Yes Virgina .999… really does equal 1.

I have recently been inundated by number of people who for some strange reason can not seem to accept the mathematical fact that .999… = 1. Not just an approximation, not just math tricks, but they are in fact *exactly* equal just as 1/2 = .5 is a objective mathematical fact. There are many things like life which are debatable…mathematical facts are simply not one of them.

If you ever find anyone thinks otherwise, challenge them answer these 4 questions I wrote:

1) ALL rational numbers by definition can be expressed in the form of a ratio of two integers p/q as ℚ = {p/q | (p,q)=1, p,q ∈ ℤ where q ≠ 0}. Since .999… is rational by definition as all repeating decimals are rational. It has to be able to be expressed in the form p/q which is the ratio of two integers. If .999… is rational by definition what is p/q = .999…?

2) .999… = lim k→∞ Σ 9/10^n, n=1 to k by the definition of a limit of a sequence.  lim k→∞ Σ 9/10^n, n=1 to k = 1. If both .999… and 1 = lim k→∞ Σ 9/10^n, n=1 to k please explain how they are not equal.

3) 1- .999… = x. What is x? infinitesimally small values are not allowed in ℝ due to the Archimedean property so given no infinitesimals can exist in ℝ please give the real value of x if x ≠ 0. (If you claim x = 0.000…1 then please construct 0.000…1 using the definition of a Cauchy sequence ε > 0 there exists N such that if m, n > N then |am- an| < ε and please give the sequence and show it is convergent upon 0.000…1).

4) By the trichotomy property you can only have x<y, x=y, x>y. If x <> y then there exists (x+y)/2 between them else as the reals are dense else x=y. If x=1 and y=.999… what is (x+y)/2?

Yes Virginia, using REAL math 1=.999… is absolutely true.

  1. Martinphipps

    .999… is not exactly equal to 1 because it is strictly less than 1.

    Consider 2 minus .999… It is strictly greater than 1. If .999… is exactly equal to 1 then .999… and 2 minus .999… are both equal to 1 but they can’t be equal if one is strictly less than 1 and the other is strictly greater than 1.

  2. Sujit

    That was a fun read, Steve.

    Here’s one more way to think about it.

    x = 0.9999…

    10x = 9.9999…
    10x − x = 9.9999… − 0.9999…
    9x = 9
    x = 1

    So, , . . 0.9999… = 1


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