O.9999999... = 1?
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Re: O.9999999... = 1?
O.9999999... doesn't equal 1, simply because it's an O in front, not 0 (Zero)
Testing colors: CYAN LIME
Testing colors: CYAN LIME
Re: O.9999999... = 1?
The problem is not 0.99999999999...
10-9.9999999999999999.... seems more confusing. Can someone explain that smallest positive number?
10-9.9999999999999999.... seems more confusing. Can someone explain that smallest positive number?
Re: O.9999999... = 1?
Can someone explain to me why “a” is essentially the same as “A”? Only don’t use the whole alphabet, only use the letters a, h, x, and z. Any answers that contain letters other than those I won’t accept for some arbitrary reason. Thanks.
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Re: O.9999999... = 1?
I disagree with the notion that 0.9999999... = 1. Usually people like to "prove" this by talking about 0.333333... = 1/3, and that 1/3*3 = 1, so 0.33333...*3 = 1, and thus 0.9999... = 1. But this fails to take into account the fact that 0.33333... is merely a decimal approximation of 1/3, albeit infinitely close, just as 0.9999... is infinitely close to 1.
The two numbers are differentiated by an amount equal to 1*10^-infinity.
The two numbers are differentiated by an amount equal to 1*10^-infinity.
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Re: O.9999999... = 1?
I'm not sure that is correct. If you ever stop the repeating sequence, THEN 0.3333333333333333333333333333 is an approximation of 1/3. But if you accept that 0.3333333333333333333333.... is a repeating rational number, then it IS EQUAL to 1/3. Its the infinite repetition that makes it equal. Any contrary argument implicitly assumes that the function stops repeating at some point.But this fails to take into account the fact that 0.33333... is merely a decimal approximation of 1/3
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Re: O.9999999... = 1?
what's up with these random forums popping back up after so long? 7 years or so before the posts today... crazy! Anyway, I saw math and felt the urge to post/correct:
0.99999... infinitely repeating is equal to 1.
10*0.999999...=9.999999...
10*0.999999...-0.999999...=9*0.999999...=9.999999...-0.999999=9
therefore: 9*0.999999...=9, divide both sides by 9, and we get 0.999999...=1
furthermore, the limit as x->-infinity of 1*10^x is 0, so 1-0.99999... infinitely repeating is 0, so 1=0.99999... infinitely repeating.
the two numbers are mathematically equal to each other. Another way to think of is as the limit of the series 9*10^-n, as n goes to infinity.
for a more worded explanation: consider 0.999... repeating. to make exactly 1, a person would need to add 0.000...1, where there are infinity 0's before the 1. the 1 will never arrive, because there are infinite 0's, which is the same as 0. so 0.9... repeating is the same 1, since their difference is 0. This explanation was how I first comprehended the equality between the 0.9... repeating and 1.
0.99999... infinitely repeating is equal to 1.
10*0.999999...=9.999999...
10*0.999999...-0.999999...=9*0.999999...=9.999999...-0.999999=9
therefore: 9*0.999999...=9, divide both sides by 9, and we get 0.999999...=1
furthermore, the limit as x->-infinity of 1*10^x is 0, so 1-0.99999... infinitely repeating is 0, so 1=0.99999... infinitely repeating.
the two numbers are mathematically equal to each other. Another way to think of is as the limit of the series 9*10^-n, as n goes to infinity.
for a more worded explanation: consider 0.999... repeating. to make exactly 1, a person would need to add 0.000...1, where there are infinity 0's before the 1. the 1 will never arrive, because there are infinite 0's, which is the same as 0. so 0.9... repeating is the same 1, since their difference is 0. This explanation was how I first comprehended the equality between the 0.9... repeating and 1.
the smallest positive number also doesn't really exist, since you could get infinitely close to 0, but there will always be number between any number you give and 0. as a direct counter example, if 1*10^-infinity is a positive number (which implies non-zero), a smaller number would be 1/2*10^-infinity. and you could keep cutting that smaller and smaller, ad infinitum.
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