What’s The Biggest Number?

My son’s in first grade now, which I still don’t quite believe. But it’s true. And, thanks to first grade, he’s really starting to get a handle on this ususual technology we call “math”. It’s just adding and subtracting right now, but he’s really getting excited about it. He’ll ask me random math questions, and ask me what my favorite number is, and so on and so forth. Then, one day, he asks me this: “What’s the biggest number?”

Well. I’m ready for this. One of the classes I took for my woefully under-used BS in Computer Science and History was on logic and number theory. “There isn’t one.”

“Yes there is!” he declares.

“No,” I tell him. “Because, no matter what, you can always add one to the number.”

“No you can’t!” he insists, shocked by his first glimpse of the concept of infinity.

“Well,” I ask him, “what’s the biggest number you can think of?”

“A googol!” he announces with some confidence.

“Do you know what that is?” I respond. His puzzled expression tells me he doesn’t, so I fill him in. “You know how one hundred is one followed by two zeros, right?” He nods. “Well, a googol is one followed by a hundred zeros.”

His eyes go wide. “What’s a googol take away one?” he asks.

“That would be…” I think for a moment. “Ninety-nine nines.”

“Wow,” he says. “What’s that called?”

“Uhm… I don’t know.”

What is the biggest number?

Like I said, there really isn’t one. You can always add one to any arbitrarily large number. A googol, for example, is one followed by a hundred zeros. It looks like this (adjust your browsers, please):

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

Pretty big, right? Well, a googol plus one is:

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,001

Clearly, all of these zeros get unwieldy in an absurdly short period of time. That’s why exponential notation is used. This, if you’re not familiar with it, is written in the format xy and means that x is multiplied by itself y times. 32 is 3*3 (equaling 9), while 23 is 2*2*2 (equaling 8). For convenience, since we count by 10s, most numbers are presented in a 10y format – sometimes called “scientific notation”. In this format, a googol is 10100, which you’ll have to admit is a whole lot easier to write than all those zeros I typed out above.

We do have names for a lot of really big numbers, though. Oddly enough, the names you use depend on whether or not you are using the “long scale” of numbering, the “short scale” of numbering, or Metric prefixes.

Hang on. I know what Metric is, but what’s short scale and long scale?

Brace yourself. The rabbit hole goes deep, here.

Short and long scale is just different ways of naming really big numbers, depending on where you live. According to Wikipedia, most English-speaking and Arabic-speaking countries use short scale, while most other countries in continental Europe and most countries that speak Romance languages use long scale. The difference sounds simple, and is all based on what you call numbers bigger than a million (106). Here goes:

  • When using short count for numbers larger than a million, you get a new name every time you get 1,000 times larger. So you count 1 million, 10 million, 100 million, 1 billion.
  • When using long count for numbers larger than a million, you get a new name every time you get 1,000,000 times larger. So you count 1 million, 10 million, 100, million, 1,000 million, 10,000 million, 100,000 million, 1 billion.

Simple, right?

No.

Yeah, it kind of confused me too. I grew up using short count, so it looks intuitive and long count looks really wierd. I’m sure that if I grew up counting long count instead, than I’d flip those attitudes. Maybe a table would help?

Probably

All right. Here goes. Brace yourself.

