What am I ever gonna need this?

I'm looking at your screenshot,

and I think the answer is never,

you are never gonna need this.

I'm professor Moon Duchin, mathematician.

Today, I'm here to answer any and all math questions

on Twitter.

This is Math Support.

[upbeat music]

At RecordsFrisson says, What is an algorithm?

Keep hearing this word.

Hmm.

The way you spelled algorithm, like it has rhythm in it.

I like it.

I'm gonna keep it.

A mathematician,

what we mean by algorithm is just any clear set of rules,

a procedure for doing something.

The word comes from 9th century Baghdad

where Al-Khwarizmi, his name became algorithm,

but he also gave us the word that became algebra.

He was just interested in building up the science

of manipulating what we would think of as equations.

Usually, when people say algorithm,

they mean something more computery, right?

So usually, when we have a computer program,

we think of the underlying set of instructions

as an algorithm,

given some inputs it's gonna tell you kind of

how to make a decision.

If an algorithm is just like a precise procedure

for doing something,

then an example is a procedure that's so precise

that a computer can do it.

At llamalord1091 asks,

How the fuck did the Mayans develop the concept of zero?

Everybody's got a zero in the sense that

everybody's got the concept of nothing.

The math concept of zero is kind of the idea

that nothing is a number.

The heart of it is,

how do different cultures incorporate zero as a number?

I don't know much about the Mayan example, particularly,

but you can see different cultures wrestling with.

Is it a number?

What makes it numbery?

Math is decided kind of collectively.

Is that, it is useful to think about it as a number

because you can do arithmetic to it.

So it deserves to be called a number.

At jesspeacock says, How can math be misused or abused?

'Cause the reputation of math is just being like

plain right or wrong and also being really hard,

it gives mathematicians a certain kind of authority,

and you can definitely see that being abused.

And this is true more and more

now that data science is kind of taking over the world.

But the flip side of that,

is that math is being used and used well.

In about five years ago,

I got obsessed with redistricting and gerrymandering

and trying to think about how you could use math models

to better and fairer redistricting.

Ancient, ancient math was being used.

If you just close your eyes and do random redistricting,

you're not gonna get something

that's very good for minorities.

And now that's become much clearer

because of these mathematical models.

And when you know that, you can fix it.

And I think that's an example of math being used

to kind of move the needle in a direction

that's pretty good.

At ChrisExpTheNews.

That is hard to say Analytic Valley Girl.

I honestly have no idea what math research looks like,

and all I'm envisioning is a dude with a mid-Atlantic accent

narrating over footage of guys in labcoats

looking at shapes and like a number four on a whiteboard.

There's this fatal error at the center of your account.

The whiteboard, like no!

Mathematicians are fairly united on this point

of disdaining whiteboards together.

So we really like these beautiful things called chalkboards.

And we especially like this beautiful fetish object,

Japanese chalk.

And then when you write, it's really smooth.

The things that are fun about this,

the colors are really vivid

and also it erases well, which matters.

You just feel that much smarter

when you're using good chalk.

One thing I would say about math research

that probably is a little known, is how collaborative it is.

Typical math papers have multiple authors

and we're just working together all the time.

It's kinda fun to look back at the paper correspondence

of mathematicians from a hundred years ago

who are actually putting all this cool math into letters

and sending them back and forth.

We've done this really good job of packaging math

to teach it,

and so that it looks like it's all done and clean and neat,

but math research is like messy and creative

and original and new,

and you're trying to figure out how things work

and how to put them together in new ways.

It looks nothing like the math in school,

which is sort of a much polished up

after the fact finished product version

of something that's actually like out there

and messy and weird.

So dYLANjOHNkEMP says,

Serious question

that sounds like it's not a serious question

for mathematician, scientists, and engineers.

Do people use imaginary numbers to build real things?

Yes, they do.

You can't do much without them

and particular you equation solving requires these things.

They got called imaginary at some point

because just people didn't know what to do with them.

There were these concepts

that you needed to be able to handle and manipulate,

but people didn't know whether they count as numbers.

No pun intended.

Here's the usual number line that you're comfortable with,

0, 1, 2, and so on.

Real numbers over here.

And then, just give me this number up here and call it i.

That gives me a building block to get anywhere.

So now I come out here, this will be like 3+2i.

So i is now the building block

that can and get me anywhere in space.

Yes, every bridge and every spaceship and all the rest,

like you better hope someone

could handle imaginary numbers well.

At ltclavinny says,

#MovieErrorsThatBugMe The 7th equation down,

on the 3rd chalkboard,

in A Beautiful Mind, was erroneously shown

with two extra variables and an incomplete constant.

Boy, that requires some zooming.

I will say though, for me and lots of mathematicians,

watching the math in movies is a really great sport.

So what's going on here is, I see a bunch of sums.

I see some partial derivatives.

There's a movie about John Nash

who is actually famous for a bunch of things in math world.

