Is 1 = 0.999... ? Really?

Philosophy Forums Science, Technology, and Math
1981

I can tell you what happens when logic and math disagree.

A question gets asked… how is that possible, and all of a sudden you aren’t so concerned with whether it is or it isn’t, but how it can be both at once.

Allow me to wax on. A coin, it is the same coin with two sides, that are as different as a head and a tail, but you can still find them on the same coin. It’s a sort of planar linear intersection. On one side of the plane pointing in one direction of the line is a reference to movement in one direction and it looks like a whole turn around and look in the other direction and you would swear its something less.

Look at it one way and it looks like one thing, look at in the opposite direction and it looks like something else so different they can not logically be the same thing.

1 = 0.9 recurring and it doesn’t. It’s a plane of reference in two directions intersecting perpendicular to an infinite line.

Previously I could only see the relationships in question as only a point along a line. One point can not be anything other then what it is. But this equation doesn’t represent the location of a point along a line. It represent the intersection of a plane and a line. A single plane has two sides that can look so different while still being two sides of the same plane. A sort of pencil stuck through a sort of piece of paper.

Thank you all and hey James.

As analogies go the arrow did hit one side of the target it just didn’t get through to the other side.

I just had to imagine it from a line parallel to the one in question with some distance of separation. The same plane intersects this line too. Funny I didn’t see this line before. I was stuck on the one with a point on it. >drum drum cymbal< or is that symbol crash.

1982

Thanks Sil, you’re a fine mathematician. And a most worthy foil. I concede. 1 = 0.9 recurring from one perspective and is not equal from another.

But 1 and 0.9 recurring aren’t the same thing while they are like two sides to the same coin.

1983

I’m just “a” mathematician, but thanks all the same back to you - for being a fine example of how not to double down in the face of reasoning that is contrary to that which you previously found more persuasive. Some more of that and this thread could get off the ground and turn into something creative.

Whilst I agree that there are arguments from multiple perspectives, as with most topics that attract philosophical analysis, I still go a step further and side with the perspective that (1=0.\dot9) rather than the perspective that suggests inequality for the same reason I side with atheism rather than agnosticism. It’s not just the quantity but also the quality of argument that makes it less like a simple fair coin with information entropy of “2”, but more like a loaded coin with a lightsource shining on one side only, giving it information entropy closer to “0”. That’s not to say this topic is a matter of probability, I’m just extending your coin analogy to better represent a fuller understanding of why the mathematical consensus is with (1=0.\dot9)
Philosophically, of course there doesn’t have to only be one answer to every possible question - whether or not there can be one answer depends on the question. The higher standard of philosophy is not just in evaluating which arguments are logically more sound and valid, but in recognising the limits of the question and why its faults are what they are - and finally in how to improve the question and how to usefully explore any seeming ambiguities that might pop up along the way, which is how mathematics itself evolves.

Mathematics explores what can exactly be said with certainty, proving exactly why, and opening up avenues for what else can be explored and potentially also proven exactly. Any differences this has with philosophy do not lie in casually wondering about the inexact, or devoutly ignoring the exact in favour of siding with the inexact. For example, infinity being undefined is not useful for mathematics as it has infinite margin for error, whereas the difference between (1) and (0.\dot9) has exactly zero margin for error. Good philosophy isn’t in wondering what happens when you treat infinity as something that can be operated upon with zero margin for error, because this is simply muddying definition with non-definition, which unequivocally takes us away from clarity. So a mathematical question explored philosophically still moves parallel to the interests of mathematics - and to say otherwise is simply to underestimate the full extent of what mathematics is. It’s not simply rule-following, that’s just the kind of attitude that the inexperienced have - akin to saying philosophy is just wondering about big questions, or all music of a certain genre sounds the same, or art is just painting accurate portrayals.

1984

While a point a line and a plane remain imaginary constructs, further application of them in the form of angles rays and vectors can be quite useful.

1985

Within a single interpretation, a proposition is either true or false. It can’t be both.

1986

A Möbius strip.

1987

There’s literally an infinitude of other examples of Magnus being wrong yet again from just the simple mathematical question of what multiplied by 0 equals 0? What single interpretation is there of the answer? Within each single interpretation of any one answer, you might try to claim it is a true one, and yet it is never just that one single interpretation of the answer! Solve for x when multiplying x by 1 equals x.
The answer can only be undefined. Just like infinity. Will we now see him trying to suggest it is true that the answer is therefore “defined” as “undefined”? :-"
Almost everything he says seems to be just another example of something I’ve already correctly observed about him - e.g. not being able to cope with the term “undefined” (aka infinity - by both derivation and definition). Certain minds deal particularly badly with ambiguity, though at least this condition often serves such people very well to becoming mathematicians - but statisically some will unfortunately roll snake eyes.

