Mathematics and precedence rules

[Socratov]

Originally Posted by noparlpf

I usually like zero. It means I can eliminate an entire piece of an equation, or reduce it to a one, or something. Except sometimes I hate zero. Like when the equation says ln(φ)=(1/RT)(integral from 0 to p (dp/p)). Because without that actually being explained in class, when you get to it on the homework it looks like you're supposed to have ln(0).

it's also the very reason why differentials are way more awesome then integreals (or worse: double, triple, circle, surface and volume integrals)

[The Extinguisher]

Originally Posted by Socratov

that depends on you expecting the caculation in integers.

For instance in integers it's the much awaited 4. However, to be overly correct one can assume that the '2's in the equation can be rounded numbers of anywhere between (for significance kept on 2 decimals) 1,50 and 2,49

this then results in a range of minimum 1,50+1,50=3,00 (or: 2+2=3 for minimal values of 2)and a maximum of 2,49+2,49=4,98 (or: 2+2=5 for maximal values of 2), resulting 2+2=y in a range of 3=<y<5 for any value of 2

When we apply rounding off again we shall see that 2+2 can be solved as 3, 4 or indeed 5.

Now if the above explanation is the correct one, the 2+2=4 is not correct (in only providing 1 solution while others exist) even though 99.000 out of 100.000 people will probably say 2+2=4.

note: this phenomenon becomes funnier if applied to slightly more complex problems. Imagine a calculation where the function is the between-2,50-and-not-quite-4-th root of (whatever). At some point you will encounter the euler root and the Pi root. have fun calculating

I'm pretty sure claiming that's not how rounding works. If you round 1.86 to 2, you're committing that value to be two later. If I have 1.86 and 2.43 and I add them, I get 4.29, but if I round them both to two and add them I get 4.

[Saposhiente]

Originally Posted by noparlpf

"Even" means it can be divided by two and not yield a fraction. So technically zero is even, because 0/2=0 and zero isn't a fraction.

(Terminology nitpick: Zero is a fraction. 0/1. 0/42. So is every other number, 3/1, 6/2, etc. What it is is an integer multiple of 2, 2*0=0.) Pendell: Dividing by zero has nothing to do with it.

[noparlpf]

Originally Posted by Saposhiente

(Terminology nitpick: Zero is a fraction. 0/1. 0/42. So is every other number, 3/1, 6/2, etc. What it is is an integer multiple of 2, 2*0=0.) Pendell: Dividing by zero has nothing to do with it.

Eh, sort of. I don't consider n/1 a fraction. Not that it makes a difference, really.

[Heliomance]

Originally Posted by noparlpf

Eh, sort of. I don't consider n/1 a fraction. Not that it makes a difference, really.

Your explanation would make pi even, because you don't get a fraction when you divide it by 2.

[noparlpf]

Originally Posted by Heliomance

Your explanation would make pi even, because you don't get a fraction when you divide it by 2.

If 3/2 is a fraction then why isn't pi/2? Three is a constant, pi is a constant. Same thing, really. And anyway, if we figured out all the decimals of pi, we could very well turn it into proper fraction form, just with arbitrarily long numbers in the numerator and denominator.

[Castaras]

You can't figure out all the numbers of pi. It's irrational.

So the only way you can have pi as a fraction is pi/1.

[Urpriest]

Originally Posted by noparlpf

If 3/2 is a fraction then why isn't pi/2? Three is a constant, pi is a constant. Same thing, really. And anyway, if we figured out all the decimals of pi, we could very well turn it into proper fraction form, just with arbitrarily long numbers in the numerator and denominator.

Nope! 3/2 is a fraction (the actual term is a rational number) because 3 and 2 are integers. The fact that they're constants doesn't factor into it, p/q is also rational provided p and q are integers, even if they aren't constant. Integers are not arbitrarily long, either, since infinity is a limit, not a number.

[warty goblin]

Originally Posted by ForzaFiori

I can tell you roughly when the dark ages are, who the US's first president was, the name of the person that discovered america is

Probably not, actually. Columbus was about four hundred years after Leif Ericson, who was the first European to make landfall in the Americas. But according to one of the sagas, Leif already knew about Vinland from a guy with the impossibly marvelous name of Bjarni Herjólfsson, who, along with his crew, had been blown off course there several years previously.

Of course other sources have it being Leif who was blown off course and wound up in Vinland, so who knows...

