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Earlier quoted context omitted.
or, x = 0.9999... 10x = 9.999... 10x = 9 + 0.999... 10x = 9 + x 9x = 9 x = 1 Presented slightly more clearly https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments
Yeah that’s way more complicated than it needs to be and I’m tempted to replace that whole section with: x = 0.9999... 2x = 1.9999... 2x - x = 1 x = 1
x = 0.999...
to
2x = 1.999..
This can be solved by using base 12 rather than base 10 to do the calculation...
Firm believer that adopting base 12 would have had a ripple affect on society, preventing many trials tribulations and wars. Pity we only have base 10 and donald trump
I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious: 1/3 = 0.333.. 3 * 1/3 = 3 * 0.333.. 3/3 = 0.999.. 1 = 0.999..
I remember WarCraft 3 official forums being torn apart by this, with probably thousands of comments in the thread. Blizzard even had to post their official stance on the issue, but that didn't calm those who insisted 0.999... was 1 minus epsilon and not exactly 1.
I'll just chime in with my completely ignorant theory that 1 - 0.999... = the infinitely smallest number, but is still, in my mind, regardless of any logic, reason, or educated calculations, greater than 0. I understand and accept this is wrong. However, somewhere in my brain I still believe it. Sort of like +0 and -0, which are also different in my head.
Mathematically you are wrong.
It is indeed true that if there was an entity that could reason beyond infinity 1-0.999... would be greater than 0.
Maybe the major source of confusion is that our decimal representation for whole numbers is supposed to be unique. Then when we extend it to rationals and reals this property fails at rationals in the form of a/10^n. Arguably the sign symbol ruins it for whole numbers as well, as +0 and -0 could be equally valid representations of the number 0. We just conventionally don't allow -0 as a representation. There are othe…
I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious: 1/3 = 0.333.. 3 * 1/3 = 3 * 0.333.. 3/3 = 0.999.. 1 = 0.999..
Earlier quoted context omitted.
Firm believer that adopting base 12 would have had a ripple affect on society, preventing many trials tribulations and wars. Pity we only have base 10 and donald trump
base 12? Them are fighting words...I call WAR!
This is 'more intuitive' if you think about it this way: If any two real numbers are not equal, then you can take the average and get a third number that is half way between them. Conversely, if the average of two numbers is equal to either of the numbers, then the two numbers are equal. (this isn't a proof, just a way to convince yourself of this) What's the average of .9999... and 1?
0.999…5 obviously.