My answer to a question on Quora:<\/p>\n
Mathematicians frequently make mistakes, but they also have instincts and skills to identify them. To locate an arithmetic mistake in a long calculation could be very difficult, but, in surprisingly many cases, it is possible to say that the result is wrong because it does not behave properly under transformation of inputs.<\/p>\n
Unfortunately these all-important skills of checking the answers are ignored in the mainstream mathematics education. I am trying to show my students at least some examples. Here is one from a recent lecture.<\/p>\n
Problem: A truck travelled from A to B with average speed\u00a060<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0km\/h and back, on the same road, with average speed\u00a040<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0km\/h. What was the overall average speed of the truck?<\/p>\n As I expected, students\u2019 answer was\u00a050<\/span><\/span><\/span><\/span><\/span><\/span> km\/h. I asked them: \u201cso you believe that if I put $u$ <\/span><\/span><\/span><\/span><\/span><\/span>and $v$ <\/span><\/span><\/span><\/span><\/span><\/span>instead of\u00a060<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0and\u00a050<\/span><\/span><\/span><\/span><\/span><\/span>, the answer should be $\\frac{u<\/span><\/span>+<\/span><\/span>v}{<\/span><\/span><\/span><\/span>2}$<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>?\u201d Their answer was affirmative \u201cyes\u201d. \u2014 \u201cOK\u201d, said I, \u201cbut what if if the truck run out of fuel at B and its speed on the way back was\u00a00<\/span><\/span><\/span><\/span><\/span><\/span> km\/h. Your formula produces the answer $\\frac{60<\/span><\/span>+<\/span><\/span>0}{<\/span><\/span><\/span><\/span>2}<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>30$<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0km\/h. But the truck will never come back! What is wrong?\u201d<\/p>\n Of course, this example is from school level mathematics, but it gives a simple example of transformation of inputs as a way of checking the answer. In research mathematics, there are many other (quite sophisticated and subject-specific) methods of checking the result without looking into details; they are not a substitution for a proof, but allow to detect 95% of mistakes.<\/p>\n","protected":false},"excerpt":{"rendered":" My answer to a question on Quora: Mathematicians frequently make mistakes, but they also have instincts and skills to identify them. To locate an arithmetic mistake in a long calculation could be very difficult, but, in surprisingly many cases, it is possible to say that the result is wrong because it does not behave properly […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-159","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/159","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/comments?post=159"}],"version-history":[{"count":7,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/159\/revisions"}],"predecessor-version":[{"id":166,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/posts\/159\/revisions\/166"}],"wp:attachment":[{"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/media?parent=159"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/categories?post=159"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/borovik.net\/selecta\/wp-json\/wp\/v2\/tags?post=159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}