Why is .9 bar equal to 1?

Answer 1

1 over 3 is equal to .33333333.... So, 3 over 3 must be equal to .999999999... However, 3 over 3 is also equal to 1. So, this is proof that .99999999999999999.... is equivalent to 1.

That is a pretty easy way to think about it. Just multiply each digit by 3, which is the same as multiplying 1/3 by 3. To convert any repeating decimal to fraction, follow these steps: Let x = the decimal. So for example x = 0.3333....

Now multiply both sides by a power of ten, equal to the number of digits that repeat. So for example there is one digit which repeats (the 3), so multiply by 10¹ which is 10. So we have 10x = 3.3333.... So now we can subtract the original equation {x = 0.3333....} from the new equation. Remember the 3's repeat forever, so every one will cancel out all the way to the end:

• 10x - x = (3.3333....) - (0.3333....) = 3
• 9x = 3 ; solve for x = 3/9 which simplifies to 1/3.

Now do x= 0.9999.....

• 10x = 9.9999.....
• 10x - x = (9.9999....) - (0.9999....) = 9
• 9x = 9, x = 9/9 = 1.