[noparlpf]

Originally Posted by Urpriest

Nope! 3/2 is a fraction (the actual term is a rational number) because 3 and 2 are integers. The fact that they're constants doesn't factor into it, p/q is also rational provided p and q are integers, even if they aren't constant. Integers are not arbitrarily long, either, since infinity is a limit, not a number.

Okay, that's fair.

[Heliomance]

You can't figure out all the digits of pi because it never ends - it has infinitely many digits, and never starts repeating. There is no way to write pi as a fraction.

[nedz]

Originally Posted by noparlpf

If 3/2 is a fraction then why isn't pi/2? Three is a constant, pi is a constant. Same thing, really.

You have answered a different question.

Originally Posted by noparlpf

And anyway, if we figured out all the decimals of pi, we could very well turn it into proper fraction form, just with arbitrarily long numbers in the numerator and denominator.

This is never going to happen, π is irrational. The term irrational means that you can never write it down as a rational number. A rational number is one which can be written down as some P/Q where P and Q are integers. This has been proven, though IIRC the proof is a little subtle.

[Socratov]

Originally Posted by The Extinguisher

I'm pretty sure claiming that's not how rounding works. If you round 1.86 to 2, you're committing that value to be two later. If I have 1.86 and 2.43 and I add them, I get 4.29, but if I round them both to two and add them I get 4.

well, rounding is only done in favor of less writing down (are you going to write every decimal? have fun with Pi) but that doesn't mean you can round off and calculate on with the rounded off numbers. You always continue with the non rounded numbers. I have been punished for not doing this in school (our teacher sometimes was slightly sadistic in making math tests like these). In maths acutally 2 things you need to know: 1) what is a function? (and how does it work and hwat can you do with it) 2)every numeral representation of a number is a range unless otherwise defined or reasoned.

[Ashery]

Originally Posted by Urpriest

Nope! 3/2 is a fraction (the actual term is a rational number) because 3 and 2 are integers. The fact that they're constants doesn't factor into it, p/q is also rational provided p and q are integers, even if they aren't constant. Integers are not arbitrarily long, either, since infinity is a limit, not a number.

Hurray, one explanation I don't have to do. Although the additional requirement that q≠0 needs to be added for the definition to be complete. Ie a rational number is defined as "p/q where p, q are integers and q≠0."

Now, for even/odd:

An even number is any integer that can be expressed as 2m, where m is any integer.

An odd number is any integer that can be expressed as 2m + 1, where m is any integer.

So, for the case of proving that odd + odd = even (To bastardize the language a bit), we have:

(2m + 1) + (2n + 1) = 2m + 2n + 2 = 2(m + n + 1) = 2k

The last "step" is unnecessary, but I figured it was better to play it safe and include it. It should be clear that k = m + n + 1 and we know k to be an integer because of the fact that integers are closed under addition.

Bit of a weak explanation, but meh.

[TheFallenOne]

Originally Posted by warty goblin

Probably not, actually. Columbus was about four hundred years after Leif Ericson, who was the first European to make landfall in the Americas. But according to one of the sagas, Leif already knew about Vinland from a guy with the impossibly marvelous name of Bjarni Herjólfsson, who, along with his crew, had been blown off course there several years previously.

Of course other sources have it being Leif who was blown off course and wound up in Vinland, so who knows...

Yup. And anyone using the term 'dark ages' is instantly suspected to have little clue of the Middle Ages.

Lesson being: if you give some examples of basic facts you know, be sure everything you list is in fact accurate and not just a common misconception you picked up somewhere.

[Mr.Silver]

Originally Posted by TheFallenOne

Yup. And anyone using the term 'dark ages' is instantly suspected to have little clue of the Middle Ages.

Lesson being: if you give some examples of basic facts you know, be sure everything you list is in fact accurate and not just a common misconception you picked up somewhere.

Just to further beat this point into the ground: technically speaking, the line "who the US's first president was", is, on account of the wording, more accurately answered by 'John Hancock' than 'George Washington'*. The reason for this being that while Washington was the first person to hold the office of 'President of the United States of America', the US's first president (note lack of capital) would have been Hancock due to him being the president of the Continental Congress when independence was declared.

Pedantic? Yes, but it seemed fitting given the main topic of the thread


*There are other potential answers that can be argued for as well, but Hancock is probably the strongest option.

[Knaight]

Originally Posted by warty goblin

Probably not, actually. Columbus was about four hundred years after Leif Ericson, who was the first European to make landfall in the Americas. But according to one of the sagas, Leif already knew about Vinland from a guy with the impossibly marvelous name of Bjarni Herjólfsson, who, along with his crew, had been blown off course there several years previously.