One of them is game theory ideas and economics.

But I do not think that's what's on the board here,

if I have to guess.

I think what he is doing is

earlier very important work of his,

this is like Nash embedding theorems, I think.

So this is like fancy geometry.

You can't tell 'cause it looks like

a bunch of sums and squiggles.

You're missing the part of the board that defines the terms.

[chuckles]

So do I agree with J.K. Vinny

that stuff is missing from the bottom row?

I don't think that I do, sorry Vinny.

[chuckles]

At ADHSJagCklub asks, Question... without using numbers,

and without using a search engine,

do you know how to explain what Pi is in words?

You sort of need pi or something like it

to talk about any measurements of circles.

Everything you wanna describe about rounds things

you need pi to make it precise.

Circumference, surface area, area, volume,

anything that relates length to other measurements

on circles needs pi.

Here's a fun one.

So what if you took 4 and you subtracted 4/3,

and then you added back 4/5,

and then you subtracted 4/7, and so on.

So it turns out that if you kept going forever,

this actually equals pi.

I don't teach you this in school.

So this is what's called the power series

and it's pretty much like all the originators of calculus.

We're kind of thinking this way,

about these like infinite sums.

So that's another way to think about pi if you like

are allergic to circles.

At cuzurtheonly1,

Bro, why did math people have to invent infinity?

'Cause it is so convenient.

It completes us.

Could we do math without infinity?

The fact that the numbers go on forever, 1, 2, 3, 4...

It would be pretty hard to do math

without the dot, dot, dots.

In other words, without the idea of things

that go on forever, we kinda need that.

But we maybe didn't have to create like a symbol for it

and create an arithmetic around it

and create like a geometry for it,

where there's like a point at infinity.

That was optional, but it's pretty.

At TheFillWelix, What is the sexiest equation?

I'm gonna show you an identity or a theorem that I love.

I just think is really pretty.

And that I use a lot.

So this is about surfaces and the geometry of surfaces.

It looks like this.

This is called Minsky's product regions theorem.

So this is the, a kind of almost equality

that we really like in my kind of math.

The picture that goes along with this theorem

looks something like this, you have a surface,

you have some curves.

This is called a genus 2 surface.

It's like a double inner tube.

It's sort of like two hollow donuts

kind of surgered together in the middle.

And so this is telling you what happens

when you take some curves,

like the ones that I've colored here

and you squeeze them really thin.

So it's the thin part for a set of curves.

And it's telling you that...

This looks just like what would happen

if you like pinched them all the way off

and cut open the surface there,

you'd get something simpler and a leftover part

that is well understood.

At avsa says, What if blockchain is just a plot

by math majors to convince governments, VC funds

and billionaires to give money to low level math research?

No.

And here's how I know.

We're really bad at telling the world what we're doing

and incidentally getting money for it.

Most people could tell you something

about new physics ideas, new chemistry,

new biology ideas from say, the 20th century.

And most people probably think

there aren't new things in math, right?

There are breakthroughs in math all the time.

One of the breakthrough ideas from the 20th century

is turns out there aren't three basic

three dimensional geometries.

There are eight.

Flat like a piece of paper, round like a sphere.

And then the third one looks like a Pringle.

It's this hyperbolic geometry or like saddle shape.

Another one is actually instead of a single Pringle,

you pass to a stack of Pringles.

So like this.

So we call this H2 x R.

Put these all together

and you get a three dimensional geometry.

And then the last three are Nil, this guy over here,

Sol, which is a little bit like Nil,

but it's hard to explain.

And then the last one, which I kid you not,

is called SL2[R] twiddle.

Really? That's what it's called.

Finally, it was proved to the community satisfaction

what is now called the geometrization theorem.

The idea of how you can build stuff

out of those eight kinds of worlds.

It's just one example of the publicity mathematicians

are failing to generate.

Did we invent blockchain to like get money for ourselves?

No, we did not.

At ryleealanza, Is geometric group theory

just anabelian topology?

And then there's this like my absolute favorite part of this

is the laughing, crying emoji

because Rylee is just like cracking herself up here.

Or Rylee's, I think, really saying here

has to do with just like, how much things commute, right?

So you're used to ab equals ba, that's when things commute.

And then you can sort of do math

where that's not true anymore,

where like,

ab equals ba times a new thing called c.

That's just not the math you learned in school.

Like, what is this new thing?

And how do you understand it?

Well, it turns out, this is the math of this model here.

[chuckles]

This is a model of what's called Nil or nilpotent geometry.

It's pretty cool, as I rotate it,

you can probably see that there's some complexity here

from some angles that looks one way,

from some angles you see different kinds of structure.

This is my favorite.

I love to think about this one.

a and b are kind of moving horizontally

and c is kind of moving up in this model.

So that really shows you something

about what Rylee's calling geometric group theory.

You start with just like the group theory

of how to multiply things and it builds geometry for you.

