“Numbers” Table of Contents

Part 1: Magnitude
Part 2: Exponentiation
Part 3: Hyper and Knuth Up Arrows
Part 4: Graham’s Number and Conway Arrows
Part 5: Extended Conway Arrows
Part 6: Limit Functions
Part 7: Larger Limit Functions
Part 8: Exponentiation Within Limit Functions

ω Cubed Plus One

So we know all the easy ways to make ω³ larger. We could go ω³+1, and use function notation. We could go ω³+ω and access any “+x”. We could go ω³+ω2 or even ω³+ω², or even more:

f_ω³+ω²(x) = f_ω³+ωx(x)

f_ω³2(x) = f_ω³+ω²x(x)

f_ω⁴(x) = f_ω³x(x)

f_ω⁵(x) = f_ω⁴x(x)

…

Remember that when we go from ω⁴ to ω⁵ we’re doing a very large jump — skipping things like ω⁴+1, ω⁴2+63, ω⁴+ω², ω⁴3+ω³16+ω²919+ω21+3, etc.

Also remember that right now we are incomparably larger to any mainstream notation for writing large numbers, and get indescribably bigger at every increase.

One Superscript to Rule Them All

The real fun is when we get to ω^ω.

f_ω^ω(x) = f_ω^x(x)

Woah!

So, for example:

f_ω^ω(8) = f_ω⁸(8) = f_ω⁷8(8) = f_ω⁷7+8(8) = f_ω⁷7+7⁸(8) =
f_ω⁷7+7(f_ω⁷7+7(f_ω⁷7+7(f_ω⁷7+7(f_ω⁷7+7(f_ω⁷7+7(f_ω⁷7+7(f_ω⁷7+7(8)))))))) = …

Yet Bigger

But there are simple ways to go even bigger, just by repeating what has already been done. ω^ω gives way to ω^ω+1, ω^ω+2, and then ω^ω+ω. That gives away to things like ω^ω+ω2 and ω^ω+ω², ω^ω+ω^6928314, and all sorts of things in between.

Then of course there’s ω^ω2 and ω^ω3… And there’s even a ω^ω2.

Here are some examples of what they all evaluate to:

fωω+1(x) = fωωx(x)

f_ω^ω+2(x) = f_ω^ω+1^x(x)

f_ω^ω+ω(x) = f_ω^ω+x(x)

f_ω^ω+ω2(x) = f_ω^ω+ω+x(x)

f_{ω^ω+ω^6928314}(x) = f_{ω^ω+ω^6928313x}(x)

f_ω^ω2(x) = f_ω^ω+ω^x(x)

f_ω^ω3(x) = f_ω^ω2+ω^x(x)

f_ω^ω+1(x) = f_ω^ωx(x)

Yet Bigger, Yet Again

So with simple logic, this system can be expanded very quickly using just a few rules. You can also keep expanding stacks of “ω” on top of each other, getting much larger numbers, very quickly. Each stack also opens a lot of areas for recursion.

For example:
f_ω^ωω(2) = f_ω^ω2(2) = f_ω^ω2(2) = f_ω^ω+2(2) = f_ω^ω+12(2) = f_{ω^ω+1+ω^ω+1}(2) = f_{ω^ω+1+ω^ω2}(2) = f_{ω^ω+1+ω^ω+ω^ω}(2) = f_{ω^ω+1+ω^ω+ω²}(2) = f_{ω^ω+1+ω^ω+ω2}(2) = f_{ω^ω+1+ω^ω+ω+2}(2) = f_{ω^ω+1+ω^ω+ω+1}(f_{ω^ω+1+ω^ω+ω+1}(2)) = …

Next

In the next article we’ll take a significantly larger jump. We started at addition, then went to multiplication, and then later to exponentiation. Just like how the initial numbers were expanded by going beyond exponentiation, we can do the same with stacking limit functions…

Onward to the summer and then Denison University!

But there are 365 possible birthdays. How is this possible?

But first…

How is it a Paradox?

