Negative Zero

Addition squared? Subtraction squared?? What the hell are you on about now?!

What is the point of me explaining things if you either never read them/ignore them/forget them??

I already explained how division is derived from subtraction and therefore by addition, and in the same way multiplication is derived from addition. And no, you don’t need to “square” or “root” anything before you want to put any more nonsense in my mouth.
Multiplication is the amount of times you need to add something to itself to get to the answer e.g. 3x4 is 3 added to itself 4 times - that’s why it’s called 3 times 4.
Division is the amount of times you need to subtract the denominator from the numerator to get to zero (or add the denominator to itself to get to the numerator) e.g. 12/3 = subtract 3 from 12 to get to zero = 4 times or add 3 to itself to get to 12 = 4 times.
They’re all grounded in addition, and interestingly enough since you mention them - but probably beyond you, squares and roots also derive from addition too. This is literally how your computer works - the very thing you’re using to argue against the concepts that make it work… I’m not making this up, it’s very real and in your life everywhere. I’m using a different “modality”??? No, I’m just telling you what maths actually is and where it actually came from. You’re making up nonsense and putting nonsense into the mouth of others to try and make something ignorant and clearly flawed seem relevant.

I mean, here you are trying to say that 1/0 is 1. Yes you can use language to misleadingly represent something like division so it sounds like it doesn’t mean what it means. The word for this is Sophistry.

I rest my case, there’s no way you can salvage what you’re trying now with any amount of linguistics. Good bye.

So how do you multiply pi times the square root of 2 by your definition?

Bonus question: In the complex numbers, how do you show that i^2 = -1? How many i’s is that? How do you reduce this problem to addition?

Fortunately for computers the solution to multiplying irrational numbers is forcibly simplified by the number of bits you have to work with. Thereby, you apply the same reduction to addition that I explained to an approximation of said irrational numbers i.e. to a limited number of decimal (or binary in this case) places - as though they were rational numbers. To deal with (non-integer) numbers that have decimal points, variables of “float” or “double” type are used to represent numbers in an alternative way to allow more decimal places to be dealt with, even though the same number of binary places are being used - at the cost of accuracy. “Float” is short for floating decimal point, which hints at how the alternative representation manipulates bits to this end, although that’s not the only trick used. Still though, the limited number of bits nevertheless results in irrational numbers being dealt with as though they were rational - and the problem posed to mathematicians of irrational numbers is circumvented artificially.

Even outside of the world of computers, the same constraints are forced by practicality. Most vulgarly, for engineers and others who use irrational numbers in calculations that apply to everyday scales, the use of more decimal places quickly becomes redundant, meaning they too use irrational numbers like they were rational. But even for quantum physicists performing experiments at the quantum level, the use of more decimal places also becomes eventually redundant beyond a certain only slightly further threshold and the same truncation is resorted to - and everything still boils down to a twist on addition.

But what about for theoretical physicists and pure mathematicians? Their solution is even simpler: algebra. Got an irrational number? Use a letter to denote it (in exactly the same way that numbers denote integer quantities). This preserves the implied infinity of decimal places throughout all calculations, sometimes being cancelled out, sometimes being eliminated by certain properties of irrational numbers when used in, for example geometry. Consider that famous identity by Euler: “e^(pi)i = -1”. This equation is most simply shown on an 2D graph of a (complex) unit circle that plots its imaginary component against its real component, where pi is in radians and e is the base of natural logs as standard. Here we have 3 non-real numbers that when related to one another in a specific way amount to a real number, an integer no less.

Technically, “i” is just another kind of anomaly like irrational numbers that presents itself when building up maths from addition upwards - and to deal with that, in answer to your bonus question, the same solution of algebra is used. Mathematicians have to bring everything back into the realm of “boils down to addition” in order to deal with it - even the apparent exceptions that emerge from building maths in such a way. The whole point of maths is that it boils down to the simplest possible concepts, such that it is robust and consistent throughout.

In short: the practical solution is to truncate to a rational approximation, the theoretical solution is to use algebra.