[Man] But is it hilarious?

No.

[laughs]

It's sort of stringing a bunch of words together

and trying to make meaning out of them.

And I think that's the joke here.

And like all jokes, when you try to explain it,

it sounds desperately unfunny.

At RuthTownsendlaw, Question for mathematicians,

Why do we solve maths problems

in a particular order of operations?

Eg, why multiplication first?

This is like asking in a chess game,

how come bishops move diagonally?

It's because over time those rules were developed

and they produced a pretty good game.

I could make about a chess game

where the bishops moved differently,

but then it would be my burden to show

that it's a good game.

We could do arithmetic differently.

And we do in math all the time,

we set up other number systems with other arithmetic.

You just have to show

that they have some internal consistency

that you can build a good theory around them.

And maybe that they're useful for modeling things

in the world, and then you're in business.

At hey_arenee, How is math supposed to be universal

when all our teachers in the same state teach different?

The thing about math being universal,

there might be like 10 different ways to do long division

and get the answer right.

We're trying to stabilize math around the world.

We're trying to take

lots of different mathematical practices

and turn them into something where we have enough consensus

that we can communicate.

At shamshandwich says, Music is just math that [beep].

I'm not quite sure what you mean by that.

But there is a lot of math in music.

If you think about constructing notes

that are gonna sound good,

to a mathematician,

you're just doing rational approximations to algorithms,

transcendental numbers again like pi,

numbers that can't be made into exact fractions,

but can only be approximate in order to decide

on the distances between keys on a keyboard.

In order to make it sound good,

we're trying to approximate something

that is a number that can't be exactly captured

with fractions.

There's a lot to say about the math that's in music.

As to the rest of your proposition,

I will just trust you on that.

At tuktukou.

Tuktukou, tuktukou?

How does math make sense?

Lots of punctuation.

Why put a fraction on top of another fraction?

When am I ever gonna need this?

That is like the thing that math people do,

like 6 divided by 2.

And that's a very basic thing we like to be able to do.

And so then math people come along and say,

Well, what if I put in different kinds of numbers?

What is 6 over minus 2?

But that's what mathematicians do,

we take a system and we just try to put in

other kinds of inputs that it wasn't expecting.

You teach me how to add,

and then I come along and I wanna add shapes.

And you're like, You don't add shapes.

You add numbers.

And I'm like, But why?

We're gonna do it every time.

We can't be stopped.

And when am I ever gonna need this?

Looking at your screenshot, and I think the answer is never,

you are never gonna need this.

At neilvaughan1st, A question for mathematicians...

Is zero an odd or even number?

Even number is any number that can be written

as 2 times K, where K is a whole number.

Zero is even if zero is a whole number.

Zero a whole number and you get down a rabbit hole.

Zero is even 'cause it's convenient for some things.

It is definitely different from the rest of the numbers.

You're not wrong about that.

At deftsulol asks,

Who is the greatest mathematician in history?

Does anybody know... and if so, explain why?

There are all kinds of incredibly interesting people

that are not well enough known.

So I'm just gonna tell you about a few of my favorites.

Felix Hausdorff, he is awesome.

He basically built the math behind fractals

and did all kinds of other creative stuff.

And nobody's ever heard of him outside of math.

Emmy Noether, you cannot go wrong with Emmy Noether.

She's so interesting.

She's a great mathematician,

and had a kind of a cult following.

Her math is great.

Her ideas are deep.

She was very powerful builder of abstraction.

And I think you can't go wrong learning about Emmy Noether.

Math is full of these really colorful characters

having like out of control, original great ideas.

It'd be great if we figured out

how to tell their stories a little better.

At jhach17 says, I have a question for math people.

If there are infinite amount on a points

between any two points,

but we can still walk from point A to point B.

Do we walk through infinite points to get there?

How do we get anywhere?

This is an old and deep question.

The idea that math is math is math

and that it's universal and that it's all the same

and that it's all figured out,

hides a lot of mess and this is a good example.

The theories that let you do that,

that let you describe how points combine to make a line,

we're actually controversial

and took hundreds and hundreds of years

to kind of work out to people's satisfaction.

The best way to explain

how math has built structure to answer this question

is calculus.

It's about the difference between durations and instance.

It's the difference between lines and points.

Calculus and what comes after it measure theory.

Those are the ways that mathematicians have built

to answer questions like this.

At alejandra_turtl says,

I have a question for mathematicians.

Why letters? In an equation.

It's kind of hell.

This is one of those great examples

where it didn't have to be this way,

but some people made some decisions

and they caught on and they traveled around the world

and people were like,

Well, it'd be kind of nice if we all did it the same way.

And so letters caught on.

This is very arbitrary.

It's just a convention,

and we kind of all agreed that we'd do it the same way.

Those are all the questions for today.

So thank you to Math Twitter.

And thanks for watching Math Support.