There are many different types of paradoxes. Most paradoxes are based on contradictions, but some paradoxes, such as the Birthday Paradox, are only called paradoxes because they defy common sense. The Monty Hall problem, which I explain in depth with wonderful illustrations over in another post, could also be called a paradox for the same reason. …But both paradoxes are completely true.

From People to Pairs

We know that there are 23 people in the group that we’re testing. But we didn’t ask for the chances of them matching you, we wanted to know the chance of any person having the same birthday as any other person.

So if we had a group of 4, containing You, Alice, Bob, and Carol. If we just see who has the same birthday as you, there are only three matches to test: You-Alice, You-Bob, and You-Carol. Bt with the actual problem matching any person to any other person, we have 6 matches to test: You-Alice, You-Bob, You-Carol, Alice-Bob, Alice-Carol, and Bob-Carol.

If you do the math, any group of size x will have x choose 2, or _xC₂, both of which is the Binomial coefficient function, or more commonly a “combination”. “x choose 2″ means that if we have x things, how many different ways can we combine 2 of them?

For example, with our previous group of 4, we could combine them in 6 different ways. That was “4 choose 2″. Note that “You and Alice” and “Alice and You” are the same combination.

For 23 people, we’re finding way more than 23 pairs:

…253 pairs.

A Sprinkle of Probability

Now, what’s the chance of two different people having the same birthday? Well, ignoring leap years, there are 365 possible birthdays. That means that if, say, Alice has the same birthday, than the chance of Carol’s birthday matching Alice’s birthday is 1/365.

So how do we check the chances for every single person?

Well, we know from statistics that 1 – [the chance of the event not happening] is the chance of the event happening. For example, if rolling a 3 on a 6-sided die happens 1/6 of the time, the chance of it not happening is 5/6. 1 – (5/6) is 1/6.

And we also know that you can multiply events to see the chance of both of them happening. If flipping a coin and getting heads happens 1/2 of the time and rolling the 3 happens 1/6 of the time, then getting heads and rolling a 3 at the same time will happen (1/2)(1/6) of the time, or 1/12 of the time.

The last trick we’ll need is exponentiation. If we wanted to know the chance of the same event happening several times in a row, we would have to multiply it a lot. For example, the chance of flipping 5 coins and getting 5 heads is (1/2)(1/2)(1/2)(1/2)(1/2), which is (1/2)^5, or one-half to the fifth power. (It evaluates to 1/32.)

Putting it All Together

So we have a group of 23 people. We know that’s 253 pairs.

The chance of any pair not sharing a birthday is 364/365. That’s 99.73%, so it’s very likely that any pair will not share a birthday. However, the real action gets in when we factor for all 253 pairs. The chance of all 253 pairs not sharing a birthday is (364/365)^253.

That’s 49.95%. And since 1 – [the chance of the event not happening] is the chance of the event happening, the chance of a group of 23 people having one shared birthday is 1 – 49.95%, or 50.05%.

What About the Group of 75?

Well, for the group of 75, we just do the same thing.

First, we find the number of pairs:

Then we do the probability for all 2775 people:

Despite starting out with a 1/365 chance for a single pair, groups make for lots of pairs, and the exponentiation works out quickly.

So the next time your in a classroom of 20 people, think of how many of them might share birthdays with each other! Or don’t, because that would be excessively nerdy…

…But what happens if he gets in a fight with “lack of journalistic integrity?”

That’s what happened on Fox News recently:

The Fox News Argument

The summary of the Fox News argument is as follows:

The SDSU “Expert” Study

The often mentioned “expert” study actually had nothing to do with Mr. Rogers at all. Here is a link to the study, that came out of San Diego State University (SDSU) and the University of South Alabama, which found “that narcissism in college students is increasing faster than previously thought, especially among college women.” The referenced study continues to simply state that out of a random sample of 20,000 college kids gathered each year, narcisism is at an all time high since 1982.

No mention of Mr. Rogers. No mention of “entitlement”. No mention of any cause at all. Anywhere.

The LSU Professor “Expert”

Since the study of college students did not include any explanation for why narcism is on the rise, Fox News felt like filling in the blanks for themselves.

