## Five Numbers That Got Munched (And One That Can't Be Proven)

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“Okay, so we need to practice. Let's munch the numbers that are not fractions!”

“Great! Like, one-half?”

“That's a fraction.”

“But it's so tasty!”

“Try to restrain yourself.”

“Two-thirds? Can I munch two-thirds?”

“That's also a fraction.”

“Negative one-fourth?”

“Still a fraction.”

“But it's got a negative sign in front of it, that doesn't count as eating at all!”

“Yes, it really does. It's like—when you eat both pasta and antipasto, they don't cancel each other out.”

“But I'm so hungry. Four? Can I eat a four?”

“No, that's still...still a rational number.”

“It's not a fraction, it hasn't got a denominator in it!”

“But you can rewrite it as four over one...those are equal...”

“2.71? I want to munch a 2.71. Please let me?”

“That's the same as 271/100. Which is a fraction.”

“All the numbers are fractions, then! I'm going to starve!”

“No, try over here.”

“The square root...of two? Is that a number?”

“Sure! If you have a square, like the one you're standing on, yeah, and its sides are one unit long, how long is the diagonal?”

“Like 1.4 something? It's not a fraction?”

“No! See, if it was, you could write it in lowest terms, right? And factor the numerator and denominator.”

“Just like any other fraction!”

“But if you look at this part, see, there would really need to be a common factor in both of them. Because the square is two.”

“Oh. Yes...”

“So we can really cancel this factor, when it appears on both the top and the bottom!”

“We eat it! Eat both copies of it, so it goes away! Yum!”

“Well, that is the general idea, yes, although we don't usually express it in quite those—”

“I love prime factors! So tasty!”

“But now you see the fraction couldn't have been in lowest terms to start with. But that's absurd, so this isn't a fraction at all.”

“Which means I can eat it?”

“Well, yes.”

“Phew, thank goodness!”

“You don't care about the mathematical insight behind this? That there are uncountably more numbers than the rationals you know and love?”

“Oh, I suppose it's a great big discovery, sure. We should sacrifice a cow or something to celebrate this day.”

“Well, I don't really know about that—”

“Yeah, you're right, that was a silly idea.”

“Okay.”

We should sacrifice a hundred cows!”

“Um...”

“And then we can eat them! All of them!”

“Maybe not.”

“Not if the Troggles get to our cows first. Let's go beat them to the punch!”

“You don't want to munch on any tasty transcendental numbers before you go?...No?...Okay then.”

“Munch the powers of two!”

“Here's a two...here's a four...here's a sixteen...ooh, the Troggles are getting that eight, I have to wait for them to move...but tell me, when are we ever going to use this?”

“No! What happened?”

“Well, he'd designed a little game on a grid sort of like this, divided into many squares.”

“Where Munchers fought Troggles?”

“Well, technically it was two teams fighting each other, composed of the same sorts of creatures, but when they ran into their enemies—”

“They'd munch them?”

“Take them off the board until the next time, but, I suppose, that's the general idea...:

“Sounds like a pretty boring game, then.”

“Well, the Muncher that Sissa showed it to didn't think so—he was a High Muckety-Muck among Munchers, and he decided that Sissa should be justly rewarded for the discovery.”

“Just desserts! I should think so!”

“Sissa asked just for one grain of rice for the first square, two for the next, and twice as many for each one after that. So, there would be four, then eight—”

“Then sixteen and then thirty-two, all those other powers?”

“Yes, exactly! And so on, through all sixty-four squares.”

“Oh, dear.”

“Yes, you see the problem.”

“Was it brown or white rice?”

“Come again?”

“Short or long-grained? Wild? Did he eat it with chicken or beans?”

“That's not the point of the story—”

“It most certainly is! What's the point of being rewarded with rice if you can't munch it?”

“He never got all that rice, don't you understand? There isn't nearly enough to keep up with exponential growth, two to the sixty-third is an unfathomably large number.”

“So did the Troggles eat him? Were they jealous?”

“No...they never got halfway there. Even two to the thirty-first, you can't have that much rice in one place.”

“You're right, I couldn't!”

“I'm glad to see you understand.”

“I'd munch it as soon as I got it. Then there would be less and less, it wouldn't stay in one place too long.”

“That's...not...”

“Muncher Sissa was being falsely modest, I get it. He should have shared with me, we'd have munched that rice in no time!”

“Okay, so can you munch the powers of two?”

“That...that's not even a multiple of two, don't you see it's an odd number?”

“Oh. Well, what if I added one?”

“Then...yes, that's 2^63, very good.”

“I'm not greedy. I'd give you one grain of rice if it meant I could munch the other quintillions!”

“Aren't you kind.”

