Name: The Map of Mathematics
Uploaded: 2022-07-02T13:07:08.000Z
Duration: 11 min 6 s
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The mathematics we learn in school doesn't quite do the field of mathematics justice.

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We only get a glimpse at one corner of it, but the mathematics as a whole is a huge and

My aim with this video is to show you all that amazing stuff.

The origin of mathematics lies in counting.

In fact counting is not just a human trait, other animals are able to count as well and

e vidence for human counting goes back to prehistoric times with check marks made in

There were several innovations over the years with the Egyptians having the first equation,

the ancient Greeks made strides in many areas like geometry and numerology, and negative

And zero as a number was first used in India.

Then in the Golden Age of Islam Persian mathematicians made further strides and the first book on

Then mathematics boomed in the renaissance along with the sciences.

Now there is a lot more to the history of mathematics then what I have just said, but

I'm gonna jump to the modern age and mathematics as we know it now.

Modern mathematics can be broadly be broken down into two areas, pure maths: the study

of mathematics for its own sake, and applied maths: when you develop mathematics to help

In fact, many times in history someone's gone off into the mathematical wilderness

motivated purely by curiosity and kind of guided by a sense of aesthetics.

And then they have created a whole bunch of new mathematics which was nice and interesting

But then, say a hundred hears later, someone will be working on some problem at the cutting

edge of physics or computer science and they'll discover that this old theory in pure maths

is exactly what they need to solve their real world problems!

And this kind of thing has happened so many times over the last few centuries.

It is interesting how often something so abstract ends up being really useful.

But I should also mention, pure mathematics on its own is still a very valuable thing

to do because it can be fascinating and on its own can have a real beauty and elegance

Okay enough of this highfalutin, lets get into it.

The study of numbers starts with the natural numbers and what you can do with them with

And then it looks at other kinds of numbers like integers, which contain negative numbers,

rational numbers like fractions, real numbers which include numbers like pi which go off

to infinite decimal points, and then complex numbers and a whole bunch of others.

Some numbers have interesting properties like Prime Numbers, or pi or the exponential.

There are also properties of these number systems, for example, even though there is

an infinite amount of both integers and real numbers, there are more real numbers than

So some infinities are bigger than others.

The study of structures is where you start taking numbers and putting them into equations

Algebra contains the rules of how you then manipulate these equations.

Here you will also find vectors and matrices which are multi-dimensional numbers, and the

rules of how they relate to each other are captured in linear algebra.

Number theory studies the features of everything in the last section on numbers like the properties

Combinatorics looks at the properties of certain structures like trees, graphs, and other things

that are made of discreet chunks that you can count.

Group theory looks at objects that are related to each other in, well, groups.

A familiar example is a Rubik's cube which is an example of a permutation group.

And order theory investigates how to arrange objects following certain rules like, how

something is a larger quantity than something else.

The natural numbers are an example of an ordered set of objects, but anything with any two

Another part of pure mathematics looks at shapes and how they behave in spaces.

The origin is in geometry which includes Pythagoras, and is close to trigonometry, which we are

Also there are fun things like fractal geometry which are mathematical patterns which are

scale invariant, which means you can zoom into them forever and the always look kind

Topology looks at different properties of spaces where you are allowed to continuously

For example a Möbius strip has only one surface and one edge whatever you do to it.

And coffee cups and donuts are the same thing - topologically speaking.

Measure theory is a way to assign values to spaces or sets tying together numbers and

And finally, differential geometry looks the properties of shapes on curved surfaces, for

example triangles have got different angles on a curved surface, and brings us to the

The study of changes contains calculus which involves integrals and differentials which

looks at area spanned out by functions or the behaviour of gradients of functions.

And vector calculus looks at the same things for vectors.

Here we also find a bunch of other areas like dynamical systems which looks at systems that

evolve in time from one state to another, like fluid flows or things with feedback loops

And chaos theory which studies dynamical systems that are very sensitive to initial conditions.

Finally complex analysis looks at the properties of functions with complex numbers.

At this point it is worth mentioning that everything here is a lot more interrelated

In reality this map should look like more of a web tying together all the different

subjects but you can only do so much on a two dimensional plane so I have laid them

Okay we'll start with physics, which uses just about everything on the left hand side

Mathematical and theoretical physics has a very close relationship with pure maths.

Mathematics is also used in the other natural sciences with mathematical chemistry and biomathematics

which look at loads of stuff from modelling molecules to evolutionary biology.

Mathematics is also used extensively in engineering, building things has taken a lot of maths since

Very complex electrical systems like aeroplanes or the power grid use methods in dynamical

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The Map of Mathematics

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