{"id":1163,"date":"2017-09-21T00:54:00","date_gmt":"2017-09-21T00:54:00","guid":{"rendered":"https:\/\/kellyfish.me\/?p=1163"},"modified":"2018-11-04T00:49:58","modified_gmt":"2018-11-04T00:49:58","slug":"magic-tables","status":"publish","type":"post","link":"https:\/\/kellyfish.me\/index.php\/2017\/09\/21\/magic-tables\/","title":{"rendered":"Magic Tables"},"content":{"rendered":"

[latexpage]<\/span><\/p>\n

Fun Math for Girls
\n<\/span>By Kelly Tan<\/span><\/span><\/p>\n

\"\"<\/span><\/p>\n

Did you know multiplication of integers can have many exquisite patterns to admire and show to others?<\/span><\/p>\n

First, look at this magic table with multiplication of integers such as 1, 12, 123, 1234, . . .<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 \u00b7 9 + 2 = 11<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a012 \u00b7 9 + 3 = 111<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0123 \u00b7 9 + 4 = 1111<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a01234 \u00b7 9 + 5 = 11111<\/span><\/p>\n

\u00a0\u00a0\u00a012345 \u00b7 9 + 6 = 111111<\/span><\/p>\n

\u00a0\u00a0123456 \u00b7 9 + 7 = 1111111<\/span><\/p>\n

\u00a01234567 \u00b7 9 + 8 = 11111111<\/span><\/p>\n

12345678 \u00b7 9 + 9 = 111111111<\/span><\/p>\n

123456789 \u00b7 9 +10 = 1111111111<\/span><\/p>\n

You can verify the results with calculator, of course, but how did we even get these results? The answer is algebra. Let’s take the third row to see what you get if you manipulate it with algebraic rules:<\/span><\/p>\n

$123$\u00b7$9+4$<\/span><\/p>\n

$=(100+20+3)$\u00b7$(9)+(4)$<\/span><\/p>\n

=($10^2$<\/span>+$2$\u00b7$10^1$<\/span>+$3$\u00b7$10^0$<\/span>)\u00b7$(10-1)+(3+1)$<\/span><\/span><\/p>\n

=($10^3$<\/span>+$2$\u00b7$10^2$<\/span>+$3$\u00b7$10^1$<\/span>)-($10^2$<\/span>+$2$\u00b7$10^1$<\/span>+$3$\u00b7$10^0$<\/span>)$+3+1$<\/span><\/span><\/p>\n

=$10^3$<\/span>$+10^2$<\/span>$+10^1$<\/span>$+1$<\/span><\/span><\/p>\n

$=1000+100+10+1=1111$<\/span><\/p>\n

So,it is not a random occurrence if you understand the algebra behind it. So, let’s practice again for the fourth row:<\/span><\/p>\n

$1234$\u00b7$9+5$<\/span><\/p>\n

=$(1000+200+30+4)$\u00b7$(9)+(5)$<\/span><\/p>\n

=($10^3$<\/span>+$2$\u00b7$10^2$<\/span>+$3$\u00b7$10^1$<\/span>+$4$\u00b7$10^0$)<\/span>\u00b7$(10-1)+(4+1)$<\/span><\/span><\/p>\n

$=(10^4$<\/span>$+2$\u00b7$10^3$<\/span>$+3$\u00b7$10^2$<\/span>$+4$\u00b7$10^1$<\/span>)-($10^3$<\/span>$+2$\u00b7$10^2$<\/span>$+3$\u00b7$10^1$<\/span>$+4$\u00b7$10^0$<\/span>)$+4+1$<\/span><\/span><\/p>\n

$=10^4$<\/span>$+10^3$<\/span>$+10^2$<\/span>$+10^1$<\/span>$+1$<\/span><\/span><\/p>\n

$=10000+1000+100+10+1=11111$<\/span><\/p>\n

Now, try this with the next row for 12345\u00b79+6 to see if you get 111111.<\/span><\/p>\n

Another magic table is:<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a09 \u00b7 9 + 7 = 88<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a098 \u00b7 9 + 6 = 888<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0987 \u00b7 9 + 5 = 8888<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a09876 \u00b7 9 + 4 = 88888<\/span><\/p>\n

