While this may be true, it is not then proper to say that these things are a part of mathematics. Mathematics exists outside these concepts, and people choose to use mathematics as a tool to better understand them. However, much of mathematics is outside of reality. The simplest example I can think of is the concept of a Klein bottle, something that cannot exist in reality but has defined mathematical properties nonetheless.

Sets are mathematics... Unless you mean that Real numbers and Integer numbers aren't part of Mathematics... And I did agree to the fact that most of Maths is outside reality, it's what I referred to as Mysticism... Moebius' Stripe, Klein's Bottle, or the infinity would all fall under Mysticism... They are "intangible" things (I do realize now that intangible wasn't the best word to describe it, since the bottle and the strip are both kinda tangible) that cannot be physically explained... And unfortunately, we have been conditioned to explain things physically...

∞

I can never choose whether I hate you or love you...

ppl do the same thing with integrals to find the area in a curved space, this doesn't mean it's anything more than an approximation.

you recognized the pattern yourself, you logically realize that using squares to chip leaves surface area behind that still belongs to the original "square". at no point would you ever cut away a TRIANGULAR portion which would create a perfect isosceles.

just an application of Zeno's all over again. at no point is a FULL distance covered. only half of the remaining. it just gets to the point to where the difference is so irrelevant in terms of scale that the approximation is enough.

There can be no smallest real number. Just take s/2 and you have yourself a smaller one. Nor are there any infinitely small or large real numbers. This is known as the archimedean property.

Infinity isn't a real number, so the operation x * infinity is not defined.
An infinite number of decimal places does not mean there's a decimal place "at infinity". The set of natural numbers is infinite. This is because, if you take any finite set of natural numbers, take the maximum element from that set, then add one to it, you get a new natural number. There are no infinities here, however. Every natural number is constructed by starting with 0 and adding 1 a finite number of times.

A decimal expansion is defined by assigning a digit to every natural number. i.e. the natural number 1 corresponds to the first decimal place. The nth natural number corresponds to the nth decimal place. Thus 0.99... is defined by assigning 9 to every natural number. But 0.00...1 is just bad notation, because there's no infinity in the set of naturals, so there's no place to assign the 1.

Zeno was an ancient philosopher who attempted to support Parmenides' claim that the existence of motion is contradictory. If you accept his argument, then it follows that motion is impossible. As it turns out, it's quite easy to resolve this particular argument because he implicitly assumes that if you sum an infinite number of finite numbers, you'll end up with something infinite. This is false. Calculus proves it's false. Though it's not hard to see it intuitively: It should be clear that the sum
0.1 + 0.01 + 0.001 +... does not sum to infinity. In fact, it sums to exactly 0.11... = 1/9.

What is the difference between a number and its value?

What you're describing doesn't apply to sets, it applies to fields. A field is a set together with two binary operations satisfying some strong properties. The set of real numbers is a field. In fact, it's the unique complete ordered field. However, if you construct a new set by taking all the real numbers, adding infinity and negative infinity as the supremum and infimum of the old real numbers, you lose a lot of structure. This set is known as the extended reals, and it isn't a field. Though as a sidenote, 1 = 0.99... still holds in this set.

Infinity isn't a number. You can't multiply by infinity because you can only multiply by numbers.

Look dude you are just making **** up because it make sense in your head - this is the anthisis of what math is.

Take my statement there is no positive smallest number. Suppose there is a smallest number, namely s. construct s' = s/2. s' < s and s' is positive, which contradicts our supposition therefor our supposition is false.

Assume that a number "inf" has the property a*inf = inf for every a in the reals. Let a be some non 1 number.
a*inf = inf
a*inf - a*inf = inf - a*inf
a(inf-inf) = (1-a)inf
a*0 = (1-a)inf
0 = inf

Thus in the reals (or any division ring) if a number has the property a*x = x it must be zero.

There are real mathematical reasons for what i'm saying. You are making up what makes sense in your brain. How you think math works is fundamentally flawed.

Take the whole .999~ = 1. .999~ is DEFINED to be a geometric ration with a common ratio of 1/10 with r = 9. This series equals 1 (not close to 1, 1 exactly). Because of how we define limits, reals, and sequences. This sequence has the properties it does because of many definitions and results in both set theory and logic, namely ZFC+ and 1-st order. We use ZFC+ and 1-st order as axioms.

That is the unbroken path of why .999~ = 1 from the axioms. If you want the details go take a set theory class, or even calc 1.

DISCLAIMER!!!

I have since retracted my approach...the numbers 1.000~ and 0.999~ are the same number represented different ways, just as "Eledhan" and my real name are just different representations or labels of the same individual. I've left my comment intact below this line for clarity, should anyone trace the conversation to here.

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I'm going to show how this concept is faulty math by substituting an integer (I'm going to use "2&quot for .999...

Using the above concepts, one would get the same "proof", but it should be obvious there is a fatal flaw in the proof's logic.



In the example I quoted from the first page, in Step F), the proof assumes that 9x / 9 equals 1. It does not. It equals whatever X was in Step A) divided by 1. X = .999..., meaning that the correct answer in Step F) should be .999.../1, meaning, .999... is the true answer to the proof. It's easy to see when you use "2" in place of .999...

TL/DR - .999... does NOT equal 1, no matter how close to infinity you get. Speaking of infinity, it's not a real number, which means .999... isn't real either, but just a number to show a concept that isn't easily converted into a decimal number system.

FYI, math doesn't work with infinity...so, any "proofs" to show any form of "infinity" as a mathematical function are invalid.

I learned it in the 5th grade in PUBLIC SCHOOL. You just don't pay attention.

The only infinity needed for 0.99... is the infinity associated with sequences and limits. If you're telling me sequences and limits are no good, then you're saying that the set of real numbers can't be constructed. The construction of the reals, after all, is done by completing the set of rationals, i.e. adding (in a completely non-trivial way) the limits of cauchy sequences of rationals.

No such assumption is made. All that's assumed is that 9x/9 is x, and 9/9 is 1.

When you get to 9x = 9, the point is that by dividing both sides by 9, equality will be preserved. So 9x/9 is equal to 9/9.

Well the left side is just x, and the right side is just 1, ergo x = 1

Your example doesn't contradict anything. It just shows that if you start with x = 2, you end up with x = 2, as one would hope.