What if 0.999... does not equal 1?



The stupid numbers, S, form a set which may or may not be a field. The size of the set S has not been determined and S may be the empty set. There are two binary operations on S, addition ( + ) and multiplication ( * ), which may or may not make the set S a commutative group under addition and may or may not make the non-zero elements of S a commutative group under multiplication. ***S ***is based on the following axioms:

Axiom of Reflexivity:  For every x, x = x.
Axiom of Symmetry:  For every x and y, if x = y then y = x.
Axiom of Transitivity:  For every x, y, and z, if x = y and y = z then x = z.
Axiom of Additive Commutativity:  For every x and y, x + y = y + x.
Axiom of Additive Associativity:  For every x, y, and z, x + (y + z) = (x + y) + z.
Axiom of the Existence of the Additive Identity:  There exists an element x such that for every y, x + y = y.
Axiom of the Existence of the Additive Inverse:  For every x there exists a y such that x + y = 0.
Axiom of Multiplicative Commutativity:  For every x and y, x * y = y * x.
Axiom of Multiplicative Associativity:  For every x, y, and z, x * (y * z) = (x * y) * z.
Axiom of the Existence of the Multiplicative Identity:  There exists an element x such that for every y, x * y = y.
Axiom of the Existence of the Multiplicative Inverse:  For every x there exists a y such that x * y = 1.
Axiom of Distributivity:  For every x, y, and z, x * (y + z) = (x * y) + (x * z).
Axiom of Stupidity:  0.999 … ≠ 1.  Note: 0.999 … = 0.9 ̅

Discussion # 1:

Are the axioms of the stupid numbers consistent or inconsistent? If the axioms are inconsistent, which axiom(s) could be removed, without the removal of the Axiom of Stupidity, to make the remaining axioms consistent? If the axioms are inconsistent, can more than one scheme of axiom removal without the removal of the Axiom of Stupidity be devised to make the remaining axioms consistent? If so, which scheme(s) create interesting or useful axiomatic system(s)?

One scheme of axiom removal to ensure that the axiomatic system is consistent would be to remove all of the axioms except the Axiom of Stupidity. Would this result in S = {0.999 … , 1} and the operators of addition and multiplication being undefined?

Alternatively, if the axioms of stupid numbers are inconsistent, could additional axioms be added to make the axiomatic system consistent?

Discussion # 2:

If the axioms of the stupid numbers are consistent:
What is size of S?
Is S a field?
Is S a commutative group under addition?
Are the non-zero elements of S a commutative group under multiplication?
Is 1 > 0.999 … ?
What are some of the other properties of S?

If axioms, other than the Axioms of Stupidity, must be removed or added to make the axiomatic system of the stupid numbers consistent, what are the answers to the above questions using the new axiomatic system(s)?

Discussion # 3:

You may note that the axioms of S do not include any axioms of order. What are some of the consequences of this omission? Without any axioms of order, does the relation > have any meaning? Can consistent axioms of order be added to the axiomatic system of the stupid numbers?

These rules are contradictory. If there exists a zero element as described (the result of an additive inverse and identity), 0 is in your set. However, there exists no y such that 0*y=1.

Perhaps the Axiom of the Existence of the Multiplicative Inverse could have been better stated as For every x other than 0, there exists a y such that x * y = 1.

If you don’t assume a priori that 0 and 1 are distinct elements (S wasn’t claimed to be a subset of the real line), the set with a single element and the trivial addition and multiplication operations satisfies the axioms.

This question keeps coming up in these forums (I’m not sure why- echoes of Zeno’s Paradox?) Your long list of axioms, modulo corrections such as have already been mentioned, are distracting from the focus of your question, which is about the expression 0.999… which you have left undefined. If, for example, you construct something like hyperreal numbers and give the expression an appropriate interpretation then, sure, you can make it strictly less than 1. This is called non-standard analysis and is a legitimate approach.

in the stupid numbers, could 0.999… be greater than 1?

0.999… are just symbols on a page.

You cannot define what it is not. You have to define what it is.

So saying 0.999… is not a cat does not really help, either.

(Reminder, as a Computer Science person I naturally know that 0.999… can not be 1 simply because they are different types. One is an infinite series and the other is a number. The limit of 0.999… is 1. The equality is a shorthand but not a formal equation. So I am perfectly fine with saying they are not equal and the universe does not collapse.)

Your problem there is that overexposure to computers has made you overlook the fact that almost no real number can be stored in or represented by a computer :slight_smile: One (imaginary) way to do it would be to store infinitely many binary or decimal digits; expressions like 0.3461169134728834… and 0.4999999… would be legitimate real numbers and not merely “infinite series”. (There is a “type” of real number that can be represented by a finite number of binary digits; these are rational numbers whose denominator is a power of 2. But most real numbers cannot even be calculated by a computer.)

That is merely a digression, though; I add my voice to yours in calling for the poster asking whether 0.999… can be bigger and/or smaller than 1 to define what he or she supposes the former expression to mean.

