<div dir="ltr"><div><div><div><div><div><div><div>Hi Sylvia,<br><br></div><div>Well written question! I hope my response matches the thoroughness of your question. :-)<br></div><div><br></div>Computers have different numbers of 'bits' and 'bytes' that are used internally to represent various number types.<br><br></div>Some languages (eg, NCL, C) will, by default, use 4-bytes for integers and 'real' numbers. Each byte is 8-bits. Hence, a default integer would be 32-bits. Commonly, one bit is reserved as a sign (- or +). The (signed) integer number spans -<span class="gmail-Y0NH2b gmail-CLPzrc">2147483647 to +</span><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc">2147483647. Usually the preceding + sign is not shown. It is 'understood' to be positive.<br><br></span></span></div><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc">Some numbers need more than 4-bytes/32-bits for proper representation. NCL supports "long" integers and "double" reals. A long integer is represented by 8-bytes or 64-bits. Again, one bit is reserved for the sign (+ or -). A 'long' 64-bit integer can span -</span></span><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-f_Text">9223372036854775808</span> to +</span></span><br><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-f_Text">9223372036854775808.<br><br></span></span></span></span></span></div><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-Y0NH2b gmail-CLPzrc"><span class="gmail-f_Text">If there is no sign-bit then numbers can span </span></span></span></span></span><span class="gmail-f_Text">0 to 18446744073709551615. Yikes!!!<br></span></div><span class="gmail-f_Text">This representation is called an unsigned long integer (ulong).<br><br></span></div><div><span class="gmail-f_Text">OK ... How to print </span><span class="gmail-f_Text"><span class="gmail-f_Text"><font face="Calibri, sans-serif">400000000 correctly in NCL? Append an 'l' (long) at the end of the number.<br><br><br></font></span></span></div><span class="gmail-f_Text"></span><span class="gmail-f_Text"><font face="Calibri, sans-serif">print(400000000l)<br></font></span><span class="gmail-f_Text"><font face="Calibri, sans-serif">print(12.0*40000000000l)<br><br></font></span><span class="gmail-f_Text"></span></div><div>print(42949l*50000l)<br>print(42949*50000l<br><br></div><div>print(<font face="Calibri, sans-serif">50000*50000l</font>) ; Note that you need just one appended 'l'.<br></div><div> ; NCL will 'promote' the result to the numeric type<br></div><div> ; that is 'bigger'<br><br>print((12.0*40000000000.0)/(12.0*40000000000l)) ; note the 'l' <= 1<br>print((12*40000000000l)/(12*40000000000l)) ; note the 'l' <= 1<br><br></div><div>Why does the following fail?<br><br>print((12.0*40000000000.0)/(12.0*40000000000 )) ; no 'l'<br><br></div><div>Because when NCL or C sees 40000000000 without an 'l', it mangles (not sure how) the 40000000000 number.<br><br> print(0.002*1.2*40000000000l*(20.52839^3)) <br><br>print(0.002*1.2*40000000000.0*(20.52839^3))<br><br>===<br></div><div>Here is what NCL currently supports:<br><br><a href="https://www.ncl.ucar.edu/Document/Manuals/Ref_Manual/NclDataTypes.shtml">https://www.ncl.ucar.edu/Document/Manuals/Ref_Manual/NclDataTypes.shtml</a><br></div><div>There is a lot of detail bt, I hink, the top table gives you the idea.<br></div><div><br></div><div>A geneic WWW link:<br><br><a href="https://docs.mql4.com/basis/types/integer/integertypes">https://docs.mql4.com/basis/types/integer/integertypes</a><br><br></div><div>Great work<br></div><div>Dennis <br></div></div><div class="gmail_extra"><br><div class="gmail_quote">On Tue, Mar 27, 2018 at 7:55 PM, Sylvia Tang <span dir="ltr"><<a href="mailto:stang471@gmail.com" target="_blank">stang471@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Dear NCL support,<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">My name is Sylvia Tang, a high school student from Syosset
High School, New York, USA. I’m doing cyclone research with Dr. Colle from
Stony Brook University in New York.<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">I have recently used the NCL program to calculate the hurricane
PDI (power dissipation index). <span> </span>I realized
that when I multiply an integer with 10 digits or more, OR two integers whose
products contain 11 digits or more, the results for both are incorrect.<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 0> PDI=0.002*1.2*40000000000*(20.<wbr>52839^3)<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 1>
print(PDI)<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span> </span><span style="color:red">2.793144e+10</span><span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">The actual
result should be: <span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:rgb(0,176,80)">8.3049E+11 <span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Or<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:rgb(0,176,80)">830492859254<span></span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:rgb(0,176,80)"><br></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:rgb(0,176,80)">==</span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:rgb(0,176,80)">
</span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">My environment is:<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">root@LAPTOP-EF1RBS4V:~# ncl -V<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">6.4.0<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">root@LAPTOP-EF1RBS4V:~# uname -a<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">Linux LAPTOP-EF1RBS4V 4.4.0-43-Microsoft<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">root@LAPTOP-EF1RBS4V:~# gcc --version<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">gcc (Ubuntu 5.4.0-6ubuntu1~16.04.4) 5.4.0 20160609<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif">Copyright (C) 2015 Free Software Foundation, Inc.<span></span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;color:rgb(34,34,34);font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px;font-size:11pt;font-family:Calibri,sans-serif"><span>==</span></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-align:start;text-indent:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px"><span style="color:rgb(34,34,34);font-family:Calibri,sans-serif;font-size:11pt;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-transform:none;white-space:normal;word-spacing:0px">I have ftped 2 files </span><font face="Calibri, sans-serif"><span style="font-size:14.6667px">integer_10_digit_issue.<wbr>txt and integer_with_10_digit_<wbr>issue.docx (same file with different format)</span></font></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-align:start;text-indent:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px"><font face="Calibri, sans-serif"><span style="font-size:14.6667px"><br></span></font></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-align:start;text-indent:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px"><font face="Calibri, sans-serif"><span style="font-size:14.6667px">==</span></font></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-align:start;text-indent:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px"><font face="Calibri, sans-serif"><span style="font-size:14.6667px">
