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):
Pretty big, right? Well, a googol plus one is:
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.
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?
All right. Here goes. Brace yourself.
|Number||Short Count||Long Count||Metric Prefix|
|109||Billion||Thousand Million (or Milliard)||giga-|
|1015||Quadrillion||Thousand Billion (or Billiard)||peta-|
|1021||Sextillion||Thousand Trillion (or Trilliard)||zetta-|
|1027||Octillion||Thousand quadrillion (or Quadrilliard)||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.
But, what about infinity? Isn’t that the largest number?
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.
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.