If the fraction answer is 2 and the decimal answer is 1.9 repeating and it says to compare and make sure theyr the same is the repeating decimal rounded to be the same as the 2?

Question

anonymous · Answer

I think you're being asked to show that,$$1.9999999...=2$$This is one of those things where the introduction of an infinity of something gives you something (here, the infinitely repeating 9's) that is true, but counter-intuitive.

To see this, let $$x=1.99999999...$$then$$10x=19.9999999...$$If you subtract x from 10x, you get$$10x-x=9x=19.9999999...-1.99999999...=19-1=18$$In the middle-ish step, we've used the fact that you're taking away the infinity of repeating 9's from each number (this is proved a lot better if using sigma notation and the definition of numbers in base 10, but I think this gets the message across).

So now you have $$9x=18 \rightarrow x=\frac{18}{9}=2$$But we said that $$x=1.9999999...$$So$$1.9999999...=2$$If you're required to show a 'proper' proof, let me know.

Hope I've explained it well :)

anonymous · Answer

no i dont need proper proof thanx for the help!(;