is 0 9 equal to 9 The short answer is yes 0 9 zero point 9 repeating is exactly 1 However a lot of people find this result counter intuitive people often feel that 0 bar 90 9
So the common proof that I have seen that 0 999 that is 9 repeating to infinity in the decimal is equal to 1 is let x 0 999 10x 9 999 10x x 9 999 0 999 9x 9 x 1 For this statement to be true there must be 0 0 followed by a 1 to satisfy the claim 0 9 repeating will always be 0 0 1 away from 1 There can not be a number following a repeated
is 0 9 equal to 9
is 0 9 equal to 9
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Does 0 999 1 The idea is that 0 9 recurring 0 999 with the digits going on forever is actually equal to 1 Is this really true You decide But here we give some nice arguments as Indeed the most sensible definition is the following 0 9 lim n inf 0 9 9 n times Another way to express 0 9 9 n times is using the following sum 0 9 0 09 0 009 0 0 09 sum 1 to n 0 9 0 1 k 1 Taking the limit as n
The two numbers are not actually equal If x 99999 then 10x 9 99999 and 9x 9 so x 1 This proof of 10x x is especially insidious as it is a simple case of algebraic trickery Enter the expression you want to evaluate The Math Calculator will evaluate your problem down to a final solution You can also add subtraction multiply and divide and complete any
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Formal proof I followed it for a while until to the 0 le 1 x ldots line then I got lost in how it was trying to prove Algebraic arguments I don t follow how this one works because Essentially as the number of 9s in the decimal expansion of 0 9 recurring increases the difference between 0 9 recurring and 1 becomes infinitesimally small
Yes both claims are true 0 9 1 is also a true claim and this has no bearing on 0 9 0 9 or 1 1 The sum is not equal to the limit Yes it is actually By definition it is The Proof The statement that we want to prove is 0 99999999 1 Note that the 3 dots after the 9s mean recurring i e that there are an infinite number of nines after the decimal point
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is 0 9 equal to 9 - 0 9 repeating is equivalent to 1 because the difference of them is 0 0000 1 or 0 1 to the power of infinity but if there is an infinite amount of zeros there s no room for the one