Thank you for asking the first not-profoundly-stupid question in this thread for far too long.

:stuck_out_tongue:

I appreciate the kind words. So now if I were to say “You’re all wet,” or, “You have no idea what you’re talking about,” I’d feel guilty. So I’ll just point out a few things. I could write a lengthy post but I’ll keep this mercifully short and just list some bullet iitems.

  • The mathematical real numbers (\mathbb R) are not anything remotely like the floating point numbers as described by IEEE-754. Yes I know what that is and they are not the real numbers. You seem to have studied computers and not math. Everything you say about computer arithmetic may be true, but totally irrelevant. The question was not, “Can computer multiplication be reduced to addition?” That’s completely different question to whether real number multiplication can be so reduced.

  • Likewise your off-topic remarks about engineering math, and all practical operations involving real numbers being reducible to rationals. Of course that’s true, but equally irrelevant. We’re talking about the mathematical real numbers. You know, the ones that require infinitely much information to represent.

  • To take you entirely out of the realm of computers, how do I multiply two noncomputable real numbers? Those are the numbers that can not be approximated by a program or a Turing machine. Most reals are noncomputable, as you can see from noting that there are uncountably many reals but only countably many Turing machines. How do you reduce the multiplication of two noncomputable reals to “repeated addition?” The idea is absurd on its face. You could not reduce a noncomputable real to a rational approximation with any finite amount of computing power or memory no matter how large, if your approximation is required to be computable.

  • Your handwaving about “algebra” is nonsense. Yes I’m mindful that you said something nice to me. I regret not being able to respond in kind other than to note that you know a lot about computers and engineering math but sadly nothing about math. Your remarks in this area were vague. What do you mean that “algebra” shows that multiplication is reducible to repeated addition? On the contrary. In algebra, multiplication is a map from pairs of real numbers to real numbers, satisfying the usual field axioms. There’s nothing in the field axioms about multiplication being repeated addition.

  • Likewise your handwaving that " “i” is just another kind of anomaly like irrational numbers …" Please friend, I really appreciate that you complimented me. I’m going to hate myself in the morning for being so churlish as to say this, but you just embarrassed yourself.

It’s ok that you studied computers and engineering instead of math. But be humble about what you don’t know.

If you’d like me to expand on anything I said, please ask. Like I said I could have written a lot more.

One more point. Here is what you meant to say, if you’d studied math: “Well we can define multiplication as repeated addition in Peano arithmetic; then we can lift multiplication to rationals using the standard field of quotients construction; and then we can lift multiplication to the reals by taking limits of Dedekind cuts of rationals; and in this way, although real number multiplication is NOT in any meaningful sense repeated addition, we can indeed find multiplication defined as repeated addition a long way back in the chain of the construction of the real numbers.” If you said that, it would be the right answer. I’ll leave the definition of multiplication of complex numbers to you.

Another simple way out would have been to say that we can view multiplication as repeated addition for natural numbers but not for rationals, reals, or complex numbers. That would be sensible. But your changing the subject to computers was wrong, because floats are not real numbers. Maybe they just don’t tell the CS students that. But you should realize, there’s a LOT they don’t tell the CS students about math. They run you through “Discrete math” and consider your mathematical education done. It’s a crime.

Ok now I feel terrible. You shouldn’t have been so nice to me!!

I would argue from the perspective of reality (rather than abstract mathematics where you are perfectly correct) which implies physical limitations on the representation of numbers where those lying without are referred to as “dark numbers”, which are those that cannot be represented within the observable universe even if written at the planck scale. There are certainly numbers that cannot be said to exist in any meaningful way other than trivial abstraction.

Math is merely a construct and not necessarily representative of what we call reality nor constrained by any limitations thereof, so our definition of the symbolic nomenclature must be in the context of what “just to the right of zero” means to the one using the information. In abstractness, there is no number just to the right of zero, but in counting apples, there is: 1. On a tape measure, the gradations are generally limited as no carpenter implementing a role-up tape could make any meaningful use of 1/100th of an inch when framing a house.