Number Short Count Long Count Metric Prefix
1 One One N/A
10 Ten Ten deca-
102 Hundred Hundred hecto-
103 Thousand Thousand kilo-
106 Million Million mega-
109 Billion Thousand Million (or Milliard) giga-
1012 Trillion Billion tera-
1015 Quadrillion Thousand Billion (or Billiard) peta-
1018 Quintillion Trillion exa-
1021 Sextillion Thousand Trillion (or Trilliard) zetta-
1024 Septillion Quadrillion yotta-
1027 Octillion Thousand quadrillion (or Quadrilliard) N/A
1030 Nonillion Quintillion N/A
1033 Decillion Thousand quintillion N/A
1036 Undecillion Sextillion N/A
1039 Duodecillion Thousand sextillion N/A
1042 Tredecillion Septillion N/A
1045 Quattudorcellion Thousand septillion N/A
1048 Quindecillion Octillion N/A
1051 Sexdecillion Thousand octillion N/A
1054 Septendecillion Nonillion N/A
1057 Octodecillion Thousand nonillion N/A
1060 Novomdecillion Decillion N/A
1063 Vigintillion Thousand decillion N/A
1066 N/A Undecillion N/A
1069 N/A Thousand undecillion N/A
1072 N/A Duodecillion N/A
1075 N/A Thousand duodecillion N/A
1078 N/A Tredecillion N/A
1081 N/A Thousand tredecillion N/A
1084 N/A Quattuordecillion N/A
1087 N/A Thousand quattuordecillion N/A
1090 N/A Quindecillion N/A
1093 N/A Thousand quindecillion N/A
1096 N/A Sexdecillion N/A
1099 N/A Thousand sexdecillion N/A
10100 Googol Googol N/A
10102 N/A Septendecillion N/A
10108 N/A Octodecillion N/A
10114 N/A Novemdecillion N/A
10120 N/A Vigintillion N/A
10303 Centillion N/A N/A
10600 N/A Centillion N/A
1010100 Googolplex Googolplex N/A

There are, of course, proposals for names for larger numbers. I won’t go into too many details here, beyond saying that the proposed name for the number 103003 is “millinillion” for short count naming conventions and “thousand quingentillion” for long count naming conventions. And now you know.

Yeah. It’s big.

But, what about infinity? Isn’t that the largest number?

Nope.

What? But, I’ve always heard…

“Infinity” isn’t the largest number, because infinity isn’t a number. Infinity is a concept, and in numbers it refers to either an arbitrarily large or undefined number (such as how the value of X/0 goes to infinity in calculus), or it represents sets of numbers. When using infinity as a set, you get two basic types of infinity: countable, and uncountable. Countable infinity is an infinite set whose “elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.” Basically, it’s a clearly defined set. All natural numbers (1, 2, 3, 4, … infinity) is a countable infinity. So is all even numbers, all odd numbers, all prime numbers, and so forth.

Uncountable infinities, on the other hand, aren’t quite so neat. An uncountable set “contains so many elements that they cannot be put in one-to-one correspondence with the set of natural numbers. In other words, there is no way that one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.” For example, the set of all real numbers between 0 and 1 is uncountable. Why? Because there will always be an infinite number of fractions between any two members of that set you care to name.

Could you try and make all that a little clearer?

Sure. A countable infinity has a finite (although possibly large) number of points between any two elements of the set. An uncountable infinity has an infinite number of points between any two elements of the set. As an example, take the countable set of all natural numbers. If you pick a starting point (let’s say the number 12) and an ending point (let’s say the number 17), then it you have to count only 5 steps to get from 12 to 17. If your starting and ending points were 2 and a googol, then you’d have to count 9,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,999,998 steps – that’s a whole lot of counting, but you could do it in a finite (but large) amount of time. It can be counted.

Now, consider the set of all real numbers between 3 and 5. It looks shorter at first glance, but remember that real numbers include fractions and decimals. So, let’s pick our starting point as 3.0001 and 3.0001001. 3.00010001 is between them. So is 3.00010002, and 3.00100000000007, and every other amount of extra decimal places that could conceivably be tacked on. And since there is no limit to the number of extra decimal places, you would never be able to count every single possible number that lie between 3.0001 and 3.0001001. Ever. It cannot be counted, so it is uncountable.

My head hurts.

Mine, too. Now, let me make it hurt worse. Uncountable infinities are larger than countable infinities, even though both infinities are infinite in size.

What?

Yes, and it’s all down to that the countable versus uncountable aspects of the sets. Let’s look at the set of all natural numbers versus all real numbers, and then count the “steps” between 2 and 3. The countable infinite has a finite distance between those two numbers, while the uncountable infinity has an infinite distance between the two numbers. So the uncountable infinity is larger because, between any two arbitrary points, it is larger than the countable infinity for the same segment of the set.

