Isn't it cool how 0.9999 (repeating 9's) is exactly equal to one?

Question

I love explaining that one to the mathematically declined, but see who can whip out a quick proof, and whip out nothing else mind you!!!

(sorry didn't mean to call anyone "declined" I was just trying to keep it light around here!)

Even more details......

Thanks for the additional answers, confirming that it is indeed cool. I reassert that indeed 0.999..... is exactly 1. By the way I've got as much math education as most, but I agree this is a real thinker.

Another approach is to agree first that:

1/3 = 0.33333.....

then

3*0.3333.... = 0.99999.... = 3*(1/3) = 1

Answer 1

This can be proven with a geometric sequence where .999... is represented as the summation of Σ9(1/10)^n=9(1/10)+9(1/10)^2+9(1/10)^3... The infinite geometric series summation is equivalent to:
Σar^k=limΣar^k where k approaches infinity = a/(1-r)..
given this formula you just plug in the values...
9(1/10)/[1-(1/10)] =1....

Answer 2

Only if you round it off - otherwise it's always going to be slightly less than 1. Simply stated proof: 0.9999999999 needs the addition of 0.0000000001 to become exactly 1 - irregardless of how many 9's you string out past the decimal point, you will always have to add an equal number of 0's followed by 1 after the decimal point to exactly equal 1. Ever wonder: if I'm given a penny for my thoughts and I give my two cents worth - what happens to the other penny?

Answer 3

I'm not sure I agree with this. Infinity is a generic term and has no specific value. I think it depends upon how one arrived at .9999 repeating infinitesimally as to whether or not it really equals one.
It's quite likely that you've heard the almost banal math problem of the infinite number of students and the infinite number of lockers. This school has a big problem. They have an infinite number of students and lockers and they are sure that currently all the lockers are full. But some new kid moved into the district from out of town and needs to enroll. Will the school have a locker available for him? As the classic solution goes yes they will. Because if the lockers are numbered from 1 to infinity then you can just have every student move to the locker with the next higher number and you'll be fine right?
Ahhh. no. if all the lockers are full then all the lockers are full. In fact it's possible, if each count of lockers and students is infinite, to have more students than lockers, more lockers than students or the exact same number of lockers and students. It's possible to have one more locker than students. It's also possible to have an infinite number more lockers than students.
The classic solution to this locker-student problem ignores the fact that both counts are infinite not just the number of lockers and that infinity is an undefined, generic term.
Stemming from this .99999 repeating might not necessarily be equal to one but it could be as long as you in some way more specifically define the ellipse or the repeating nature of your base ten number to be so.
∞ + 1 = ∞ iff the ∞ in the left side of the equation is defined to be exactly the same as the ∞ on the right side - 1 unless you accept both ∞'s to be undefined. Technically they are a different number though.

Answer 4

Convert 0.99999.. to a fraction.
9/9
simplify
1

Answer 5

The question is:

Does 9.999... - 0.999... = 9 ?

let x be 0.999... = 1 - infinitesimal
10x = 10 - 10(infinitesimal)
9x = 9 - (10(infinitesimal) - 9(infinitesimal))

now the whole thing is meaningless because you cannot operate on the infinitesimal.

Answer 6

Let's say 1 is the number of planets in the universe.
Then how many are we uncertain about whether or not life exists on them or not.
I say .99999999....
So the last 1 does does count.

Answer 7

It cannot be true therefore the proof must be in error. When you multiply by 10 you move the decimal place over one. Therefor the number now has one less 9 after the decimal point. So when you subtract you do not get exactly 9 but 8.999....1
Q.E.D.

Answer 8

Certainly not a mathematician here, but I didn't think that 0.999 (repeating) would be EXACTLY equal to one. Wouldn't it always be just a little bit less than one, no matter how many times the 9 repeats?

Answer 9

Do you just think you're amazing because you can outsmart the majority of the population?

10x = 9.99999
-
x = .99999
________

9x=9

Answer 10

just one of the many interesting results of playing with infinities and infinitesimals