Why does 1/3=0.333333333333333... repeating?

It doesn’t , it stops repeating after a few dozen decimals and then transforms into pictures of butterflies and flowers and then starts to make a musical sound and becomes a beautiful, mind altering experience —- the issue is most people are too lazy to fully calculate it

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.

It’s Drunk

1/3 ammonia scrubber time