0.9 repeater = 1 exactly

This blew my mind.  If you asked me a few days ago, I’d have said that 0.999999 (with 9’s repeating to infinity) could never exactly equal 1, since it would always be that little bit less than one.  Sure, it would tend towards one asymptotically, but it would never quite get there.  But here’s a proof someone told me to show that it does actually equal precisely one:

1/9 = 0.111111 (with 1s repeating to infinity)

Multiplying both sides by 9 gives:

1 = 0.999999 (with 9s repeating to infinity)

This seems to make much and little sense, simultaneously!  I guess infinity means an awful lot of nines.  My brain hurts.

One thought on “0.9 repeater = 1 exactly”

1. Innes on said:

Equals signs are overused – being mathematically pedantic the expression below is false:

1/9 = 0.111111 (with 1s repeating to infinity)

These two things are not equal. Strictly speaking 0.1111111111. . . is a decimal approximation of the vulgar fraction 1/9.

Therefore, it is accurate to write:
1/9 ≈ 0.11111111 (where the squiggly equals sign means almost equal to).

And it does indeed follow that (multiplying by 9)
1 ≈ 0.99999999. . .

It is often the case that ‘=’ is used incorrectly in place of ‘≈’ where impossible proofs are performed.

Obviously, for any practical application this misuse of the equals sign is unimportant. But I suppose that strictly speaking the ‘=’ button on my calculator should really be a ‘≈’ – well, only when I’m performing a calculation where the answer cannot be precisely expressed in decimal notation. Maybe I need another button, and to know the answer before I perform the calculation, so that I know which button to use. Maybe that’s a silly idea. Anyways, hope this stops everyone’s head from hurting.

Innes