Sure, there is nothing wrong with attacking Mr. Rogers. He certainly was no child psychologist. My interpretation of his message was simply that in a world where everyone is out to get you, it is very important to stick to your guns and be an individual.

However, Fox News has found a new interpretation. They borrowed their interpretation from acclaimed “expert” Louisiana State University professor Don Chase. His opinions were written in a Wall Street Journal article — that Mr. Rogers is responsible for a feeling of entitlement.

But what is the evidence for this argument? Well Professor Chase noticed that every end-of-term, Asian students would simply accept their grade if it was very close to an A, but not an A. However, non-Asian students would — gasp — ask the Professor to bump their grade to an A.

This is obviously because of Mr. Rogers’s influence.

Was this connection well researched? No. The Mr. Rogers / entitlement connection “dawned on [Professor Chase] last spring” and, according to Chase himself — “[I]t just hit me. We can blame Mr. Rogers.”

Was this connection even supported statistically? No. It was a simple observation by a single university professor.

Was this professor even credentialed? No! Fox News acclaimed “expert” Professor Chase was actually an expert in finance, not child care! His background in economics certainly doesn’t give him the credibility to attack Mr. Rogers without relying on outside expert information, which he does not.

All it allows him to do is pedal his racism as if it is science.

Journalistic Integrity

So Fox News basically took a professor with no credibility who spout out an opinion backed up by no data, no evidence, and no science and put him on air as if he was right, calling him an “expert” several times. Then they threw in a study that had nothing to do with Mr. Rogers simply because it gave them a second “expert” and made them sound twice as smart.

Certainly people are allowed to have opinions, but Fox News is a news organization. Their slogan is “Fair and balanced”. If you’re going to be a news organization, you can’t just present opinion like it is fact. That isn’t journalistic integrity at all — that isn’t even close!

Fox Commentator Brian Kilmeade actually used the word “evil” to describe Mr. Rogers, and criticized him for his “optimistic message where everyone was special even if they didn’t deserve it”, which is “ruining kids with a sense of entitlement”, even going as far as to state that Mr. Rogers “unintentionally did a whole generation or two a disservice.” Then one of them actually starts stating that children should go back to churning butter and making their own sweaters!

Fox News is Not News, it is Lies

Tony Burman, ex-editor-in-chief of CBC News, once said that “every news organization has only its credibility and reputation to rely on.” If that is true, than Fox News has nothing.

Certainly Fox News can suspend the news portion of their title and shout unfounded opinions, but they are not entitled to pretend their opinions are facts. When they cite two “expert” sources that are 1.) irrelevant and 2.) not expert sources, they are the ones doing us a disservice. They are the ones with the arrogant sense of entitlement, thinking you can just throw whatever you want on TV in front of the American public.

Maybe if Fox News stopped lying to our generation, we would be less like the way we are.

And not only that, but I have as much data supporting the idea that Fox News is to blame as Fox News has data supporting that Mr. Rogers is to blame.

At least Mr. Rogers had a train that warned us that we are about to enter the land of make-believe. Fox News should consider getting one for their own channel.

If you didn’t see it, here is a collection of Jon Stewart clips ripping on Fox News for some other completely unethical things that they do.

1.) I made a new theme due to the fact that very few people were exploring my subpages. Now, every single piece of content on this website is in the two sidebars to your right and left. Everything will be cleaned up over the course of the weekend and the next week.

Regular posts should still resume on May 19.

2.) I got my first death threat in the comments section as a response to my series on the Death Star:

Well I hope I never see this blog again because you guys no nothing of the Sith and what we do and if you post again I will destroy you not the planet.
- RW

…Uh oh.

Fox News and Fox Opinion Are One in the Same

Fox’s Use of Sweeping, Hypocritical Generalizations

Fox’s “Creative Editing”

Fox News Creatively Selects What They Choose to Support

Fox News Only Likes Their Own Protests

Fox News Makes the Conclusion, You Make the Choice

These assumptions, fundamental to our use of the internet, are not actually guaranteed. They actually could be stopped by scary company interests at any point in time.