All I can determine so far is that 0.999… is an element of the set S, if S is not the empty set, and that 0.999… does not equal 1. Given the proposed axioms for S, I was hoping someone could derive some of the properties of the stupid numbers and 0.999… in particular. Remember, the stupid numbers are not the real numbers (although I borrowed most of the axioms from the real numbers).

What is 1 - 0.999… in this system?

If I were defining the axioms to create this system, I would at least say that, rather than 1 != .999…, I would say that 1- 0.999… is equal to an undefined but positive number. Maybe, as long as we are redefining numbers, turn 0.000…1 into a valid number?

So 1=0.999…=0.000…1.

Though, I do not think that this would make any sort of math problems easier, probably make them harder.

Is there anything that this could be used for beneficially?

At least you’re not one of the weirdos that think we should switch to using Tau, that’s just bonkers. :slight_smile:

You can explode this by saying that 1 is simply how we conventionally write 1.000… so they’re the same type again, and therefore exactly equal. This makes sense, in fact, because if you want 1 and 0.999… and 0.5 to exist on the same number line, they have to be comparable, and the comparison operators can’t compare things which aren’t of the same type.

Finally, as a Computer Science person, I’d view a type system which didn’t treat 0.999… and 1 as both being Numeric and, therefore, of the same type in one sense even if one’s an Integer and the other’s a Float or Double as being crippled. But then, I want my type systems to reflect semantics, not underlying implementation issues, unless I explicitly use implementation-defined types.

The only beneficial use I can think of for stupid numbers is a response to people that insist that 0.999… does not equal 1 in the real numbers. If we accept that 0.999… does not equal 1 as an axiomatic truth, what kind of number system would we have left?

If 0.9999… and 1 are distinct real numbers, then |1 - 0.9999…| > 0, and so is |1 - 0.9999…|/2. Let X = |1 - 0.9999…|/2.

Assuming 0.9999… < 1, 0.9999… + X = 1 - X. And 1 - X in decimal form is…?

As has been explained, this depends on a lot more than 0.999… just being unequal to 1. For example, we could end up with the hyperreals; in that case, 0.999… would be shorthand for the sequence (9/10, 99/100, 999/1000…), or a number which gets infinitesimally close to 1 without being equal to it.

The hyperreals are perfectly well-defined and used to to calculus, however, so proving that your system was equivalent to them wouldn’t destroy the arguments of those who claim that 0.999… is not equal to 1.

I love these threads! I have a limited math education that ended with right angle trig. I’ve always hated math, so I can declare without hesitation that any number that begins with a decimal point is always less than one. So there, and phooey upon any disagreement. :stuck_out_tongue:

When I saw the thread title, I said “Noooooo…”

There have indeed been numerous discussions about 0.999… here before, including this thread, started in 2000, that went on for 44 frickin’ pages and is now closed, and this more recent 11-page thread—not to mention the Straight Dope column An infinite question: Why doesn’t .999~ = 1?

When I read the OP, I found it to be more intelligent and mathematically literate than I had feared. But what jumped out at me was what other replies have already noted: “0.999 …” is never defined. If you’re trying to break things down to axioms, you need to establish what “0.999 …” is—what does that notation refer to?

The “Discussion” questions then might be phrased as “Is there a way of defining 0.999… such that…” all your axioms are satisfied? etc.

You are missing the closure axiom: if x and y are stupid numbers, the x+y and xy are stupid numbers.
If the closure axiom is included, the inverse axioms imply that 0 and 1 are stupid numbers.
How is a number defined to be stupid if 0 and 1 are stupid?

To move things along, can anyone suggest an expansion of the Axiom of Stupidity or a new axiom that more clearly defines “0.999…”. It appears that we may be heading towards something like the hyperreal numbers. Also, what about axioms of closure and axioms of order? Will it be necessary to add these as well?

To the OP: If you want to get serious first read up about Dedekind cuts, from which you will want to learn about completeness and linear continuums. Then to really get to work, start reading about infinitesimals.

There’s a lot of stuff, a lot, that has to hold for a usable number system akin to the Reals. Commutativity and associativity are trivial by comparison.

The real problem is that 0.999… acts as a shiny object for math deniers. But it’s simply a special case of a more general function.

0.999… is 9/10 + 9/10[sup]2[/sup] + 9/10[sup]3[/sup] + 9/10[sup]4[/sup] + … = 1

But any two adjacent numbers produces a similar infinite string that adds to one.

1/2 + 1/2[sup]2[/sup] + 1/2[sup]3[/sup] + 1/2[sup]4[/sup] + … = 1
2/3 + 2/3[sup]2[/sup] + 2/3[sup]3[/sup] + 2/3[sup]4[/sup] + … = 1
11/12 + 11/12[sup]2[/sup] + 11/12[sup]3[/sup] + 11/12[sup]4[/sup] + … = 1

All of them get closer and closer to one, but differ from 1 at the last calculated step by a different amount. (Actually any value from 0.000…1 to 0.000…9.) So 1-X can have ten different values.

How does the stupid axiom account for this? Why is 0.999… singled out? Why not any of the other infinite number of cases?

That’s the difference between real math and these silly workarounds. Infinity will sneak in and bite you on the ass every time.