How can 1 = 0.9999999...?

Question

I am new to proofs, and I am trying to prepare for my discrete mathematics course that I will take next semester. The professor made videos in the past (corona-seasons) and he told us to do this proof on our own.

$$1 = 0.\bar{9}$$

Can anyone help me with this please? I've seen this in my textbook since grade 4 but I could never understand smh.

Welcome to Math S.E. Do you know the formula for the sum of a geometric series $0.9+0.09+0.009+\dots$? — J. W. Tanner, Mar 08 '23 at 22:07
If needed, there are many proofs here: https://math.stackexchange.com/questions/11/is-it-true-that-0-999999999-ldots-1 — Scene, Mar 08 '23 at 22:08
Thank you everyone for your help! The proof that helped me was multiplying it by ten then distributing, subtracting 0.99... on both sides, then dividing it by at the end. — Joseph Choi, Mar 08 '23 at 22:22
@J.W.Tanner That would be $$\sum_{i = 1}^{\infty} \frac{9}{10^i}$$ — Joseph Choi, Mar 08 '23 at 22:28

score 0 · Answer 1 · answered Mar 08 '23 at 22:50

There are several ways to prove it. You could use the geometric sum, which needs some analytic knowledge. Without this, you could do the following: $$ \begin{align} x &= 0.\overline{9} &|\cdot 10\\ 10\cdot x&=9.\overline{9} &| -x\\ 9\cdot x &=9 &|\text{since $x = 0.\overline{9}$} \\ x &= 1 \end{align} $$

How can 1 = 0.9999999...?

1 Answers1