# Mathematics

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.

Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of the theory under consideration.

Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications. The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks.

Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements. Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new areas. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than 60 first-level areas of mathematics.

Source: Mathematics (wikipedia.org)

#### Collatz Conjecture in Color - Numberphile

The Great Courses Plus (free trial): http://ow.ly/RqOr309wT7v This video features Alex Bellos. More info and links in full description. Extra footage with Alex and coloring: https://youtu.be/w8nc8wbgXPU Or real-time video of the coloring: https://youtu.be/wH141HLD57o Our previous Collatz Conjectur

#### UNCRACKABLE? The Collatz Conjecture - Numberphile

Catch David on the Numberphile podcast: https://youtu.be/9y1BGvnTyQA Professor David Eisenbud on the infamous Collatz Conjecture, a simple problem that mathematicians may not be "ready" to crack. More links & stuff in full description below ↓↓↓ Extra footage from this interview: https://youtu

#### The unexpected maths problem at work during the women's World Cup

There was something strange about the recent Women's World Cup in Australia. If you were paying close attention, you might have spotted it. Many of the international teams had players who were born on the same day of the year – they shared birthdays. What was going on?

#### Benford's law

Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.

#### Roger Penrose explains Godel's incompleteness theorem in 3 minutes

good explanation from his interview with joe rogan https://www.youtube.com/watch?v=GEw0ePZUMHA

#### Elliptical Pool Table - Numberphile

A game to play on the elliptical table: http://youtu.be/3WHBlPvK3Ek More links & stuff in full description below ↓↓↓ And more extra footage: http://youtu.be/pulp55gTKGE Alex Bellos' Loop Table website: http://www.loop-the-game.com Alex discusses the topic in his book Alex Through the Looking

#### The numbers that are too big to imagine

What's the biggest number you can think of? When I was a child, it's the kind of question we'd ask each other in the school playground.

#### Explaining the SECRET of Penrose Patterns

The first 200 people to https://brilliant.org/minutephysics get 20% off an annual premium subscription to Brilliant. Thanks to Brilliant for their support. This video is about a better way to understand Penrose tilings (the famous tilings invented by Roger Penrose that never repeat themselves but s

#### Why Is 1/137 One of the Greatest Unsolved Problems In Physics?

Thank you to Squarespace for supporting PBS. Go to ​https://www.squarespace.com/pbs for a free trial, and when you are ready to launch, go to Squarespace.com/PBS to save 10% off your first purchase of a website or domain. PBS Member Stations rely on viewers like you. To support your local station

#### The Reciprocals of Primes - Numberphile

Matt Parker explores the work of William Shanks - and boots up the ShanksBot. More links & stuff in full description below ↓↓↓ Matt Parker's 2022 Pi Day Video: https://youtu.be/dtiLxLrzjOQ Discussing William Shanks on Objectivity: https://youtu.be/7yTXMeiVBCc Prime Number playlist: https://b

#### Why π^π^π^π could be an integer (for all we know!).

Check out the Jane Street programs if you're considering a mathematics/finance/programming job: https://www.janestreet.com/join-jane-street/our-programs/ Here is Tim Gowers's reply to the original tweet: https://twitter.com/wtgowers/status/1346212151581700096 Start your Schanuel's Conjecture journ

#### Euler's Formula - Numberphile

Tom Crawford shows us some cool things about Euler's Formula... Check https://brilliant.org/numberphile for Brilliant and get 20% off their premium service (episode sponsor) More links & stuff in full description below ↓↓↓ Tom Crawford's website, with links to his work and other outreach: htt

#### Twin Proofs for Twin Primes - Numberphile

With Ben Sparks... Check https://brilliant.org/numberphile for Brilliant and get 20% off their premium service (episode sponsor) More links & stuff in full description below ↓↓↓ Twin primes with James Maynard: https://youtu.be/QKHKD8bRAro Ben Sparks on the Numberphile Podcast: https://youtu.

#### Goldbach Conjecture - Numberphile

Professor David Eisenbud on the famed Goldbach Conjecture. More links & stuff in full description below ↓↓↓ Catch David on the Numberphile podcast: https://youtu.be/9y1BGvnTyQA Extra footage from this interview: https://youtu.be/7D-YKPMWULA Prime Playlist: http://bit.ly/primevids Prime Num

#### Big Factorials - Numberphile

Large factorials and the use of Stirling's Approximation. Featuring Professor Ken McLaughlin. More links & stuff in full description below ↓↓↓ Professor McLaughlin is based at Colorado State University: https://www.math.colostate.edu/~kenmcl/ We filmed this during his time at the Mathematica

#### All the Numbers - Numberphile

Matt Parker talks about numbers - as he often does. His book "Humble Pi" is at: http://bit.ly/Humble_Pi More links & stuff in full description below ↓↓↓ The book on Amazon: https://amzn.to/2NKposg Numberphile podcast is on your podcast player. Or the website is: https://www.numberphile.com/p