Pragmatically, “to the right of zero” would mean the smallest number you find meaningful and its symbol would be 0+.

Yes I think you’re right.

+1 is the opposite of -1, but not-1 means any number that is not 1.

Infinity is undefinable :wink:

A negative beaver is a tallywacker. They live in the forest and visit the bush during mating season. :slight_smile:

That’s prescient! :wink:

Neither zero nor infinity exist.

If you have zero apples, you have a non-existence of apples. If you have infinite apples, you have a non-existence of apples.

The only thing that can be ubiquitous is nothingness and the only thing there can be infinite amount of is nothing.

If there were infinite apples, there would be nothing that’s not-apple, and therefore there would be no context in which the apples could manifest.

I’m going to reply to this by simply stating: the issue at hand is zero, and strange properties of zero relative to non zero numbers.

To use non zero numbers as analogies is to miss the argument completely.

Actually that charge silloutte is is brining against me is not “sophistry” as claimed, but rather “equivocation”.

I stated outright that these are modalities of math and that mine is grammatically correct … to silhouette, this is putting words on people’s mouths, to which I replied, these are different modalities.

I perfectly well stipulate that I know nothing of reality. I’m not even convinced there is one. We could be a brain in a vat, an experiment in an alien grad student’s AI lab, or a program running in God’s own Turing machine. I try to clarify mathematical issues when they come up. But I certainly have no idea of what’s “out there.” Personally I don’t think the standard real numbers are literally how reality works. To me, the real numbers are a wonderfully interesting mathematical abstraction that probably have no relation at all to the world. Or maybe they do. It’s fun to talk about.

Absolutely. I feel like I spend my life trying to explain this to people online. Then other people try to explain it to me, as if they think I’m arguing the opposite.

Ah sorry part with you here. When we try to formalize our vague intuitive notions of infinitesimals, we find that there is not a smallest positive real number in ANY conceivable model of the real numbers, in particular the hyperreals, which are everyone’s go-to example of a system of numbers with infinitesimals.

So if you are making a philosophical point, you still can’t violate known math. If I’m understanding you correctly.

Ok.

Ah I see where you’re going with this. No, there is no smallest number or infinitesimal just to the right of zero. Because if you say there is, and you call it x, then I’ll just point out that 0 < x/2 < x and x was not the smallest after all.

That’s math, brotha.

But zero is nothing… absence of things. How can nothing have properties?

Sophistry implies mal-intent, right?

What you’re on about reminds me of this video:

Start “just to the left of” 5:00 :wink:

[youtube]https://www.youtube.com/watch?v=emlcwyvnsg0[/youtube]

You’re perfectly correct in math, but in what we colloquially regard as reality there are limits to numbers.

Check out this video. Start at 10:00

[youtube]https://www.youtube.com/watch?v=WabHm1QWVCA[/youtube]

Here’s a longer lecture if you’re really interested:

[youtube]https://www.youtube.com/watch?v=p9xX-Jpsr_E[/youtube]

You are conflating the mathematical with the physical
Zero is a number and numbers are not actual things
The properties in question are purely mathematical

Some video some guy made is not proof of anything. Even if I spent 30 minutes of my life watching it, if I disagreed with anything he said I’d just be arguing with some guy who made a video. What is the point of that?

If you can state an argument relevant to my post we can have a discussion. You and I, not me and some guy on the Internet with an opinion. I can’t argue with every Youtube video that’s out there. I’d have to make my way through the flat earthers and the moon landing deniers long before I got to the infinity opinionators.


That second video you posted looks very interesting so I will watch it later today

Some guy?

