I> + <I>h</I> + <I>i</I> + <I>j</I> +</TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD>&nbsp;</TD> <TD ALIGN="LEFT" NOWRAP><I>k</I> + <I>l</I> + <I>m</I> + <I>n</I> + <I>o</I> + <I>p</I><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">3</TD> <TD WIDTH=10 ALIGN="RIGHT"> (2)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

<BR> <DIV ALIGN="CENTER"> <!– MATH

x $\displaystyle \ll$ y1 + ... + yn  
  z  

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>x</I></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG STYLE="" SRC="img9.png" ALT=" $\displaystyle \ll$"></TD> <TD ALIGN="LEFT" NOWRAP><I>y</I><SUB>1</SUB> + <SUP> ... </SUP> + <I>y</I><SUB>n</SUB></TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>&#8804;</TD> <TD ALIGN="LEFT" NOWRAP><I>z</I></TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

<BR> <DIV ALIGN="CENTER"> <!– MATH

y = a + b + c + d + e + f + g + h + i + j  
      + k + l + m + n + o + p  

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>y</I></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><I>a</I> + <I>b</I> + <I>c</I> + <I>d</I> + <I>e</I> + <I>f</I> + <I>g</I> + <I>h</I> + <I>i</I> + <I>j</I></TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD>&nbsp;</TD> <TD ALIGN="LEFT" NOWRAP>&nbsp; + <I>k</I> + <I>l</I> + <I>m</I> + <I>n</I> + <I>o</I> + <I>p</I></TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

<BR> <DIV ALIGN="CENTER"> <!– MATH

w + x + y + z =    
    a + b + c + d + e + f + g + h + i + j +  
    k + l + m + n + o + p  

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>w</I> + <I>x</I> + <I>y</I> + <I>z</I></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD>&nbsp;</TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD>&nbsp;</TD> <TD ALIGN="LEFT" NOWRAP><I>a</I> + <I>b</I> + <I>c</I> + <I>d</I> + <I>e</I> + <I>f</I> + <I>g</I> + <I>h</I> + <I>i</I> + <I>j</I> +</TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD>&nbsp;</TD> <TD ALIGN="LEFT" NOWRAP><I>k</I> + <I>l</I> + <I>m</I> + <I>n</I> + <I>o</I> + <I>p</I></TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

<P> If <I>f</I> (<I>x</I>) = <I>x</I> + 1, then we will have <P><!– MATH

f ([x + 1]/[x + 2]) = {[x + 1]/[x + 2]} + 1 = (2x + 3)/(x + 2)

–> </P> <DIV ALIGN="CENTER"> <I>f</I> ([<I>x</I> + 1]/[<I>x</I> + 2]) = [<I>x</I> + 1]/[<I>x</I> + 2] + 1 = (2<I>x</I> + 3)/(<I>x</I> + 2) </DIV><P></P> <BR> <DIV ALIGN="CENTER"> <!– MATH
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12      
+13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 190  

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12</TD> <TD>&nbsp;</TD> <TD>&nbsp;</TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">+13 + 14 + 15 + 16 + 17 + 18 + 19 + 20</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP>190</TD> <TD WIDTH=10 ALIGN="RIGHT"> &nbsp;</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

<P> <BR> <DIV ALIGN="CENTER"> <!– MATH

max(f, g) = $\displaystyle {\frac{{f+g+\left\vert f-g\right\vert }}{{2}}}$,$\displaystyle \tag$4 (3)
max(f, - g) = $\displaystyle {\frac{{f-g+\left\vert f+g\right\vert }}{{2}}}$.$\displaystyle \tag$5 (4)

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">max(<I>f</I>, <I>g</I>)</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><IMG STYLE="" SRC="img10.png" ALT=" $\displaystyle {\frac{{{f+g+\left\vert f-g\right\vert }}}{{{2}}}}$">,<IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">4</TD> <TD WIDTH=10 ALIGN="RIGHT"> (3)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">max(<I>f</I>, - <I>g</I>)</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><IMG STYLE="" SRC="img11.png" ALT=" $\displaystyle {\frac{{{f-g+\left\vert f+g\right\vert }}}{{{2}}}}$">.<IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">5</TD> <TD WIDTH=10 ALIGN="RIGHT"> (4)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

