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More on .9999... and 1

Read the com­ments. I am ashamed of math­e­mat­i­cal ed­u­ca­tion, right now. If these peo­ple has passed any math­e­mat­ics tests (and some even claim to have gone to col­lege), maths are hope­less­ly dif­fi­cult.

Some choice quotes:

1/3 is a sym­bol for a set of 4 word­s, it is not a NUM­BER.

On­ly a sin­gle num­ber CAN POS­SI­BLY = 1. Oth­er num­bers may ADD UP to 1, but they don't EQUAL 1. Since 1 clear­ly = 1, .99999 re­peat­ing sim­ply can­not equal 1.

.33(repet­ing) is ir­ra­tional.

.99999 does not equal 1. It might in the CUR­RENT UN­DER­STAND­ING of math­e­mat­ic­s, but that don't make it true.

Math­e­mat­ics can­not even prove that .99999 ... is not equal to 1.

Right now, math re­al­ly can't deal with in­fi­nite num­bers

.9 re­peat­ing, an ir­ra­tional num­ber, is AB­SO­LUTE­LY EQUAL to the ra­tio­nal num­ber 1. Can this be used as proof to show there is no such thing as ir­ra­tional num­ber­s?

I'm half tempt­ed to say there is­n't re­al­ly a right or wrong an­swer

I think I've come to the con­clu­sion that .999... = 1 in the same sense that .333... = 1/3. Which is to say, it does­n't, quite, but we treat it like it does be­cause our dec­i­mal sys­tem has prob­lem­s.

0.9 re­cur­ring does not equal 1. Why? Be­cause it's 0.9 re­cur­ring.

1 = .9 re­peat­ing IF WE WANT IT TO.

This is an ex­ploita­tion of our nu­mer­ic sys­tem, to ar­rive at an out­come that is in­deed very close to be­ing true, but the clos­er it gets to be­ing true the fur­ther away it ac­tu­al­ly is.

2.9 re­peat­ing plus 2.9 re­peat­ing equals 5.9 re­peat­ing 8

And this last one is amaz­ing. The poster pro­pos­es a num­ber that is a 5.9 (an in­fi­nite num­ber of 9s... and an 8). Right. An 8. Af­ter in­fi­nite 9s. At the end of them. Right there. Go to in­fin­i­ty po­si­tion, then one more. There's the 8.

My mind bog­gles. And it's a mind that ac­tu­al­ly ac­cepts .99(re­peat­ing) is 1.

How satanic messages work (with video)

Ev­ery­one knows about the hid­den sa­tan­ic mes­sages in songs.

You take a song, you play it back­ward­s, and in cer­tain places, you will have the singer say­ing some­thing evil, like "I like eat­ing pup­pies with cin­na­mon".

I have al­ways as­sumed that this hap­pens be­cause our brains try to rec­og­nize pat­terns in the sounds they get, and they are a bit too good in that job, but now I have proof.

Here's a video Rosario (my wife, hi dear!) sent me:

In it you can hear pieces of pop songs, in eng­lish (and lat­in), and sub­ti­tles of what they seem to say in span­ish.

Now, un­less you be­lieve Avril Lav­i­gne ac­tu­al­ly says "Lei­va quiso ven­derme el Ford" (Lei­va tried to sell me a Ford), and Mar­ley sings about "Where is Ju­li­a", the "pick­ing too much sig­nal" the­o­ry seems true.

Spe­cial­ly, if you are told what you should hear, it works much bet­ter!

I had heard these songs a mil­lion times, and I had nev­er thought they said that, but with the sub­ti­tles... some of them are pret­ty close :-)

The is­sue of why sub­lim­i­nal mes­sages en­cod­ed back­wards in songs make no sense in the first place is an­oth­er top­ic.

Infinite Amateurism on Maths

I stum­bled up­on Thomas Thur­man's post on his blog (here) where he com­ments about a dis­cus­sion on The Guardian about how .99(re­cur­ring) is or is not the same as 1.

Of course to any­one who knows what a ra­tio­nal num­ber is, .99(re­cur­ring) is sim­ply a very long way to write 1. Hel­l, to any­one who both­ered learn­ing his frac­tion­s, that should be ob­vi­ous!

But any­way, one of the com­ments men­tions Hilbert's Hotel, which is a pet toy of mine.

If you are un­com­fort­able or an­noyed by the con­cept of in­fin­i­ty, you may want to avoid the rest of this post.

Hilbert's Ho­tel is this para­dox (Thanks Wikipedia!):

A ho­tel with an in­fi­nite num­ber of rooms (1, 2, 3 and so on, so it's a nu­mer­able in­fin­i­ty) is ful­l. A guest ar­rives. Yet he still gets a room. How?

