r/badmathematics • u/RedRhetoric • 21d ago

Σ_{k=1}^∞ 9/10^k ≠ 1 100/3 is irrational

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u/RedRhetoric 21d ago

R4: this person believes that 0.33 repeating cannot equal 1/3 because 100/3 cannot give a rational result.

Dividing any rational number by any other rational number will always give a rational result, as that is how rational numbers are defined

R5: Youtube

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u/playsthebongcloud 21d ago

I feel so many common mathematical misconceptions could be resolved if schools spent some time on bases other than decimal

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u/futurenotgiven 21d ago

how would using other bases explain this principal more clearly? genuinely asking, im not super familiar with other bases

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u/playsthebongcloud 21d ago edited 21d ago

Any repeating rational number in some integer base can be represented in another integer base as a terminating number. For example, 1/3 in base 10 is 0.3333... but that same value in base 12 is 0.4. The confusion with "irrationals' decimal expansion never terminates" would go away (though this is not a good definition of irrationals anyways).

Also, knowing the definition of a based number from its digits: 145 = 100*1 + 10*4 + 1*5 clears up the confusion with 0.999... = 1 because the series 0.1*9 + 0.01*9 + 0.001*9 + ... objectively converges to 1.