So, in summary, there is no “largest number”. You can always add one to any number you choose (a millinillion and one, a millinillion and two, a millinillion and three,…), and infinity doesn’t count because it’s not a number. Even when it’s countable.  And a googol take away one would be called, in short count, “a googol minus one” because I couldn’t find a short count name for numbers that big.

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Why Was The United States Underwater?

Several months ago, I wrote about the fossil my son found and what it most likely was. What I didn’t talk about in either article was the trip we took to the Trammel Fossil Park here on the north side of Cincinnati. It’s really just the exposed rocky side of a hill, with signs posting the various stratigraphic layers so you know where you’re looking and other signs showing you the fossils you’re likely to find at each level. There’s no cost to go, and you’re allowed to keep any fossil-bearing stones you find that you care to haul down the hill and back to your car. I found some brachiopods.

My son was extremely disappointed with the trip, at least for the first ten or fifteen minutes we were there. We’d told him we were going fossil hunting, after all, and he wanted to find a Tyrannosaurus rex skeleton. Which, lets be honest, would have been extremely unlikely even if the park had exposed strata from the Albian. But he was six at the time, and he wanted a dinosaur. So I reminded him that the layers we were looking at were from an ocean, because Ohio was underwater at the time.

I don’t think he asked today’s question at that point, but it helped inspire it. Because, eventually, he asked me this: “Why was the United States underwater?”

Well? Why?

Uhm. Something to do with plate tectonics, I guess? And maybe changes in climate?

Can you do better than that?

Of course I can. This’d be a pretty lame blog post, otherwise.

What are plate tectonics?

That’s a great question, and to understand it we’ll need to cover the structure of the Earth itself. The Earth is comprised of multiple layers, rather like an onion. These layers are the:

  • Lithosphere: the outermost rocky shell of a rocky planet (our own, for instance).
  • Asthenosphere: the hot, viscous layer that the lithosphere floats on.
  • Mesosphere (or mantle). Geologists have an explanation for why this is distinct from the asthenosphere and the outer core, and it has something to do with temperature and pressure causing one type of mineral to decompose into another type of mineral. I didn’t quite follow the explanation, and I think I’ll save trying to understand it for the day when my son asks “what is the mesosphere?”
  • Outer Core, a sea of liquid iron and nickel.
  • Inner Core, an extremely hot ball of (mostly) iron and nickel kept solid by pressure.

The lithosphere is the layer we live on – the high parts are the continents and the lower parts are covered with water. And it isn’t a solid shell. It’s broken up into (depending on who you ask and the definitions they use) seven or eight major tectonic plates and a bunch of minor ones. And the plates move.

Why do they move?

The tectonic plates move because the Earth is hot.

Let’s start with an analogy. When you boil water, you get an uneven distribution of heat Heat rises, after all, but the source of the heat is at the bottom. So the hot water rises and the cool water sinks. But then the hot water at the surface cools and sinks, and the cool water at the bottom heats up and rises. This gives rise to something called convection currents. this effect isn’t limited to water, though. All liquids do it – our atmosphere, for instance (which functions a lot like a liquid).

The Earth, when you get below the lithosphere, is pretty much a liquid as well. The mesosphere has convection currents in it, and the tectonic plates can be thought of as the “cool water” part of the current in the boiling water analogy. Magma pushes up from the mesosphere into the lithosphere at the Ocean Ridge (a planet-circling chain of mid-ocean ridges), pushing and expanding the plates. The plates then sink back down towards the mesosphere at subduction zones. These currents also push around the solid chunks of the lithosphere, in much the same way that ice cubes floating in boiling water will be pushed and shoved around.

Now, even the “minor” tectonic plates are massive structures. So, when they get moving, there’s a lot of force built up. When they collide, something has to give. And frequently, what gives is the structure of the plate itself – it will buckle and crumple, throwing up mountain ranges and pushing parts of the plate below sea level. If water, in the form of the oceans, gets access to that portion of the plate below sea level, it will begin to fill the depression. That’s what happened in the theorized Zanclean Deluge, for instance. 5.33 million years ago, the Mediterranean was a depression in the Eurasian plate (bordered by the African and Arabian plates) that was below sea level. It had been a sea previously, until shifting plates cut off access to the Atlantic and the waters dried out. Then the plates shifted further, access to the Atlantic reopened, and the basin refilled in a period of approximately 2 years (with water gushing in at a flow rate 1,000 times greater than that of the Amazon River).