Yet there’s literally an infinitude of examples of Magnus being right from just the simple mathematical question of what integer added to 1 equals the next integer in the ascending sequence? There’s only a single interpretation for any given case, and each is as true as the last, and also in no interpretations ever false.

So is Magnus infinitely right or infinitely wrong?

Some simple boolean logic can resolve this one!
“A proposition is either true or false” is a logical “OR”, and the addendum of “It can’t be both” makes it into an exclusive “OR” (or “XOR”/ ⊕).
Since the correct answer was easily demonstrable as “AND” (or ^), the overlap is:
(x ⊕ y) ^ (x ^ y) = FALSE (across the entire truth table!)

Give yourself a break from humiliation, Magnus, and stop presenting yourself as so certain and binary :laughing:

1988

When math and logic disagree.

To approach the question as a problem of logic was creative, and I haven’t seen it before. The logic in the truth table and what it revealed hasn’t been challenged yet.

The best challenge of it by you has been “Sounds Zeno” followed by the insistence that it does.

Logic didn’t depend on the application of axioms for it’s result, but the math does.

An imaginary infinite line in space intersected with a perpendicular plane, and notions of how it may look from different perspectives was purely imaginary. That a single plane has two surfaces that my not be mirror images of each other speculation. Playing with imagination cause no one else has seriously addressed the question “what happens when logic and math disagree?”

I am reminded of the infinite recurrence that appears when standing between two parallel facing mirrors.

I didn’t see the contradiction that you saw.

If the decimal point was a mirror… but the decimal point doesn’t perform like a mirror, that is the role of the symbol denoting equality which sort of does.

1 (\dot0) . (\dot0) 1

When one imagines the equal sign as a mirror, a mirror copy is not the result. How dissimilar is a mirror to an equals sign?

1 (\dot0) . (\dot0) 1 appears valid, from the same perspective 1 = 0.9 recurring does not. The application of an axiom not required.

1 (\dot0) . (\dot0) 1 but this decimal notation represents an infinite number of “whole” numbers to the left of the decimal point and that a whole can be divided infinitely into fractions on the right side. Seems to me like there isn’t a contradiction present.

A fraction of a whole is not equal to a whole. Integers and incomplete fractions of integers are not equal.

1989

Seems like an irrelevant question. Are you the master of asking irrelevant questions? (Beside being the king of giving unnecessarily complicated answers?)

The irrelevant question you are asking has more than one correct answer e.g. “any number”, “1”, “2”, “Pi”, “1000000” and so on. But that’s a QUESTION, it’s not a PROPOSITION. Questions have answers. Answers can be correct or incorrect, and of course, there can be zero, one or more correct answers. But when it comes to PROPOSITIONS these are either true or false – they cannot be BOTH true and false.

So yeah, nothing to do with what I said, nothing to do with what Mowk said.

Mr. Stay Irrelevant.

Let us recall:

Mowk said that (0.\dot9 = 1) is both true and false. By doing so, he violated the Law of Excluded Middle.

I responded by saying that WITHIN A SINGLE INTERPRETATION a proposition cannot be both true and false. A single proposition can be interpreted in many different ways. Depending on how you interpret it, it can be true or false. But once you interpret it, it’s either true or false.

“Life is a rollercoaster” is true if you interpret it figuratively but false if you interpret it literally. Once you pick a single interpretation, however, the proposition is either true or false. It can’t be both.

1990

This is true when the question actually involves a possible middle. What constitutes a middle when referring to a plane that has two surfaces yet is lacking in any dimension between them?

1991

And while we are at it, “let” us return to this notion of the use of axiom in practice.

How certain is such an outcome of such a proof dependent on such a method? Yeah, sure it’s true, it is equal because of the absolute truths in the axioms used to prove it?
But wait, what is an axiom again?

Let’s debate that as well.

1992

Within the application of decimal notation there are certain conventions applied that are dependent on axioms as well.

Follow along if you are so inclined.

An axiom is presented that assumes the following are true. That is to say we will accept/let them operate as truth, for the sake of an argument. (forgive the cutting to the chase, as this issue alone, deserves some introspection.)

