Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of nines:<\/p>\n\n\n\n
0.99999…<\/p>\n\n\n\n
It may come as a surprise when you first learn the fact that this real number is actually EQUAL to the integer 1. A common argument that is often given to show this is as follows. If S = 0.999…, then 10*S = 9.999… so by subtracting the first equation from the second, we get<\/p>\n\n\n\n
9*S = 9.000…<\/p>\n\n\n\n
and therefore S=1. Here’s another argument. The number 0.1111… = 1\/9, so if we multiply both sides by 9, we obtain 0.9999…=1.<\/p>\n\n\n\n
Presentation Suggestions:<\/strong> The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong> We can try to clear that up by explaining what a decimal representation means. Recall that the digit in each place of a decimal expansion is associated with a (positive or negative) power of 10. The k-th place to the left of the decimal corresponds to the power 10^k. The k-th place to the right of the decimal corresponds to the power 10^(-k) or 1\/10^k.<\/p>\n\n\n\n If the digits in each place are multiplied by their corresponding power of 10 and then added together, one obtains the real number that is represented by this decimal expansion.<\/p>\n\n\n\n So the decimal expansion 0.9999… actually represents the infinite sum9\/10 + 9\/100 + 9\/1000 + 9\/10000 + …<\/p>\n\n\n\n which can be summed as a\u00a0geometric series\u00a0to get 1. Note that 1 has decimal representation 1.000… which is just 1 + 0\/10 + 0\/100 + 0\/1000 + … so if one realizes that decimal expansions are just a code for an infinite sum, it may be less surprising that two infinite sums can have the same sum.<\/p>\n\n\n\n Hence 0.999… equals 1.<\/p>\n\n\n\n How to Cite this Page:<\/strong> Fun Fact suggested by: <\/strong> Consider the real number that is represented by a zero and a decimal point, followed by a never-ending string of...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,10],"class_list":["post-323","page","type-page","status-publish","hentry","tag-easy","tag-numtheory"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/323","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=323"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/323\/revisions"}],"predecessor-version":[{"id":1701,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/323\/revisions\/1701"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=323"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=323"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
You might also mention that by similar arguments, every rational number with a terminating decimal expansion has another expansion that ends in a never-ending string of 9’s. So, for instance, the rational 7\/20 can be represented as 0.35 (the same as 0.35000…) or 0.34999…<\/p>\n\n\n\n
When seeing these arguments, many people feel that there is something shady going on here. Since they do not have a clear idea what a\u00a0decimal expansion\u00a0represents, they cannot believe that a number can have two different representations.<\/p>\n\n\n\n
Su, Francis E., et al. “Why Does 0.999… = 1?.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"