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DrWatson · 70-79, M
One way to think of it is to see the infinite decimal as a sequence of increasingly better approximations to 1/3.
We know that 3 times 1/3 equals 1.
1/3 is approximately .3, since 3 x .3 = .9. We fall short by .1.
1/3 is better approximated by .33, since 3 x .33 = .99. Now we are short by .01
We can keep doing this:
1/3 is approximately .333333, since 3 x .333333 = .999999, and we are short by only .000001
Finitely many 3's , no matter how many, will never get you to 1/3 exactly. But the error (the amount we fall short by) will "approach" 0 as the number of 3's "approaches" infinity.
That last sentence is the "formal" way mathematicians express this.
Informally, it takes infinitely many 3's to get an error of 0, and thus to arrive exactly at 1/3.
We know that 3 times 1/3 equals 1.
1/3 is approximately .3, since 3 x .3 = .9. We fall short by .1.
1/3 is better approximated by .33, since 3 x .33 = .99. Now we are short by .01
We can keep doing this:
1/3 is approximately .333333, since 3 x .333333 = .999999, and we are short by only .000001
Finitely many 3's , no matter how many, will never get you to 1/3 exactly. But the error (the amount we fall short by) will "approach" 0 as the number of 3's "approaches" infinity.
That last sentence is the "formal" way mathematicians express this.
Informally, it takes infinitely many 3's to get an error of 0, and thus to arrive exactly at 1/3.