\u00a0\u00a0\u00a098765 \u00b7 9 + 3 = 888888<\/span><\/p>\n

\u00a0\u00a0987654 \u00b7 9 + 2 = 8888888<\/span><\/p>\n

\u00a09876543 \u00b7 9 + 1 = 88888888<\/span><\/p>\n

98765432 \u00b7 9 + 0 = 888888888<\/span><\/p>\n

Again, if you work out the algebra, the third row becomes:<\/span><\/p>\n

$987$\u00b7$9+5=(900+80+7)$\u00b7$(9)+(5)$<\/span><\/p>\n

$=(9$\u00b7$10^2$<\/span>$+8$\u00b7$10^1$<\/span>$+7$\u00b7$10^0$<\/span>)\u00b7$(10-1)+(8-3)$<\/span><\/span><\/p>\n

$=(9$\u00b7$10^3$<\/span>$+8$\u00b7$10^2$<\/span>$+7$\u00b7$10^1$<\/span>)$-(9$\u00b7$10^2$<\/span>$+8$\u00b7$10^1$<\/span>$+7$\u00b7$10^0$<\/span>)$+8-3$<\/span><\/span><\/p>\n

$=9$\u00b7$10^3$<\/span>$-(10^2$<\/span>$+10^1$<\/span>$+10^0$<\/span>)$-1$<\/span><\/span><\/p>\n

$=9000-111-1$<\/span><\/p>\n

$=8889-1=8888$<\/span><\/p>\n

Now, try to analyze this table by yourself. Remember any integer like 1234 can be written as $10^3+2$\u00b7$10^2$+$3$\u00b7$10^1$+$4$\u00b7$10^0$<\/span>.<\/span><\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 \u00b7 8 + 1 = 9<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a012 \u00b7 8 + 2 = 98<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0123 \u00b7 8 + 3 = 987<\/span><\/p>\n

\u00a0\u00a0\u00a0\u00a01234 \u00b7 8 + 4 = 9876<\/span><\/p>\n

\u00a0\u00a0\u00a012345 \u00b7 8 + 5 = 98765<\/span><\/p>\n

\u00a0\u00a0123456 \u00b7 8 + 6 = 987654<\/span><\/p>\n

\u00a01234567 \u00b7 8 + 7 = 9876543<\/span><\/p>\n

12345678 \u00b7 8 + 8 = 98765432<\/span><\/p>\n

123456789 \u00b7 8 + 9 = 987654321<\/span><\/p>\n

As a bonus, the following are two more magic tables. See if you can figure the algebra that generates the rows:<\/span><\/p>\n

987654321\u00b7 9 = \u00a0\u00a0888888888 9<\/span><\/p>\n

987654321\u00b718 = 1 777777777 8<\/span><\/p>\n

987654321\u00b727 = 2 666666666 7<\/span><\/p>\n

987654321\u00b736 = 3 555555555 6<\/span><\/p>\n

987654321\u00b745 = 4 444444444 5<\/span><\/p>\n

987654321\u00b754 = 5 333333333 4<\/span><\/p>\n

987654321\u00b763 = 6 222222222 3<\/span><\/p>\n

987654321\u00b772 = 7 111111111 2<\/span><\/p>\n

987654321\u00b781 = 8 000000000 1<\/span><\/p>\n

 <\/p>\n

12345679\u00b7 9 = 111111111<\/span><\/p>\n

12345679\u00b718 = 222222222<\/span><\/p>\n

12345679\u00b727 = 333333333<\/span><\/p>\n

12345679\u00b736 = 444444444<\/span><\/p>\n

12345679\u00b745 = 555555555<\/span><\/p>\n

12345679\u00b754 = 666666666<\/span><\/p>\n

12345679\u00b763 = 777777777<\/span><\/p>\n

12345679\u00b772 = 888888888<\/span><\/p>\n

12345679\u00b781 = 999999999<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

[latexpage] Fun Math for Girls By Kelly Tan Did you know multiplication of integers can have many exquisite patterns to admire and show to others? First, look at this magic table with multiplication of integers such as 1, 12, 123, 1234, . . . \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01 \u00b7 9 + 2 = 11 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a012 \u00b7 9 + […]<\/p>\n","protected":false},"author":1,"featured_media":1264,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[28],"tags":[],"class_list":["post-1163","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-articles"],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/kellyfish.me\/wp-content\/uploads\/2017\/09\/tables.png?fit=373%2C201&ssl=1","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1163","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/comments?post=1163"}],"version-history":[{"count":17,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1163\/revisions"}],"predecessor-version":[{"id":1291,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/posts\/1163\/revisions\/1291"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/media\/1264"}],"wp:attachment":[{"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/media?parent=1163"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/categories?post=1163"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kellyfish.me\/index.php\/wp-json\/wp\/v2\/tags?post=1163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}