So. Plate tectonics is the answer?

Not completely.

Really? What else is there?

There’s changing climates. See, the Earth was – on average – a whole lot warmer back before the continents had moved into the form we’d recognize today. At present, our average global temperature is about 60 degrees Fahrenheit. During the Paleocene-Eocene Thermal Maximum (55-56 million years ago) the average got up to about 73 degrees F – there were no ice caps at the poles then, and there were palm trees and crocodiles above the attic circle.

Now, estimates are that if the ice caps melted then global sea levels would rise about 70 meters. So that’s not really enough to make an ocean out of (say) the Great Plains, although it would completely reshape the coast and drown Houston and New Orleans. But since the plates were buckled differently back then, the extra water would have increased the odds of flooding taking place.

But, ultimately, North America being underwater had far more to do with plate tectonics than changes in climate.

Oh, as a bonus, the Paleomap Project has a series of great maps of the Earth in different geologic epochs. Here’s what the Earth looked like during the age of the dinosaurs:

Yep.  It was a different world, back then.

Do Kangaroos Eat Meat?

I don’t remember the exact circumstances that brought this one up, but out of the blue my son asked the following question: “Do kangaroos eat meat?”

“Uhm…” I temporized, racking my brains. I know a few odd facts about kangaroos – for example, their legs have evolved so that once they start hopping they barely have to use any energy to continue hopping in a straight line at a steady pace, and females have two uteruses – but I suddenly realized that I have [i]no idea[/i] what they eat. “I don’t know,” I admit. “I think they’re herbivores, though.”

“No, they don’t eat meat,” my wife adds. “They do eat bugs, though.”

“Bugs are meat!” my son retorts.

Well played, son. Well played.

What do kangaroos eat?

This looks pretty straight forward. Kangaroos are herbivores. Most of them are “grazing herbivores”, meaning that they eat kind of like cattle – grass, leaves from shrubs and low-hanging trees, fungus if they find it, that sort of thing. They crop the plant matter with their sharp front teeth and grind it with their molars before swallowing. As the molars wear down, new molars grow at the back of the mouth and push forward, shoving out the worn front molars.

Also like cattle, kangaroos have two stomach chambers and chew their cud. The plant matter they swallow is held in the front chamber, where it is fermented by bacteria and fungi until it is regurgitated and chewed and swallowed again. Unlike cattle, though, they don’t fart. Cows fart because their stomach bacteria produces methane as they ferment their cud. Kangaroo stomach bacteria, however, produce hydrogen that is changed into acetate which is used for additional energy.

That seems pretty straight forward

It did, didn’t it? But then I saw this:

Wait. What the heck was that?

That was a kangaroo eating a bird on a beach.

But… but… wait! You said they were herbivores!

Yeah, that’s what I said. And it makes sense to say that, because they’re adapted to a plant-eating diet. And we all learn from an early age that animals fit neatly into one of three categories: herbivores (who eat plants), carnivores (who eat meat), and omnivores (who eat meat and plants). But it turns out that nature is messy and doesn’t really care at all about the neat little categories human beings come up with to explain things.

The animals we call “herbivores” are best thought of as “animals that are adapted for, and primarily consume, a plant-based diet”. Let’s start with the easy proof of that: do you really think that a cow would spit out a worm or a mouse if it picked it up while tucking in to a bale of hay? The answer is, it might. Or it might keep chewing. Cows have to eat a lot, after all.

But it goes beyond simple things like “oops, I ate a mouse”. Deer will actively prey on bird nests, consuming the eggs and even the hatchlings within. They’ll also rummage through animal remains, both to eat partially digested foods in the remains and to eat the meat left behind – going as far as eating human remains.  Deer aren’t the only predatory “herbivores”, either. Cows have been observed eating chickens and giraffes have been observed eating animal bones. Hippos have been observed hunting, killing, and eating other animals.