The an­swer is that you ask the guest in room 1 to move to room 2, from 2 to 3 and so on. Then the new guest goes to room 1, which is now free.

Be­cause of the na­ture of in­fin­i­ty, this work­s, while on a fi­nite ho­tel it would­n't.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: If you add any num­ber to it, the re­­sult is the same in­­fin­i­­ty.

Now as­sume an in­fi­nite (nu­mer­able) num­ber of guests ar­rives. You have to ask the guest in room 1 to go to room 2, the guest in room 2 to go to room 4, and guest n to go to n*2.

Now you have an in­fi­nite (nu­mer­able) num­ber of free room­s: all the odd room­s.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: If you mul­ti­­ply it by any num­ber, the re­­sult is the same in­­fin­i­­ty.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: If some­thing is in­­finite, it con­­tains a part the same size as the whole (fol­lows from the pre­vi­ous two things, and is, in fact an "if and on­­ly if").

How­ev­er, here's where it gets trick­y. You could get a cer­tain num­ber of new guests and there could be no way to fit them in the rooms even if the ho­tel was emp­ty.

Be­cause there's in­fin­i­ty, and then there is in­fin­i­ty. You saw that when­ev­er I men­tioned the num­ber of rooms I men­tioned they were in­fi­nite (nu­mer­able)? That's be­cause you can put an in­te­ger num­ber to each, and num­ber them al­l.

There are in­fi­nite sets of things that are big­ger, they are lit­er­al­ly un­count­able. You can't put a num­ber to each, even with an in­fi­nite amount of time (and yes, I know that in­fi­nite amount of time there is a big prob­lem).

The sim­plest set imag­in­able that large is that of the re­al num­ber­s. The re­al num­bers are all the num­bers you can imag­ine, al­low­ing for in­fi­nite dec­i­mal­s, and al­low­ing that those dec­i­mals may not ev­er be re­cur­ring (so you have things like 2, 1/3, and pi).

Show­ing there are more of those that there are in­te­ger num­bers is not sim­ple enough for this but go along with me for a while.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: There are dif­fer­­ent sizes of in­­­fi­nite. Go blame Georg Can­­tor.

Now it gets re­al­ly weird. Sup­pose we call the size of the in­fi­nite in Hilbert's Ho­tel A0 (I have no idea how to do an Ale­ph, sor­ry), and the size of the re­al num­bers C. Can­tor showed how to build, once you have an in­fi­nite set, a larg­er in­fi­nite set called his pow­er set.

That means we now have a whole in­fi­nite (nu­mer­able) "sizes" of in­ifi­nite things. Those are called the trans­fi­nite num­ber­s.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: There are in­­­fi­nite dif­fer­­ent sizes of in­­­fi­nite. Go blame Georg Can­­tor some more.

Which brings a lot of ques­tion­s:

  • Are there on­­ly those? Is­n't there some­thing be­tween A0 and C which is some in­­-­be­tween size?

  • Is there an in­­­fi­nite set that's smal­l­­er than the in­­te­ger­s?

  • Ok, in­­­fi­nite in­­finites... in­­­fi­nite (nu­mer­able) in­­finites, or in­­­fi­nite (some­thing else) in­­finites?

Well... I have no idea. And last I checked, which was long ago, and my mem­o­ry is no good, noone else knew.

This is the kind of things that will tell you whether you could be a math­e­ma­ti­cian. Do you find all this talk about trans­fi­nite num­bers in­trigu­ing and mys­te­ri­ous, or just dull and bor­ing and im­prac­ti­cal?

If you find it dull and bor­ing, it may be my writ­ing, or you may be un­suit­ed for math­s.

If you find it in­trigu­ing and/or mys­te­ri­ous, it cer­tain­ly is not my writ­ing, and you would prob­a­bly en­joy maths in your life. Where else are you go­ing to run in­to "noone knows that" this quick­ly?

The prob­lem is, of course, that most of the fun math has al­ready been done at least a cen­tu­ry ago, but there is al­ways a chance of some­thing fun and in­trigu­ing and new com­ing along.

The last I know of was Gödel's the­o­rem, which is re­al­ly sim­ple enough for any­one with knowl­edge of arith­metic to fol­low, but weird enough for 99.99% of the peo­ple to go crazy about (and for those who don't re­al­ly un­der­stand it to write whole books about it ap­ply­ing it to to­tal­ly im­prop­er sub­ject­s).

But you know, noone re­al­ly had thought of such thing as "larg­er than in­fin­i­ty" quite as Can­tor did, and noone thought about Gödel's sub­ject quite as he did be­fore him.