The only way to promise these freedoms is through a policy called Net Neutrality. Net Neutrality is essentially a promise for a level playing field. Net Neutrality makes it so the only thing an internet service provider can do is move data in a non-biased manner, and they cannot pipe their paying customers into higher-priority channels, cutting out the little websites… like mine.

But No One Really Is Threatening Net Neutrality!

This is actually the biggest argument about Net Neutrality — whether there is a real threat that needs to be responded to. Well, actually, they is.

When Comcast recently decided to intentionally and artificially slow down access to Torrent websites in an effort to “save bandwidth”, the Federal Communications Commission took Comcast to court, arguing that Comcast was violating existing Net Neutrality legislation. The resulting case reached the District of Columbia Court of Appeals in Comcast v. FCC (2010), where the court ruled that “Comcast can limit the ability of certain types of information to be easily shared over the Internet”, a process effectively discriminating against BitTorrent users.

But that’s not all:

In 2005, the Canadian company Telus blocked access to a pro-labour union website.

In the same year, Time Warner’s AOL service blocked all emails that mentioned “dearaol.com”, a website opposing AOL’s pay-to-send email scheme.

Also in 2005, BellSouth announced that “that Internet service providers should be allowed to strike deals to give certain Web sites or services priority in reaching computer users, a controversial system that would significantly change how the Internet operates.”

Last but not least, In 2006, phone companies stated their intentions to “start charging Google Inc., Vonage Holdings Corp. and other Internet content providers for high-quality delivery of music, movies and the like over their telecommunications networks”.

Net Neutrality is Just Excessive Regulations!

Well, actually, no.

Net Neutrality is actually a very simple legislation: all it means is “no discrimination”. Net Neutrality prevents network operators from blocking, speeding up, or slowing down content based upon source, ownership, or destination.

And not only is it simple, it is already in place. Net Neutrality has existed from the beginning as a part of FCC regulations insuring consumer protection.

Network Discrimination will Benefit Consumers with Higher-Quality Services!

Well, that’s not entirely true either, unless you like having less choice. Network Discrimination is also a simple concept — the Network owner intervenes between the consumer and the content to charge additional fees for delivery.

Anti-Net Neutrality-ites argue that this would allow the creation of additional “fast lanes” for better access to content, but this would simply sort the content into payers and not-payers. The poor non-company websites would be left in the slow-lane, unable to compete.

Without Network Neutrality, the network operator has total control. They can charge different fees based on what you are doing — voice, video, or data — and based on the provider, based on if you are accessing services provided by a competitor, etc. This is horrible — not only is it double dipping, by charging you for the service and then again for what you do with the service — it is blatant anti-consumerism, cutting out the competitor and preventing market operations from taking place on a level playing field.

They get to create “fast lanes” and “slow lanes” and decide who gets to be in each. There is nothing to stop exclusive deals between Time Warner and Comcast to stop access to Verizon. There is nothing to stop AT&T from slowing down Google and speeding up Yahoo, altering the market for search engines. They would be manipulating the market at it’s highest level.

It’s a lot more than getting you better access to your content. It’s borderline fraud. And only Net Neutrality can fully stop it.

Competition Would Punish Any Anti-Competitive Behavior!

Unfortunately, there are not enough individual companies to push out anti-competitive behavior. And if there was, we would see it. Even with active net neutrality regulations, companies are still trying to act like they own the entire internet. Like power and utilities, broadband providers do not spread out and give people hundreds of options — 98% of houses only have two broadband providers — one for cable, and one for DSL. These two systems dominate.

And the share of the internet by all other technologies combined — satellite, mobile, broadband over power lines — is actually decreasing.

Net Neutrality Saves Me On Monthly Bills!

Network companies are happy to state that cutting “useless regulations” will save you on monthly bills. But this again is not true. The arguments is that because network companies will be paying the internet companies fees to be in the “fast lane” and to have guaranteed content delivery.