“You look like you could use something tasty! Really, it's the least I could do.”

“Munch a fair number of coins to pay for your meal with!”

“I'd munch any of these numbers! And I guess I'd pay any of them...”

“Well, you can't have a fractional number of coins. And you shouldn't munch negative numbers.”

“I'd love to munch food and pay negative money for it! People would pay me for the joy of munching things, that sounds perfect!”

“But is it fair?”

“It's fair to me!”

“Have you ever heard about Muncher Ali's views of fairness?”

“No...what does he say?”

“Muncher Ali once saw two hungry Munchers journeying across the board. They'd both brought some loaves of bread to share along the way, but Muncher Ali didn't have any food of his own. So they invited him to eat with them.”

“And give away their own bread?”

“We're not Troggles, we share our food together.”

“But sharing is hard.”

“Well, Muncher Ali promised to pay them back.”

“Oh, so it was fair then!”

“That makes eight?”

“Yes, you have learned to add, I guess those drills weren't worthless. Anyway, Muncher Ali joined them, and they all shared the bread together.”

“And then what happened, Troggles came and devoured them?”

“No, actually, they finished their meal in peace. Muncher Ali gave the other two eight coins to split between them, to pay for his share.”

“You can't eat coins. Not really a fair trade.”

“Be that as it may, the first Muncher thought she should get five coins, and the other should get three, because she'd brought five out of the eight loaves of bread. The second Muncher thought they should each get four coins, because that would be fair.”

“What else did they munch? Fruit? Meat? Drinks?”

“Well, for the purposes of this story, just bread.”

“So the second Muncher appealed to Muncher Ali, to make a fair division, and he said he'd come up with a fair solution, but she wouldn't like it.”

“I wouldn't like it either, nothing lives up to a good meal.”

“Do you understand why that is? Think about how much of their portion they contributed to Muncher Ali.”

“Where were the Troggles?”

“They each ate two and two-thirds loaves, which means that the first Muncher gave away seven times as much bread...”

“Yeah, this story sounds really unfair. They couldn't find anything else to munch on, not even tasty numbers?”

“Are you going to find the right answer?”

“Zero? Zero is the right answer! All the munching should be free!”

“Never mind.”

“Munch a gravitational constant!”

“Aw, gee...”

“Exactly!”

“Why do I care about gravity?”

“It keeps you from falling off the edge of the board.”

“Off the edge? There's no way for us to get off the board, that's where the Troggles come from. What are you talking about?”

“Er...”

“The world is flat.”

“Never mind.”

“Okay, so why am I looking for the gravitational constant? Negative 9.8, is that it?”

“I don't care, I'm munching it anyway, I'm hungry.”

“Did you hear how Muncher Isaac learned about gravity?”

“Were there numbers to munch below the grid? Did he go exploring?”

“Not exactly. The story goes, he was hit by an apple that fell to the earth...”

“And then he munched it, right?”

“And then he realized that there's a common force that attracts objects together, whether it be a small item falling to earth, or the moon staying in orbit around the planet.”

“But the effect of those forces isn't the same!”

“No, because there's one constant that measures the effect of gravity everywhere in space, and then it's proportional to the masses of the objects. So if we're just looking at the motion of objects falling to Earth, we can combine those terms into another special constant just for our world.”

“But by our world you mean the flat grid, right?”

“Well, sort of. Close enough.”

“It even works for things like the Earth and the moon?”

“Yes!”

“But I'm not attracted to the moon. I can't munch it!”

“Your mass is relatively insignificant compared to the mass of the world...”

“So I should eat more, to get more massive?”

“So the force is really measured between the Earth and the moon as a whole.”

“The whole Earth is attracted to the moon?”

“Yes.”

“No it's not.”

“Yes it is! That's why it's so attractive. We need to get there so we can munch it.”

“I encourage learning science so we can support space travel, but maybe not for this reason.”

“When you say proportional, that means, if the mass of one object goes up, so does the force?”

“Right!”

“Of course! That's why, when I'm attracted to one number to munch, if you give me two numbers instead I'll be twice as attracted! Because they're all so tasty.”

“Sort of...”

“And if I'm twice as close, I can smell it in the air, so I'm, like, four times as attracted!”

“That doesn't...actually, it does, okay, you learned something. Well done.”

“Munch the angles that equal each other!”

“Ooh, this is a cute angle!”

“It's not acute, it's obtuse.”

“I know. But it looks so cute and it's going to taste even better.”

“If you say so.”

“What are these angles? They're not degrees.”

“No, we sometimes use radians to measure angles.”

“But degrees are so useful! I love heating up my oven to many degrees, so I can bake tasty things.”