Maybe we are miss­ing some­thing ab­so­lute­ly sim­ple, in­cred­i­bly el­e­gan­t, awe­some­ly shock­ing some­where in ba­sic math­s. Not like­ly. But pos­si­ble. Would­n't it be fun to find it?


BTW: Gödel starved him­self to death and Can­tor "suf­fered pover­ty, hunger and died in a sana­to­ri­um".

Happy post

I have not post­ed in a few days, be­cause I have been very busy.

I have not even been able to see all match­es (I missed US­A/Czech, Italy/Ghana).

But man, did watch­ing Ar­genti­na/S&M pay of­f! :-)

A very good match by Ar­genti­na, which of course brings up the ob­vi­ous ques­tion­s...

  • Is the S&M de­fense made of be­­head­­ed chick­­en­s?

Al­leged­ly it's one of the best de­fens­es in Eu­rope, but their best play­er was out. So it's not con­clu­sive. My bet is that they are not be­head­ed chick­en­s, and that Ar­genti­na made them look bad. Or rather: 25% be­head­ed chick­en­s, 75% Ar­genti­na's mer­it.

On the oth­er hand, the play­er that said they should have at­tacked against the dutch was wrong. It's not called be­ing a cow­ard, it's called know­ing you can't score.

  • Was the good Dutch per­­for­­mance against S&M re­al?

Who knows. We will fig­ure it out for sure af­ter they play Ivory Coast. My bet? Tough match for the dutch, pos­si­bly a tie.

Now for the oth­er team­s:

Eng­land has noth­ing. They suck much more than you think now. Wait un­til they play a team that ac­tu­al­ly has a for­ward who can score (and is not play­ing 5 like Yorke). Set pieces for Beck­ham at mid­field and Crouch try­ing to head it in is not some­thing that's go­ing to work a lot. For their sakes, Owen and Rooney bet­ter start show­ing some­thing com­plete­ly dif­fer­ent than against T&T. Which still has a chance. I would pre­fer them and not Swe­den to stay in the cup!

Ger­many has some­thing. The de­fense was im­proved (although Poland has a lame at­tack­), and their for­wards are not bad.

Spain has a lot. Spe­cial­ly go­ing for­ward, and they made Ukrayne look like am­a­teurs. We'll see if that was Ukrayne's fault of Spain's mer­it lat­er on.

Brazil has some­thing big. His name is Ronal­do. Re­move him and they are go­ing to kick as­s.

Poland-Ecuador-England-Paraguay-Sweden-T&T-IvoryCoast-Argentina

Poland is... enough about Poland.

Ecuador played ok, they tried to stay sim­ple, do their thing, make the oth­er team beat them. Which Poland had no in­ter­est in, so Ecuador scored a cou­ple of goals just in case, and hey, they won! Poland is per­haps the worse team so far, and that in­cludes Trinidad and To­ba­go. And Cos­ta Ri­ca.

Eng­land, I have heard, are slow starter­s. I hope for their sake that's the case. Ash­ley Cole has a se­ri­ous case of be­liev­ing he is way more skilled than he needs to be ( I thought Rio Fer­di­nand was the one with that prob­lem?) And any­way, Paraguay start­ed weak, and had huge trou­ble at­tack­ing. So, good win for Eng­land, hope­ful­ly Paraguay was just ner­vous. Gamar­ra is a re­al­ly un­lucky guy.

Swe­den... what a frus­trat­ing match. T&T was play­ing on guts alone, their best for­ward played as a de­fen­sive mid­field­er, and still they had the clear­est shot at goal, out of a pass by the goal­keep­er! Swe­den was just in­cred­i­bly un­luck­y, or their for­wards were blind­fold­ed.

Ivory Coast is a pret­ty good at­tack­ing team. They are a be­low av­er­age de­fense, though, which ex­plains why Ar­genti­na scored twice on three chances, and the oth­er one was saved 90% be­hind the goal.

Savi­o­la played well, Riquelme did­n't. The de­fense... they played waaaay back, and Ab­bon­danzieri sim­ply gave away ev­ery ball he touched with a long kick to an ivo­rian.

But hey, it's a win... no ob­vi­ous trou­ble... need to play with some con­fi­dence... ok, we suck a lit­tle. But it can get bet­ter.

Pala­cio for some rea­son could­n't stand up in the field, was fall­ing when­ev­er he tried to run.

But I think ev­ery­one agrees: the Ivory Coast is a scary team. They are skilled, they are ath­let­ic, they are not too bad­ly or­ga­nized... they can be a chore for any team.

So far, of the "good" team­s... none has looked very good. No can­di­dates to win yet, IMVHO.


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