But we all know how fees on companies work — just like when grocery companies have to pay sales tax — it’s passed right on to the consumer. Google, Amazon, and eBay are not going to simply take the extra costs that are levied upon them — they will pass them onto you somehow, perhaps by raising advertising rates. These higher advertising rates will result in higher consumer prices because those who sell the advertised goods will have to raise their prices to account for the extra charges. And some of your favourite websites that can’t afford to stay around with the new fees might close down.

You will get hit in the wallet one way or another. Would you rather do it the free market way, or by hoping Comcast will keep their promise and really lower prices?

If Net Neutrality Already Exists, Why Are You Upset?

Well, Net Neutrality exists, but only through the FEC. Existing Congressional legislation does not guarantee Net Netutrality. This is exactly why the case Comcast v. FEC could be decided in Comcast’s favour.

It may be a doomsday scenario, but tiered service still could be a reality. We need to do something to fight it.

You can find out more on what to do and how to help Net Neutrality at savetheinternet.com.

Comic by Matt Davies

The first is a video for Student Body Elections:

And then a satire video about Swine Flu hysteria for English class:

(PS: Post #2345 win!)

Wednesday, April 7, 2010 11:00pm: Jon Stewart’s show, The Daily Show airs, with this segment about Michael Steele.

Note the similarity in the quotes used! Of course I’m not stupid and I’m not accusing Jon Stewart of plagiarism, but I am claiming the chops of putting something together at a similar quality as the amazing Jon Stewart.

Now, if only I could blog post with muppets…

There is this character Michael Steele. He looks like this:

Who is This Guy?

Michael Steele is the Chairman of the Republican National Committee, which provides leadership for the Republican Party’s interests in donations and funding Republican candidates.

Michael Steele is also the guy I ranted about in this earlier post, where he said that he didn’t want to be Chairman, despite deciding to run for chariman of the RNC, announce his candidacy, run his campaign, win the election against five other candidates in a series of six run-off elections, make a killer acceptance speech with quotes such as “this is awesome” and “God bless our party”.

But that, and the other lies and half-truths, aren’t what we’re covering right now. Rather, it’s the other quote from the same post:

The problem that we have with this president is we don’t know him. He was not vetted, folks… He was not vetted, because the press fell in love with the black man running for the office.
– Michael Steele

Which is also known as “playing the race card”.

Steele’s Hatred of The Race Card

But Michael Steele hates playing the race card, which is an idiomatic phrase that refers to the act of bringing the issue of race or racism into a debate, usually with an intention to obfuscate the matter. For example, when Senate Majority Leader Harry Reid (D-NV) invoked slavery in a healthcare debate, Steele had quite the response:

I’m kinda sick and tired of the left and Democrats in this country when they get in trouble and don’t get their way and their backs are up against the wall on legislation or whatever it is their trying to do, they go to that card, they play that race card, that slavery card, that civil rights card.

And when Jimmy Carter discusses potential racism in the Republican Party, Michael Steele writes an entire article on Politico, in which he comes out very strongly against the race card, with statements like:

Voicing opposition to the president’s policy proposals is not being a racist. It is being an American. Such assertions by leaders of the Democratic Party are a pathetic attempt to shift attention away from the president’s wildly unpopular government-run health care plan that the American people simply oppose. Injecting race into the debate over critical issues facing American families [...] only divides Americans rather than uniting us to find solutions to challenges facing our nation.

Michael Steele Would Never Play The Race Card…

And Michael Steele not only hates the race card, he’s the first one to tell you that he’d never play the race card. For example, in a discussion about Glenn Beck’s claim that Obama hates white people, Steele declined to refute Beck’s statements, instead saying:

I don’t play the race card, I don’t play the race game, the way some tend to want to do.

So when Michael Steele is faced by crushing criticism — such as the news that RNC donor money was used to fund a $2000 trip to a Los Angeles bondage-themed strip club, and then later the news that a fundraising letter “mistakenly directed would-be donors to call a telephone number belonging to a phone-sex operation”, how do we expect Michael Steele to react? Would he play the race card? Never!