Of course other sources have it being Leif who was blown off course and wound up in Vinland, so who knows...

This, of course, doesn't actually matter: When they found America, there were already people living there. It had been discovered by them first, and while either Leif, Bjarni, or some other viking did make a legitimate discovery, they were by no means the first people to do so (had they been, they wouldn't have found people there, though there is technically a possibility for that to happen that simply didn't).

[razark]

Originally Posted by Mr.Silver

The reason for this being that while Washington was the first person to hold the office of 'President of the United States of America', the US's first president (note lack of capital) would have been Hancock due to him being the president of the Continental Congress when independence was declared.

That would depend on when the United States actually started to exist as an entity. Is the current United States under the Constitution the same as the country that existed under the Articles of Confederation or before? The Declaration of Independence mentions "the united States of America" (note lack of capital).

Originally Posted by Mr.Silver

Pedantic? Yes, but it seemed fitting given the main topic of the thread

[Mr.Silver]

Originally Posted by razark

That would depend on when the United States actually started to exist as an entity.

Quite so. 4th of July 1776 is traditionally considered the date when the USA started officially existing as an entity, although you could also argue for 1783 (when Britain formally recognised its independence) - which would make the first president Thomas Mifflin. 1781 is a possibility, which would make the first president John Hanson.

Is the current United States under the Constitution the same as the country that existed under the Articles of Confederation or before? The Declaration of Independence mentions "the united States of America" (note lack of capital).

The capital 'u' was in use by pretty much everyone at the time though, and was set when the Articles of Confederation were ratified in 1781. Following that line would therefore be at least as likely to give you John Hanson as it would Washington.
Basing a nation's age on a constitution however can be a bit awkward in any event, e.g. one could use it to make the argument that France has only existed since 1958. Things get very wonky in the cases of nations that don't have a written constitution, such as the UK*.

Regardless of what the most accurate answer actually, the existence of this discussion just goes to show that "who the US's first president was" cannot really be considered as basic a fact as ForzaFiori assumed it could


*it also leaves the proverbial door open to the 'type identity vs. token identity' discussion and how/whether that particular philosophical distinction does/should apply to countries.

[TheFallenOne]

Originally Posted by Mr.Silver

Regardless of what the most accurate answer actually, the existence of this discussion just goes to show that "who the US's first president was" cannot really be considered as basic a fact as ForzaFiori assumed it could

Well, I'd say that ambiguity is still better than just being plain wrong like about who discovered America or the Middle Ages being dark
Now I do wonder about the 'capital of Australia' bit; many, or even most, mistakenly believe it to be Sydney. Including me to be honest until I was corrected in, dunno, 8th grade maybe?

That said, the little excursion on who was the first president of the USA under what criteria was quite interesting. But given the presidents are numbered, shouldn't there be an 'official' answer to it or at least one endorsed by the government?

[warty goblin]

Originally Posted by Heliomance

You can't figure out all the digits of pi because it never ends - it has infinitely many digits, and never starts repeating. There is no way to write pi as a fraction.

Only when defined for Euclidean geometry in a perfect plane. Since the universe is not perfectly Euclidean* due to pesky things like gravity, the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

*Although apparently it seems to be basically Euclidean, which is too bad. I'd much rather live in a hyperbolic universe. I want my pentagon with five interior right angles.

Originally Posted by Knaight

This, of course, doesn't actually matter: When they found America, there were already people living there. It had been discovered by them first, and while either Leif, Bjarni, or some other viking did make a legitimate discovery, they were by no means the first people to do so (had they been, they wouldn't have found people there, though there is technically a possibility for that to happen that simply didn't).

A valid point, which is why I stipulated 'European.'

(Strangely, the Norse may well not have been the first people to discover Iceland. There's some evidence that Irish monks used it as a prayer retreat before the Norse colonization. However they were certainly the first to *settle* it, as one can hardly count a seasonal population of celibate men as any sort of colonization.)

[Saposhiente]

Originally Posted by warty goblin

the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

There are infinitely more irrational numbers than rational numbers (greater cardinality), therefore the probability of pi being rational at any given location is zero. (Though it's technically possible)

[nedz]

Originally Posted by warty goblin

Only when defined for Euclidean geometry in a perfect plane. Since the universe is not perfectly Euclidean* due to pesky things like gravity, the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

*Although apparently it seems to be basically Euclidean, which is too bad. I'd much rather live in a hyperbolic universe. I want my pentagon with five interior right angles.

Assuming Einstein was right: then the universe has a parabolic geometry.

