(adsbygoogle = window.adsbygoogle || []).push({}); I will prove that 1=0.999999... This one IS NOT a troll (no sneaky tricks like the last one i.e. dividing by zero).

The Fractions Proof

This one requires little to no math knowledge. Note that You must login or register to view this content. represents 0.999999...

One third is equal to 0.33333...

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Three times one third is equal to 1:

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Three times 0.33333... is equal to 0.999999...

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Therefore 1=0.999999...

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The Infinite Series Proof

A number with a repeating decimal can be represented as a sum of an infinite series:

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The formula to find the value of an infinite series where |r|<1 is as follows:

S is the sum, t1 is the first term in the series, and r is the rate.

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Therefore:

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The Algebraic Proof

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Multiply both sides by 10 (remember that when you multiply by ten it shifts the decimal place to the right by one place):

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Subtract x=0.9999... from both sides:

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Simplify:

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Divide by 9:

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The Midpoint Proof

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What is the midpoint (average) between a and b? The formula to find m where m is the midpoint is:

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Using the values for a and b:

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You can use long division to divide 1.9999... by 2 to get m=0.99999...

Since the midpoint between the two points IS one of the two points, the two points MUST be equal. If you are unconvinced that this is true, you could say from the information that:

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Multiply both sides by two:

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Subtract b from both sides:

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Simplify:

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Substituting numerical values:

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Holy shit.

ok i just wanted to share my view on this matter because it really, really bothers me lol. You claim somewhere in here that since .9 is repeating it has infinite zeros so it will never have the .000..1 round off. BUT you are looking further into it then logic allows. .9 repeating is simply just .9 repeating. It will Never actually reach one no matter how close it comes. I forget the simple parabola equation (maybe its 1/x^2) where no matter how close the parabola curve comes to reaching 1 it will actually never hit 1. Now back to our equation. Regardless of how many times 9 repeats it will never ever reach 1. Especially since it is INFINITE, it means that it will never round up to 1 also making this in my eyes unsolvable because there will never be an answer due to the fact that their will never be an end to the number to make it completely accurate. I also find a problem with converting fractions and decimals a little inaccurate because the fact that we can make 1/3 exactly into 1 by multiplying it 3 times. but when you convert 1/3 into a decimal you do not get a complete decimal you got a repeating decimal. so even if you multiply .3 repeating by 3 you will get .9 repeating. Going with my logic above then .9 repeating will never equal 1.

Please let me know what you think or where i went wrong

This is pretty standard to be honest, I learnt the Algebraic proof when I was 13.

Surely this just proves that the 3s, or 9s recur so much that the overall number is so close to 1 you just may as well use 1. There's no significant loss of accuracy when you get to a few thousand decimal places down the line.

If youre going to buy something from a dollar store its better just to give 1 dollar bill instead of 99 cents.there's no big difference.it proves 1=.999999.........lol.

No I'm not, I'm simply asking what 1-0.9999... is. The difference is an infinite string of zeros, therefore they are the same number.



Algebra may have taught you that an asymptote means that the function will never reach that value, however calculus tells us something different. Just like a limit can tell us EXACTLY the slope between two points as the difference between them approaches zero, a limit can also tell us the value of a function as x approaches ∞. The value of the function f(x)=1/x DOES eventually converge to zero when x→∞ or when x→-∞.

The algebraic concept of an asymptote is taught for simplicity, which is why you are told in school that the value never converges to the meet the asymptote.



You only need to do a bit of long division to find that the decimal representation of 1/3 is indeed accurate.

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ok i understand the bottom two but with the 1-.999 their is an infinite number of zeros making it seem that it is equal to zero because of the fact that an infinite amount of zeros would never bring the one but i just feel like regardless of the infinite amount of zeros that comes it has to have that added .000..1 on the end of it to make it equal one. Im sure you are right it just makes my brain itch a little lol. I haven't taken calculus since freshman year of college lol Idk just doesnt make sense to me common sense wise but i just love questioning things. lol

1 over 3 does not equal 0.3333333333, its 0.3 followed by a infinite amount of 3's. There is a huge difference. You are rounding off, thats like saying 2.5= 3, no shit if you round off it will.

^He didn't round the number he just used ... to represent that it goes on forever. How else do you expect him to write an infinite number or 3's?