We know that Michael Steele is not new to bad weeks. He’s been criticized of having a spending spree that has angered donors, and he sparked a GOP feud when he called abortion an “individual choice”.

Michael Steele’s chairman missteps have gotten so bad that Tony Perkins, president of the conservative Family Research Council, urged members to stop donating to the RNC. Then, Sen. John Kyl (R-AZ) complained about Steele in public, additionally refusing to say whether Steele “should step down or not.” Last but not least, a recent National Journal poll found that 71% of Republican “insiders” categorized Steele as a “liability.”

It’s the same as when Sean Mahoney, a businessman who has been thinking about running for Congress, expressed his dissatisfaction by resigning his position on the New Hampshire’s Republican National Committe, decrying the national Republican Party’s “out-of-touch, free-spending culture” in a message to RNC Chairman Michael Steele, which also stated that “the recent scandal involving RNC funds being used to entertain a small crowd at a Los Angeles strip club is the straw that broke the camel’s back.”

And then to add insult to injury, one of the Republican Party’s top strategists Alex Castellanos called on Steele to resign, saying “I think a change in the direction now, at this point, would do the party good.”

So how do we expect Steele to react to all this criticism? Would he play the race card? Never!

…Except When He Plays The Race Card

Well, apparently he actually would play the race card. In an interview about the criticism, George Stephanopoulous asked Michael Steele if his race might be responsible for the criticism:

STEPHANOPOULOS: We’ve got questions on my blog for you, this morning. One came in from Myron, he asked, do you feel as an African-American, you have a slimmer margin for error as another chairman would?

STEELE: The answer is yes.

STEPHANOPOULOS: Why is that?

STEELE: It just is. Barack Obama has a slimmer margin. A lot of folks do. It’s a different role for, you know, for me to play and others to play. That’s just the reality of it. But you take that as part of the nature of it. It’s not — it’s more because you’re not someone that they mow. I’m not a Washington insider, even though I grew up here in D.C. my view on politics is much more grassroots oriented. It’s not old boy network oriented. And so, you know, I tend to come at did stronger, streetwise, if you will..

You can watch it here:

But That’s Not All!

And that’s not even the first time race-card-hating Michael Steele has played the race card to fight back when he faced criticism. After Michael Steele refused to deny Glenn Beck’s accusation that Obama “hates white people”, Steele immediately changed the topic by saying that the real racism is in the black neighborhoods represented by Democrats:

You know, I’m looking at cities around the country where black folk live. Now, you showed me the Congress with all these wonderful black Democrats and white Democrats. But I go to black neighborhoods that are run by those same Democrats. And you tell me where racism really exists.

Racism In Reverse

But what makes Steele’s accusations of racism and having a slimmer margin when facing criticism is especially ironic as he has consistently faced the accusation that he got his position largely on his race, and the accusation that the only reason he hasn’t been fired over the criticsm is because of his race. Joshua Green, a senior editor for The Atlantic since 2003, writes:

Steele is hapless, solipsistic, and incompetent. When he isn’t embarrassing his party with his personal antics, or his staff’s, he’s setting it up for failure by driving away its top fundraisers and not keeping pace with Democrats. It’s impossible to imagine his magisterial display of buffoonery going unpunished in almost any circumstance–but it is going unpunished, and Steele appears to be in no danger of losing his job. Far from being a problem, his race is all that’s standing between Steele and a pink slip.

First Read confirms that Michael Steele is keeping his job despite his overwhelming failures and slip-ups, but that he will not be returning as chairman for the 2012 cycle.

We’ll miss you, Michael Steele.

I think that it is a very silly comment to make. I think Michael Steele’s problem isn’t the race card, it’s the credit card.
- Robert Gibbs, White House Press Secretary (Source)

As someone who has worked extensively in both the club and film side of the Adult Entertainment Industry, I know from experience that a mere $1900 outlay at a club with the reputation of Voyeur is a clear indication of a frugal investment with a keen eye toward maximum return.
– Stormy Daniels, Louisiana Senate Race Hopeful and porn star (Source)