{"id":168,"date":"2019-06-26T22:53:06","date_gmt":"2019-06-26T22:53:06","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=168"},"modified":"2019-12-09T22:17:04","modified_gmt":"2019-12-09T22:17:04","slug":"one-equals-zero","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/one-equals-zero\/","title":{"rendered":"One Equals Zero!"},"content":{"rendered":"\n

The following is a “proof” that one equals zero.<\/p>\n\n\n\n

Consider two non-zero numbers x and y such that<\/p>\n\n\n\n

x = y. 
Then x2<\/sup> = xy. 
Subtract the same thing from both sides: 
x2<\/sup> – y2<\/sup> = xy – y2<\/sup>. 
Dividing by (x-y), obtain 
x + y = y. 
Since x = y, we see that 
2 y = y. 
Thus 2 = 1, since we started with y nonzero. 
Subtracting 1 from both sides, 
1 = 0.<\/p>\n\n\n\n

What’s wrong with this “proof”?<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
This Fun Fact is a reminder for students to always check when they are dividing by unknown variables for cases where the denominator might be zero.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
The problem with this “proof” is that if x=y, then x-y=0. Notice that halfway through our “proof” we divided by (x-y).<\/p>\n\n\n\n

For a more subtle “proof” of this kind, see\u00a0One Equals Zero: Integral Form.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “One Equals Zero!.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
R. Vakil, A Mathematical Mosaic<\/a><\/em>, 1996. p. 199.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
James Baglama <\/p>\n","protected":false},"excerpt":{"rendered":"

The following is a “proof” that one equals zero. Consider two non-zero numbers x and y such that x =...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[20,7,3,19,12,18],"class_list":["post-168","page","type-page","status-publish","hentry","tag-analysis","tag-calculus","tag-easy","tag-false-proof","tag-other","tag-paradox"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/168","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=168"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/168\/revisions"}],"predecessor-version":[{"id":1553,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/168\/revisions\/1553"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=168"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}