</span></font></p><p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><font face="Calibri, sans-serif">Below is my analysis.<span></span></font></p><font face="Calibri, sans-serif">
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>1.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>9 digits, is correct<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 1> print(400000000);<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span>
</span><span style="color:rgb(0,176,80)">400000000</span><span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>2.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>10 digits, becomes a negative number<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 0> print(4000000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">-294967296<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 1> x=4000000000<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 2> print(x)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>-294967296<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Note, by using calculator,
4000000000/294967296=13.56<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>3.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>11 digit, becomes a positive number, but near 30
times smaller<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 3> print(40000000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">1345294336<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Note, by using calculator, 40000000000/1345294336=29.73<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>4.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>11 digit number divided by another number,
becomes 1<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 1> print(40000000000/1345294336)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">1<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 2> y=40000000000/1345294336<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span>ncl 3> print(y)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>1<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>5.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>9 digit number times an integer, correct<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 5> print(5*400000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span>
</span><span style="color:rgb(0,176,80)">2000000000</span><span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 6> print(5.9999999*400000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:rgb(0,176,80)">2.4e+09<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>6.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>9 digit number times an integer, if product
greater than 2.4e+09, become a negative number <span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 7><span>
</span>print(6*400000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:red">-1894967296<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:black">ncl 8> print(-6*400000000)<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:black">(0)<span> </span></span><span style="color:red">1894967296</span><span style="color:black"><span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:red"><span> </span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>7.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>9 digit number, product greater than 2.4e+09, result
is correct if one is integer and one is float<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 9> print(6.0*400000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:rgb(0,176,80)">2.4e+09<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:rgb(0,176,80)"><span> </span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>8.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>Multiply 2 integers with a predicted 10 digit
product, correct<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 10> print(100000*10000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>1000000000<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">But if the product is greater than
<a href="tel:(214)%20745-0000" value="+12147450000" target="_blank">2147450000</a>, result will be wrong.<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 55> print(42949*50000);<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraph" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><a href="tel:(214)%20745-0000" value="+12147450000" target="_blank">2147450000</a><span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 51> print(42950*50000);<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span>
</span><span style="color:red">-<a href="tel:(214)%20746-7296" value="+12147467296" target="_blank">2147467296</a></span><span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>9.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>Multiply 2 integers with a predicted 11 digit
product, wrong<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 11> print(100000*100000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">1410065408<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Note, by using calculator,
(100000*100000)/1410065408=7.<wbr>09<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>10.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>11 digit integer multiplied by another integer,
becomes a negative number<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 12><span>
</span>print(12*40000000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">-1036337152<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>11.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>11 digit integer multiplied by a float, becomes a
