Chapter 2

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Grobner Basis & Little Sister -

part 2:

"...what do you mean by

'number', exactly?"

"Well, the truth is, any polynomial that gives a

unique result when input to another polynomial can be considered to..."

As I said that, I suddenly felt like I was a pigeon and my sister's face

was a peashooter that hit me.

[My sister suddenly came in my room and

wanted to know about the Grobner basis! This isn't the sort of thing that

happens in real life, right?!]

"One, um, one more time please!"

[I can appreciate that attitude, but even if I repeat the same thing

it'll probably still be gibberish to her.]

"Why do you want to know

about this?"

"Um, that's..."

Her eyes looked watery,

reflecting light from the window.

"...for now, don't ask, please."

Seeing her like that, I couldn't bring myself to push her, and my

counterattack deflated.

"It can't be helped then."

At that,

Kanna was suddenly rejuvenated.

[Now what do I do? Do I start with

monomial order,

polynomial

division,

ideals,

Hilbert's basis theorem...?]

[...no, my sister is a sophomore in

high school; even multivariate polynomials are questionable. If I remember

right, she's not particularly good at math, either.]

I sat there

thinking, while my sister stared at me like a puppy waiting for food.

Eventually, I decided to start with 'computers' and 'numbers'.

"What do you know about numbers?"

My sister was now sitting on a

chair taken from her room next door. I could smell a faint citris scent from

her.

"Numbers? You mean, what you count with?"

"Right."

"1, 2, 3, etc?"

"Right, right."

I considered rubbing her

head and saying "good job" but managed to stop myself. Instead, I wrote in a

notebook on my desk:

<1, 2, 3 ...>

"Do you know what this type of

number is called?"

"Umm, integers?"

"Right. More specifically,

positive integers, also called natural numbers. Well, there's a religious

war about whether 0 is a natural number, so let's ignore 0 for now."

[Hashtag

#0IsANaturalNumber.]

"So, what can you do with natural numbers?"

"Not much, you can't even make a line..."

[What? She says the

weirdest things sometimes.]

"What's 1 + 1, then?"

"Eh?"

"Adding 1 to 1 gives..."

"...2?"

"Correct! In other words,

with natural numbers we can do addition."

"Um, isn't that obvious?"

"That's true."

"I'm not in elementary school, you know..."

Kanna started pouting. Seeing that, any older brother would want to

bully her a little bit.

"Then, 215389 + 574194 is...?"

"What?"

"215389 + 574194."

"Um..."

"The answer is 790583."

"...did you do that on an abacus?"

"No, I used this."

I pulled

out a calculator from a drawer.

"When you went to get a chair, I

wrote down the result here."

I showed her the small numbers in a

corner of my notebook.

"You cheater!"

[She looks pretty mad, I

should probably stop teasing her.]

"Sorry about that...anyway, if you

use a calculator, you can solve difficult math problems."

"Right."

"Then, with natural numbers, can we subtract?"

"Like, 2-1 is 1,

7-3 is 4, you can, right?"

"1-2?"

"Ah."

"Right,

negative 1 isn't a natural number."

"So it's out of bounds."

"On the other hand, if you include 0 and negative numbers, then you can do

subtraction."

"I see."

To the left of the earlier numbers in

my notebook, I add:

<...-2, -1, 0,>

"And so, ahem, the addition of

negative numbers expands the natural numbers to the integers. But anyway,

the important thing here is that the calculator can handle negative

numbers."

I typed in an example on the calculator:

< 139583 −

4487205 = −4347619 >

"We can do complicated calculations on this

little computer."

"That's a computer?"

"Well, we call this a

calculator, but the way it works is the same as smartphones and laptop

computers."

"Huh..."

"But of course, there's a limit to the

size of numbers this calculator can handle. Like an abacus, it's a physical

object with limited size. This calculator is a bridge between the real world

and an abstract world of mathematics."

"A bridge, huh..."

"Do

you understand so far?"

I watched her as she replied, worried I'd

lost her.

[Yeah, I got it. Basically, you can do math calculations on

a computer! Right?]

"Well, that's right."

I figured that was

good enough, and continued.

"The world of integers is useful, but I'd

like to expand it even more."

"More?"

"Right. Last time, we

went from natural numbers to integers with subtraction. What can we do now?"

"Ah, multiplication?"

"What happens if you multiply 2 integers?"

"...ah, it doesn't really change anything."

"Right, you get

another integer."

"Then, division?"

"Right!"

I added a

line in the notebook:

< −2, −1, −2/3, 0, 1, 2, 5/2, 8/3 >

"What

do we call these numbers that include fractions?"

"Rational {yuuri}

numbers!"

"You knew that, huh?"

"But that name makes me think

of those {yuri} novels with a lot of girl characters..."

"Ehh, I

don't know about those."

"Eh, you don't? Even though you're studying

math?!"

{TL note: I could have turned this into a joke about

"rational" fiction, but the little sister here is biologically female.}

"Ahm, anyway, now let's try to calculate a rational number using this

calculator."

I handed Kanna the calculator, then wrote in the

notebook:

< 71/131 + 37/297 >

"Try calculating that with the

calculator."

"Ah, hold on a sec."

Kanna reached in her skirt

pocket and pulled out her smartphone.

"Um...is this OK?"

She

showed me the default calculator app.

"That's fine, but why?"

"That looks hard to use."

While Kanna entered the numbers on her

phone, I used the calculator.

[To me, smartphones seem harder to use

than calculators; I guess that's the generational difference.]

"Umm,

0.66656385?"

"That's about right."

"What do you mean? Didn't

you calculate it?"

"Actually, I calculated it more accurately than

that."

"What?"

"There aren't enough digits on your phone or my

calculator for the full answer."

Kanna thought about that for a

little while.

"So 0.66656385 keeps going. How many digits do you

need, then?"

"It goes on forever."

"But the calculator has

limited space."

"Right, but it's close enough. However, I'll show you

now how to get the exact answer using a computer. So, to solve the earlier

problem..."

"Wait, I can do this."

Kanna took the pen from me and

started writing in my notebook. I pretended not to notice how close she was.

"First, multiply the denominators..."

She wrote < 131 × 297 =

38907 > in the notebook, while operating her smartphone with her left hand.

"Then, we take 71 × 297 and 37 × 131..."

< 71 × 297 = 21087 >

< 37 × 131 = 4847 >

"...and we add those..."

< 21087 + 4847 =

25934 >

< 25934/38907 >

She smiles happily at me.

"Got it!"

"By the way, if you enter 25934/38907 in the calculator, you'll get the

same 0.66656385."

"So the fraction is more mathematically accurate?"

"Well, yes."

"So, can you think of a set of numbers that's bigger

than the rational numbers?"

"Eh? Does that exist?"

"Yes,

that's the 'real numbers'."

"So, are you going to do some

calculations with real numbers?"

"..."

I thought for a moment,

and noticed that the sun had set.

"Calculating with real numbers

is...something humanity is not yet ready for."