In a less bloody example of “eating outside your assigned category”, alligators and crocodiles will eat fruit and vegetables.  And anyone who’s been around cats or dogs has seen them eat grass and houseplants.

So, uhm, why?

Sadly, I couldn’t find any specific explanation for this behavior. I assume that it happens for the same reason any animal eats anything at all: either it tastes good, or there’s something in the meat the animal needs, or both. Some nutrients may just be easier to get from animals – calcium, for instance, is easier to obtain in large quantities if you eat a bone. And a chicken has a lot more protein than the same weight of grass.

So, yes. Kangaroos do eat meat. So do a lot of other animals you wouldn’t expect to be eating meat. But they’re not predators, not really. Primarily herbivorous animals just aren’t adapted for a meat-heavy diet, or to hunt down and kill other animals.

Except for hippos.  They’ll probably just kill you for fun.

Who Is Your Favorite President Riding A Dinosaur?

“Dad?” my son asks, looking at me from the other end of the couch, “who’s your favorite president?”

I look up from my book. It’s out of nowhere, and pretty obviously a “dad, pay attention to me” kind of question, and it’s hit me out of left field. “Well, I don’t…” I start to say.

“Only he has to be riding a dinosaur!” my son adds. “And it can be any dinosaur you want, as long as it’s real!”

How on earth do you answer that?

We’re dying to find that out, really.

All right. So, clearly this is a two-part question. “Who is my favorite president?”, and then “what is my favorite dinosaur that someone could read?” How hard could this be?

The internet never fails to deliver

Who is your favorite president?

That… is a really good question. I could easily list my top two or three least favorite presidents, but I’ve never really thought about which one of them is my favorite. I will say I’ve got an unreasonable soft spot in my heart for Ronald Reagan, mostly because the 1978 Carter vs. Reagan election is the first one I really paid any attention to, and I was seven years old (the same age as my son), and my mom was a Reagan Republican and my dad was a Reagan Democrat. So, yeah. There was an influence there, because when you’re seven years old your parents are the smartest and best people in the whole planet.

My political views have shifted since I was seven. Still, it’s hard not to remember him fondly through the lens of childhood memory.

So, it’s Ronald Reagan?

I don’t think so, not really.  Despite that picture up there of him riding some variety of velociraptor.

Then who?

See, that’s tricky. It’s easy to look back in history, and see problems – often huge problems – with any or all of the presidents. They were, after all, human beings with an interest in becoming (arguably) the most powerful single man in the country (and, particularly in the late 20th century, the world). That sort of man tends to have a number of less-than-admirable characteristics. Particularly if you don’t agree with his policies.

But, clearly, I need to make a choice. So, after doing some thinking, I’ll go (this time, at least) with Lyndon B. Johnson because of the things he accomplished as president.

Lyndon B. Johnson?

Lyndon Johnson was the 36th president of the United States, born August 27, 1908 near Johnson City, Texas. He was elected to the House of Representatives in 1937, served in the Navy in World War II, and then was elected to the Senate in 1948 (becoming the Senate minority Leader in 1953, and then the Senate Majority Leader in 1954). He served as John F. Kennedy’s running mate in 1960, and became president on November 22, 1963 after President Kennedy was assassinated. As president, he ran again and was reelected with 61% of the popular vote.

What did he do, to earn the position of my current favorite president? Well, here’s what his bio on Whitehouse.gov says:

First he obtained enactment of the measures President Kennedy had been urging at the time of his death–a new civil rights bill and a tax cut. Next he urged the Nation “to build a great society, a place where the meaning of man’s life matches the marvels of man’s labor.” In 1964, Johnson won the Presidency with 61 percent of the vote and had the widest popular margin in American history–more than 15,000,000 votes.