positive number, but result is wrong<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 13> print(12.0*40000000000)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">1.614353e+10<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Note, by using calculator,
(12*40000000000)/<a href="tel:(614)%20353-0000" value="+16143530000" target="_blank"> 16143530000</a>=29.73<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>12.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>11 digit float multiplied by another float, result
is correct<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 14> print(12.0*40000000000.0)<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:rgb(0,176,80)">4.8e+11<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>13.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>11 digit correct result divided by incorrect
result, results in nearly a 30 times difference<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 15>
print((12.0*40000000000.0)/(<wbr>12.0*40000000000))<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:red"><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">29.73327<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>14.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>Product is 11 digits, correct result divided by
incorrect result, 7 times difference <span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 16>
print((100000*100000.0)/(<wbr>100000*100000))<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:red"><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">7.09187<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>15.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>12 digits, correct result divided by incorrect result,
704 times difference<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 17>
print((12.0*400,000,000,000.0)<wbr>/(12.0*400000000000))<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:red"><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">704.1739<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>16.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>13 digits, correct result divided by incorrect
result, 2887 times difference<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 18>
print((12.0*4000000000000.0)/(<wbr>12.0*4000000000000))<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">2887.154<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:red"><span> </span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>17.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>Square is OK<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-indent:0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 20> print(50000^2);<span> </span>square<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-indent:0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span> </span>2.5e+09<span> </span>--correct<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-indent:0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 21> print(50000*50000);<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraph" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:red"><span>(0)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span><span style="color:red">-1794967296 –wrong<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:black">ncl 75> print(50000*50000.0); float<span></span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span style="color:black">(0)<span> </span>2.5e+09<span> </span>--correct<span></span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpFirst" style="margin:0in 0in 0.0001pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span><span>18.<span style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span></span></span>Square root more than 15 digital number integer<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">print(sqrt(100000000000000))<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span>
</span><span style="color:red">-nan</span><span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">print(sqrt(1000000000000000.0)<wbr>); float<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpMiddle" style="margin:0in 0in 0.0001pt 0.5in;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span>
</span>3.162278e+07 --correct<span></span></p>
<p class="m_-4607131141454096826gmail-MsoListParagraphCxSpLast" style="margin:0in 0in 8pt 60.75pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Therefore, I use float 40000000000.0 instead of integer
40000000000 to produce a correct PDI result.<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 0> PDI=0.002*1.2*<b>40000000000.0</b>*(<wbr>20.52839^3)<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 1>
print(PDI)<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span>(0)<span> </span><span style="color:rgb(0,176,80)">8.30493e+11</span><span></span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 8pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">The correct PD is near 30 times larger than incorrect PD<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">ncl 2>
print((0.002*1.2*40000000000.<wbr>0*(20.52839^3))/(0.002*1.2*<wbr>40000000000*(20.52839^3)))<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">(0)<span> </span><span style="color:red">29.73327</span><span></span></p><br></font><p></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-align:start;text-indent:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px"><font face="Calibri, sans-serif"><span style="font-size:14.6667px"><br></span></font></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;text-align:start;text-indent:0px;text-decoration-style:initial;text-decoration-color:initial;line-height:15.6933px"><font face="Calibri, sans-serif"><span style="font-size:14.6667px">
</span></font></p><p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><font face="Calibri, sans-serif">Thank you<span></span></font></p><font face="Calibri, sans-serif">
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><span> </span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">Sylvia Tang<span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif"><a href="mailto:stang471@gmail.com" style="color:rgb(5,99,193);text-decoration:underline" target="_blank">stang471@gmail.com</a><span></span></p>
<p class="MsoNormal" style="margin:0in 0in 0.0001pt;line-height:107%;font-size:11pt;font-family:Calibri,sans-serif">3/27/2018<span></span></p><br></font><p></p>
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