I am a professor of mathematics at UNSW Sydney. I was educated at Adam Scott High School in Peterboro Ontario, Richmond Hill High School in Richmond Hill Ontario, University of Toronto (BSC 1979) and Yale University (PhD 1984). I taught at Stanford University (1984-1986) and the University of Toronto (1986-1989) before coming to UNSW (University of New South Wales), Sydney, in 1990. web.maths.unsw.edu.au/~norman/

My PhD thesis was Quantization and Harmonic Analysis on Nilpotent Lie Groups. I have worked in representation theory, harmonic analysis, combinatorics, and geometry. I have developed finite hypergroups and duality, Pell’s equation and Diophantine equations, and introduced Rational Trigonometry and chromogeometry (download my book here!) I have reformulated hyperbolic geometry to make it more algebraic, general and beautiful. I have a YouTube channel: njwildberger with 600+ math videos. researchgate.net/profile/Norman_Wildberger

It would be like calling Stephen Hawking “some guy”.

I told you FWD to 10:00. It’s like 1 min and if you’re interested, then listen to the other. If not, then don’t.

Stay ignorant. I don’t care.

It’s as if you said “If you do not feed me the information I want in the form I want it, I’m going to stay uninformed! So there!” ← Not scary.

I did. The universe presents a limitation on math. Watch the accomplished and passionate mathematician say it again on the video along with his reasoning.

You went so far into left field that you’re in the parking lot.

Why would you have to watch flat-earthers and moon landing deniers before watching 1 min of the video I posted?

But even in math it’s nothing and for a long time, zero wasn’t even regarded in math. livescience.com/27853-who-i … -zero.html

There were other things that were once unknown in maths such as irrational numbers and negative integers and complex numbers
But once they were discovered they were seen to be incredibly useful so the fact that zero was also unknown means nothing at all

You need zero because apart from anything else without it you would have no base ten
It is also unique as it is the only non negative / non positive integer on the number line

Idk, it wasn’t that zero was unknown so much as I think it was rejected because why do I need to write down that I have no cows? People didn’t need to keep track of nothing. Maybe zero only became relevant when negative cows came about: I owe 2 cows to my neighbor and one day hope for zero cows instead of -2.

Every number has an opposite: n and -n. That is true with the exception of zero, which has no polar mate. But the opposite of nothing is the ubiquitous or infinity. So zero does have a mate even though neither exist.

This thread is a lot of ado about nothing lol

Not at all, it’s possible from what I can tell from your words that you might actually have something to teach me. That will involve pointing out legitimate flaws in my argument, so feel no guilt for it. To be honest, I was tired of the repetitious illegitimate criticism that I was receiving. It frustrates me so when it’s so clear to me that the respondent is so very misguided - and by contrast, I seem to be recognising you as somebody who actually has knowledge about that which you’re talking.

Wetness aside, by all means let’s challenge this assumption and see if we can’t fall out as in all proper internet arguments :wink:

Some clarification though: my maths background actually extends much further than my computing one, though I won’t deny that my computing has had an influence on it and given me some insight into maths that I did not have before. Actually though, I am mostly basing my arguments loosely on a history of mathematics: starting with natural numbers, through subsequent discoveries of other categories that did not accord with the previous ones. It all started off with addition forwards and backwards (subtraction) - although it wasn’t necessarily the case that it was thought of in this way, “backwards addition” is more of a reductive perspective as I’ve already said. More than likely it was just thought of as “this much more” vs “this much less”: backwards addition is just a way to think of it that was inspired by my education in computing. With integers, multiplication and division ought to be fairly straight-forwardly a shorthand for multiple additions/subtractions. Of course this runs into problems are you start branching out past natural numbers through integers and fractions to irrationals and beyond (what I am referring to as “anomalies” compared to what preceded them) - I’m just explaining how we got there. I assume that your post was to catch me out by shifting the foundation from the beginning of mathematics to something more contemporary where things are much more complicated. Perhaps you recognised my approach and wanted to change the context to something more modern - and why not? I admit conceptions have moved on from my explanation of how maths started. My contribution to this thread, though, was in the certainty that at no point has it ever been appropriate to equivocate by equating “NOT” with “negative” - nor will it ever be. That was the whole reason I brought computing into it: what was being described was the NOT function, as in computing, only it was erroneously being called the “negative” operator.