0 = 0< Is this true or false?
0. = 0.< Is this true or false?
.0 = .0 < Is this true or false?
.0 = 0.< Is this true or false?
0. = .0< Is this true or false?
0.0 = 0.0< Is this true or false?

Do you see a contradiction implicit that is a dependent assumption of decimal notation? What can be termed an axiom?

.x = x.< is false for all infinite integers.
x.= .x< is false for all infinitely divided fractions.

One whole does not equal a fraction if a whole is a whole and a fraction is a fraction. The ontological difference between parts and wholes.

An integer expressed in the form of a fraction, is itself divided by itself which equals a whole as result, not a fraction. A fraction divided by itself, a second order division, in other words… a division of a division equals 1. A fraction of itself, equals 1, while an integer over any other integer, unbound by a secondary division is not equal to 1 whole.

As a statement in decimal form, 0.9 recurring is defined as an infinitely large fraction. A 1 does not exist in the column decimal notation reserves for 1’s.

infinity is unbounded, It is not bounded by odd even occurrences while a pattern of effect appears in application to fractions. Operated upon once the result is a fraction operated upon twice the result is a whole and a third and a fouth and a fifth… when do you imagine that pattern will change. While infinity makes no claims to be bounded by odd or even multiples.

Math as a proof is little more then one assumption applied to another and another and another…such that 1 = 0.9 recurring makes any sense at all.

Put that herb in your proverbial pipe and puff, puff, pass. Don’t be a proverbial boggart.

1993

Wanna dance the bump and grind all over again?

Ok, but we have done this before.

The practice of the same method in anticipation of a different result is needlessly redundant.

1994

Stay on point. Egos presences can hardly be denied yet it is irrelevant… straw men.

1995

Think about decimal notation.

0.0 is the deciaml notation for zero 1’s and zero fractions. When the column on the right side is filled with anything but a zero a fraction exists. When the column to the left is filled with anything but a zero some amount of 1’s exists. If the column to the left has been filled with anything other than 0 and the column on the right is filled with anything other than zero. Decimal notation is telling us we have some combination of wholes and parts.

Given the rules of decimal notation, the equality between the two as having the same value is not true.

1996

1.0 = 0.(\dot9) (by definition the largest possible fraction) + .(\dot0)1 (by definition the smallest possible fraction) is a true statement.

Through the addition of the largest possible fraction and the smallest possible fraction you get a whole, with no fractional parts left over, which is in decimal notation indicated by a zero amount in the column to the right of the decimal point. The condition of a whole has been met placing a 1 to the left of the decimal point and a zero to the right.

By the very definition of wholes and fractions… decimal notation, this statement >1.0 = 0.(\dot9)< is false.
And
1.0 = 0.(\dot9) + .(\dot0)1 is a true statement.

A whole is not equal to any fraction of itself.

1997

What happens when logic and math disagree?

Still waiting on a response to this question. Just for a second “assume” the argument expresses logical truth, even if you haven’t proved it.

Become a mathematician and assume it is true to explore the question further.

In the process of exploration I have come upon conjecture why some people just can’t except the mathematical proofs. Not one conjecture that states because it isn’t logical to do so.

It is not a math question it’s a question of reason and logic.

The writers of the series Star Trek missed a great opportunity to demonstrate Vulcan Logic.

1998

But
$$ 0.\dot{9} =1/3 + 1/3 + 1/3 $$
$$ 0.\dot{9}=0.\dot{3}+0.\dot{3}+0.\dot{3} $$
So somehow an extra tiny amount $$ 0.\dot{0}1 $$ appeared out of nowhere.

If that tiny amount is equal to zero, then there is no problem.

1999

1/3 + 1/3 + 1/3 = 3/3 = 1

Such that this:
$$ 0.\dot{9}=0.\dot{3}+0.\dot{3}+0.\dot{3} $$

is an expression of the same problem of logical application that 1.0 = 0.9 Recurring expresses? Can you prove you are doing the math of adding recurring decimal points accurately with out relying on math, or have you simply assumed you have?

2000

The parts of math that I’m using is long division and addition. Pretty simple stuff. If that doesn’t work consistently, then all of mathematics is broken.

I can’t throw out all of math and rely only on logic or philosophy. Then the equations themselves would lose all meaning. Arguing that 1=0.9… or 1=/=0.9… requires accepting some parts of mathematics.