Euclidean geometry is hyperbolic. (See Euclid:postulate 5)

[Heliomance]

Originally Posted by Saposhiente

There are infinitely more irrational numbers than rational numbers (greater cardinality), therefore the probability of pi being rational at any given location is zero. (Though it's technically possible)

PROBABILITY DOES NOT WORK THAT WAY!

An event with P(x) = 0 cannot happen. Ever. No, not even then. If it's technically possible, the probability is not zero. It can be infinitesimally small, but not zero.

[GnomeFighter]

Originally Posted by TheFallenOne

Yup. And anyone using the term 'dark ages' is instantly suspected to have little clue of the Middle Ages.

Originally Posted by TheFallenOne

Well, I'd say that ambiguity is still better than just being plain wrong like about who discovered America or the Middle Ages being dark

Um... The Dark Ages are not The Middle Ages... Dark Ages is still a perfectly valid name for the Early Middle ages. Normally from the Fall of Rome to around 1050. It's really a much more defined period than the Middle ages.

Originally Posted by Mr.Silver

Basing a nation's age on a constitution however can be a bit awkward in any event, e.g. one could use it to make the argument that France has only existed since 1958. Things get very wonky in the cases of nations that don't have a written constitution, such as the UK*.

Poor example. The date the UK came in to existence is very easy. 1 January 1801, the date the twin 1800 Acts of Union came in to force (and in 1922 in its current state).

[nedz]

Originally Posted by GnomeFighter

Um... The Dark Ages are not The Middle Ages... Dark Ages is still a perfectly valid name for the Early Middle ages. Normally from the Fall of Rome to around 1050. It's really a much more defined period than the Middle ages.

The Dark ages was so called because:
For Historians: there are very few written documents.
For the Church: Christianity retrenched as 'pagan' hordes swept through Europe.

Archaeologists don't like the term and it is considered outdated.

The term Migration period is generally preferred, which overlaps with the, so called, Age of Invasions in Britain (Romans to 1066).

The Dark ages were typically regarded as being from the end of the western Roman empire to the rise of the Carolingians. We now call that period late antiquity / early middle ages.

[Mr.Silver]

Originally Posted by TheFallenOne

That said, the little excursion on who was the first president of the USA under what criteria was quite interesting. But given the presidents are numbered, shouldn't there be an 'official' answer to it or at least one endorsed by the government?

The 'official' numbering scheme in the USA counts holders of the office 'President of the United States of America' (which are also easier to keep track of than the earlier 'presidents' because they have fixed terms). Unfortunately it doesn't actually provide the accurate figure for how many different people have held this position either, because it counts Grover Cleveland's two terms separately.
So yeah, just because it's 'official' doesn't mean it's technically correct

Funnily enough there is a similar 'official' numbering discrepancy in the British monarchy. Because monarchs only started being numbered after the Norman conquest, King Edward I was actually the second monarch to be called Edward (Edward the Confessor having been the first) and so on with all subsequent Edwards.

Originally Posted by GnomeFighter

Poor example. The date the UK came in to existence is very easy. 1 January 1801, the date the twin 1800 Acts of Union came in to force (and in 1922 in its current state).

Yes, in much the same way that the USA is considered to have come into existence on the 4th of July 1776 with the Declaration of Independence.

[Jay R]

Originally Posted by warty goblin

Only when defined for Euclidean geometry in a perfect plane. Since the universe is not perfectly Euclidean* due to pesky things like gravity, the actual value of pi - the ratio of circumference to diameter fluxuates depending on where you are. Somewhere, it's rational.

Pi is a constant, originally defined as the ratio of the circumference and the diameter of a circle on a Euclidean plane.

Its value doesn't change, even though there are non-Euclidean "circles" that are not on a Euclidean plane.

[nedz]

Originally Posted by Mr.Silver

Funnily enough there is a similar 'official' numbering discrepancy in the British monarchy. Because monarchs only started being numbered after the Norman conquest, King Edward I was actually the second monarch to be called Edward (Edward the Confessor having been the first) and so on with all subsequent Edwards.

Its slightly worse than that since after the act of union we have two numbering systems: one English, one Scottish. Hence James I/VI etc.

Originally Posted by Jay R

Pi is a constant, originally defined as the ratio of the circumference and the diameter of a circle on a Euclidean plane.

Its value doesn't change, even though there are non-Euclidean "circles" that are not on a Euclidean plane.

See below

[Heliomance]

Ah, but consider a 10' blast. It consists of 12 squares which have an area of 300 sq ft. Thus Pi must clearly be 3.