The Great Society program became Johnson’s agenda for Congress in January 1965: aid to education, attack on disease, Medicare, urban renewal, beautification, conservation, development of depressed regions, a wide-scale fight against poverty, control and prevention of crime and delinquency, removal of obstacles to the right to vote. Congress, at times augmenting or amending, rapidly enacted Johnson’s recommendations. Millions of elderly people found succor through the 1965 Medicare amendment to the Social Security Act.

Johnson also signed the Outer Space Treaty in January 1967, banning the use of nuclear weapons in earth orbit, on the moon, on other planets, and in deep space. He further signed the Nuclear Non-proliferation Treaty in 1968, committing the US to a policy of prohibiting the transfer of nuclear weapons to other nations, and began the negotiations that returned the Panama Canal Zone to Panama.

Pretty awesome, in my opinion, but not perfect. He dragged out the Vietnam War as well, and sent US Marines into the Dominican Republic to support the leader of a coup against the leaders of a coup against the democratically elected president of that country, when the people of that country rose up against the leaders of the coup. Neither of those things were what you’d call good things, but any president you care to name will have similar blemishes on his record – with the possible exception of William Henry Harrison, who didn’t manage to serve long enough to do anything particularly bad. As president, at least.  So, at this time, I’ll say Johnson still did some pretty good things for the country and say he’s my (current) favorite.

Fair enough. Now, what about dinosaurs?

Microraptor!

This is even trickier, because I love dinosaurs. All of them. Fortunately, I can narrow the field a little. The question, after all, states that the president has to ride the dinosaur, so I can rule out something like microraptor. The four wings are pretty cool, but it’s the size of a smallish chicken. You’d need hundreds of the things pulling a chariot, and that loses something quickly.

Actually, I tell a lie. It’s not tricky in the slightest. Because, no matter what, my favorite dinosaur is and always has been the Tyrannosaurus Rex.

Ah. The Tyrant Lizard King.

Oh, yeah.

Tyrannosaurus was always my favorite, from an early age. I mean, sure. I liked Triceratops and Stegosaurus as well, and I love some of the new dinosaurs that have been found over the past few decades, and Deinonychus caught my attention because my very first ever issue of National Geographic had a big article about the fossils of that brand new (to me, at least) dinosaur. But, at the end of the day, the idea of being a massive bipedal carnivore named the Tyrant Lizard King was awesome to my youthful mind and that feeling has never gone away.

Tell us about it.

Tyrannosaurus rex, based on recovered specimens, was a massive bipedal carnivore. Like, 12.3 meters (40 feet) long, 3.66 meters (12 feet) tall at the hips, and anywhere from 8.4 to 14 metric tons in weight. It was one of the largest land predators ever, and most paleontologists agree that it was an active predator that – like modern active predators – wouldn’t turn down the opportunity to chase other animals away from carrion and chow down as well. Their skulls are nearly as long as I am tall (I’m 6′ 5′ and their skulls were 5 feet long), and their teeth were foot long “lethal bananas”.

Well, maybe not every single tooth. But they still had some giant teeth.

They probably had feathers as well.

“I’m an utterly FABULOUS Tyrant Lizard King, baby!”

Yes, yes, I’m aware of the recent paper in Biology Letters (Tyrannosauroid integument reveals conflicting patterns of gigantism and feather evolution), indicating that they had scaly skin.  Early tyrannosauroids did have feathers, but the abstract states that “extensive feather coverings observed in some early tyrannosauroids were lost by the Albian”(the Albian being a stratigraphic layer and a period of time roughly 113 to 100 million years ago).  So, it’s unlikely at this point that T. rex was that fabulously fluffy thing in the image above, but the Smithsonian article on the paper (since the paper itself is paywalled) states that the authors indicate that T. rex still had plumage on it’s back.  Much more punk than New Wave, in other words. Still, I like to imagine them as 12 meter birds of paradise – which tells you everything you need to know about the way my mind works.

Which means..?

In answer to my son’s question, let’s saddle up Lyndon B. Johnson on a fabulous black and electric blue feathered Tyrannosaurus rex and do an elaborate ritual dance to celebrate the passage of the Civil Rights Act.