<BR> <DIV ALIGN="CENTER"> <!– MATH

(a + b)n+1 = (a + b)(a + b)n = (a + b)$\displaystyle \sum_{{j=0}}^{{n}}$$\displaystyle \binom{n}{j}$an-1bj$\displaystyle \tag$6 (5)
  = $\displaystyle \sum_{{j=0}}^{{n}}$$\displaystyle \binom{n}{j}$an+1-jbj + $\displaystyle \sum_{{j=1}}^{{n}}$$\displaystyle \binom{n}{j-1}%
$an-1bj$\displaystyle \tag$7 (6)
  = $\displaystyle \sum_{{j=0}}^{{n}}$$\displaystyle \binom{n+1}{j}$an+1-jbj.$\displaystyle \tag$8 (7)

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">(<I>a</I> + <I>b</I>)<SUP>n+1</SUP></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP>(<I>a</I> + <I>b</I>)(<I>a</I> + <I>b</I>)<SUP>n</SUP> = (<I>a</I> + <I>b</I>)<IMG STYLE="" SRC="img12.png" ALT=" $\displaystyle \sum_{{{j=0}}}^{{{n}}}$"><IMG STYLE="" SRC="img13.png" ALT=" $\displaystyle \binom{n}{j}$"><I>a</I><SUP>n-1</SUP><I>b</I><SUP>j</SUP><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">6</TD> <TD WIDTH=10 ALIGN="RIGHT"> (5)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><IMG STYLE="" SRC="img12.png" ALT=" $\displaystyle \sum_{{{j=0}}}^{{{n}}}$"><IMG STYLE="" SRC="img13.png" ALT=" $\displaystyle \binom{n}{j}$"><I>a</I><SUP>n+1-j</SUP><I>b</I><SUP>j</SUP> + <IMG STYLE="" SRC="img14.png" ALT=" $\displaystyle \sum_{{{j=1}}}^{{{n}}}$"><IMG STYLE="" SRC="img15.png" ALT=" $\displaystyle \binom{n}{j-1}%
$"><I>a</I><SUP>n-1</SUP><I>b</I><SUP>j</SUP><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">7</TD> <TD WIDTH=10 ALIGN="RIGHT"> (6)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><IMG STYLE="" SRC="img12.png" ALT=" $\displaystyle \sum_{{{j=0}}}^{{{n}}}$"><IMG STYLE="" SRC="img16.png" ALT=" $\displaystyle \binom{n+1}{j}$"><I>a</I><SUP>n+1-j</SUP><I>b</I><SUP>j</SUP>.<IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">8</TD> <TD WIDTH=10 ALIGN="RIGHT"> (7)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

<P> Test the overriding of automatic equation labels: <BR> <DIV ALIGN="CENTER"> <!– MATH

x = y + z$\displaystyle \tag$A–1 (8)
  = k + m$\displaystyle \tag$A–2 (9)

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>x</I></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><I>y</I> + <I>z</I><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$"><I>A</I>&ndash;1</TD> <TD WIDTH=10 ALIGN="RIGHT"> (8)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><I>k</I> + <I>m</I><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$"><I>A</I>&ndash;2</TD> <TD WIDTH=10 ALIGN="RIGHT"> (9)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

Test the overriding of automatic equation labels, suppressing annotation: <BR> <DIV ALIGN="CENTER"> <!– MATH

x = y + z$\displaystyle \tag$*A–1 (10)
  = k + m$\displaystyle \tag$*A–2 (11)

–> <TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>x</I></TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><I>y</I> + <I>z</I><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">*<I>A</I>&ndash;1</TD> <TD WIDTH=10 ALIGN="RIGHT"> (10)</TD></TR> <TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">&nbsp;</TD> <TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD> <TD ALIGN="LEFT" NOWRAP><I>k</I> + <I>m</I><IMG STYLE="" SRC="img8.png" ALT=" $\displaystyle \tag$">*<I>A</I>&ndash;2</TD> <TD WIDTH=10 ALIGN="RIGHT"> (11)</TD></TR> </TABLE></DIV> <BR CLEAR="ALL">

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