“Did you not learn what radians are?”

“No, are they tasty?”

“Well, we use them in calculus. Think about...okay, think about cutting a slice of pizza.”

“What kind of pizza?”

“Any kind.”

“Can it be pepperoni? I like pepperoni.”

“Sure, that's fine. Okay, so you slice from the center out to the edge of the pizza, you've just made a cut equal to the length of the radius.”

“I hope I didn't cut through any pepperoni, I don't like breaking them apart.”

“Now, you're going to make another cut, again from the center out to another point on the edge. This'll give you your slice.”

“Can't I cut across and make it a grid? Some of the other Munchers like to cut their pizza this way.”

“We're cutting it in slices, this time.”

“Aw, okay. If you say so.”

“Now, try to imagine—this'll be a little bit of a stretch—but that you like to be picky about your food.”

“But I'll munch any kind of pizza!”

“I know, I know. But just say, obviously the two straight edges are going to be the same length—the radius of the circle. Then you'll have a piece with two edges, but it's not a triangle—the third side of the pizza is—”

“The crust!”

“Yes, the crust, but it's curved because it comes from the original circle of the pizza. It's still an arc.”

“Okay.”

“Let's just imagine you wanted the crust to be the same length as the other two sides. When you measure along the circle...not as the crow flies, but curved.”

“Well, if it really was a triangle...an equilateral triangle, where all the sides are the same length, has angles of sixty degrees each.”

“Right.”

“But this pizza...since the crusty edge is curved, it'll be longer than that. So I'd really need to cut it at an angle a little bit less than sixty degrees, so things even out, and the crust is just as long as the other two.”

“Exactly! Just a little bit shorter...more like fifty-seven degrees and a little bit.”

“Okay. Since sixty degrees would be...one-sixth of the whole circle, this will be just less than a sixth?”

“Right, and we call that angle a radian! Because now the crust is just as long as the radius.”

“Okay, but if I munch that slice, can I have another?”

“Yes, I suppose...”

“And a third? I really like pizza.”

“If you insist, but you understand this is just a thought experiment...”

“And then if I ate three of those, there'd still be a little more to go before I was halfway through the pizza. I'll save the second half for the next day.”

“Probably a good idea.”

“So there's three radians and a little bit, for the angle making up half a circle?”

“Right! Half the distance around the circle is a little more than 3.14 times the length of the radius. We call that number pi.”

“Pizza pi?”

“In this case, yes! Maybe you've heard of it from the area of a circle—pi r squared.”

“No, no. Pie is round. Brownies are square.”

“Munch all the numbers that you're not going to munch!”

“...What? That doesn't make sense!”

“It's a little subtle, isn't it!”

“Those numbers are huge. Is that like powers of two?”

“And then some. See, you can factor each number uniquely, right? So if you had the time and energy, you could see how many twos it's a multiple of, how many threes, how many fives, same for every prime.”

“Sure.”

“Then you look at the list of those numbers you came up with, and translate them back into symbols that match. It's a code—we have a number that means 'plus,' one that means 'times,' a couple parentheses for keeping track...so it secretly is a code for a sentence.”

“What kind of sentence?”

“Something like 'I'm not going to munch a number that looks like this.'”

“You can talk about munching, using math?”

“I can't, because I have taste, but for you, the set of 'numbers you want to munch' looks basically like 'the set of all the numbers', so it's not too hard to represent that.”

“I should say!”

“But then pulling some tricks, and with some more complicated codes, we can have a number that says 'You will munch this number if, and only if, we can't prove that you'll munch this number.'”

“That's ridiculous!”

“It's pretty wild, yeah.”

“Here, I'm gonna eat the big number right now, just watch me!”

“I'm not sure that's...”

“I ate it! I most assuredly ate it! Is that enough proof for you, or do you need more evidence? Because if you can wait a couple hours...”

“Well, I don't think you're aware of how consistent you are, so it's not a contradiction...”

“Am I not consistent? I'll consistently munch all the numbers, if you let me.”

“That's good, but—”

“Or did you poison them?”

“No!”

“What if the Troggles poisoned them? They wouldn't need to eat us if we just dropped dead from poisoned numbers! Oh no, this is horrible, I'd better stop.”

“You're okay. Here, come with me, you'll be all right.”

“You're sure they're not poisoned?”

“I'm pretty sure.”

“But can you prove it?”

“Just experimentally. Here, come split a nice abundant number with me!”

“Like a dozen!”

“Sure.”

“A baker's dozen?”

“Close enough.”

“If the numbers were poisoned we could feed them to the Troggles?”

“I think you need to get back to the safe zone for a while and relax.”

“Only if I can munch the number there, too.”

“It's a deal!”