#### What is a Number? - Numberphile

Featuring Asaf Karagila. More links & stuff in full description below ↓↓↓ Asaf is a UKRI Future Leaders Fellow. Asaf's blog - http://karagila.org Asaf's Twitter - https://twitter.com/AsafKaragila Numberphile podcast featuring Asaf - https://youtu.be/b6GLCTh5ARI All the Numbers with Matt Pa

#### The Most Wanted Prime Number - Numberphile

Featuring Neil Sloane. Visit https://gift.climeworks.com/numberphile to give the gift of CO₂ removal. Use code NUMBERPHILE10 for 10% off your purchase in December (sponsor) More links & stuff in full description below ↓↓↓ Neil Sloane is the founder of The OEIS: https://oeis.org More videos

#### Witness Numbers (and the truthful 1,662,803) - Numberphile

Featuring Matt Parker - more Parker links below. Check out Brilliant (get 20% off their premium service): https://brilliant.org/numberphile (sponsor) More links & stuff in full description below ↓↓↓ MATT PARKER STUFF Stand-Ups Maths on YouTube: https://www.youtube.com/user/standupmaths Matt's

#### What is the factorial of -½?

Check out https://KiwiCo.com/StandUpMaths to get 50% off your first month of any crate! Here is all the integration you ever wanted over on my second channel: https://youtube.com/mattparker2 Thanks to Ben Sparks for helping with all of the plots for this video. Ben's companion video on their chann

#### Infinitely Many Touching Circles - Numberphile

Featuring Matt Henderson. Check out https://www.kiwico.com/Numberphile and get 50% off your first month of any subscription. (sponsor) More links & stuff in full description below ↓↓↓ Matt Henderson: https://twitter.com/matthen2 More videos with Matt Henderson: https://bit.ly/MattHendersonPl

#### The Doomsday Algorithm - Numberphile

Featuring James Grime. Check out https://www.kiwico.com/Numberphile and get 50% off your first month of any subscription. (sponsor) More links & stuff in full description below ↓↓↓ More James Grime videos: http://bit.ly/grimevideos James Grime: https://www.singingbanana.com John Conway: http

#### Protecting Privacy with MATH (Collab with the Census)

This video was made in collaboration with the US Census Bureau and fact-checked by Census Bureau scientists. Any opinions and errors are my own. For more information, visit https://census.gov/about/policies/privacy/statistical_safeguards.html or search "differential privacy" at http://census.gov.

#### The Simplest Math Problem No One Can Solve

The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve. This video is sponsored by Brilliant. The first 200 people to sign up via https://brilliant.org/veritasium get 20% off a yearly subscription.

#### The Volume of a Sphere - Numberphile

Johnny Ball discusses Archimedes and the volume of a sphere. Check out https://www.kiwico.com/Numberphile and get 50% off your first month of any subscription (sponsor) More links & stuff in full description below ↓↓↓ Johnny Ball: https://johnnyball.co.uk More Numberphile videos with Johnny

#### Mandelbrot set

The Mandelbrot set (/ˈmændəlbroʊt, -brɒt/)[1][2] is the set of complex numbers c{\displaystyle c} for which the function fc(z)=z2+c{\displaystyle f_{c}(z)=z^{2}+c} does not diverge to infinity when iterated from z=0{\displaystyle z=0}, i.e.

#### Parabolas and Archimedes - Numberphile

This video features Johnny Ball. Check out Brilliant (get 20% off their premium service): https://brilliant.org/numberphile (sponsor) More links & stuff in full description below ↓↓↓ Johnny Ball: https://johnnyball.co.uk More Numberphile videos with Johnny Ball: http://bit.ly/Johnny_Ball Jo

#### How they found the World's Biggest Prime Number - Numberphile

Featuring Matt Parker... More links & stuff in full description below ↓↓↓ See part one at: https://youtu.be/tlpYjrbujG0 Part three on Numberphile2: https://youtu.be/jNXAMBvYe-Y Matt's interview with Curtis Cooper: https://youtu.be/q5ozBnrd5Zc The previous record: https://youtu.be/QSEKzFGpCQs

#### Matt Parker: Stand-up Maths Routine (about barcodes)

Matt Parker performs a stand-up maths routine about barcodes at the Hammersmith Apollo, as part of the 2011 Uncaged Monkeys national tour. http://standupmaths.com/

Matt Parker’s comedy routine about spreadsheets. From the Festival of the Spoken Nerd DVD: Full Frontal Nerdity Buy Full Frontal Nerdity as a DVD or Download: http://shop.festivalofthespokennerd.com/ See where Festival of the Spoken Nerd are performing live: http://festivalofthespokennerd.com/bu

#### What was the first (known) maths mistake?

Thanks to Waterstones for choosing Humble Pi as a 'top 100' paperback of the year. https://www.waterstones.com/book/humble-pi/matt-parker/9780141989143 Signed copies are available at Maths Gear. It's cheaper at Waterstones, but not signed. You choose! https://mathsgear.co.uk/collections/books/produ

#### Tribonacci Numbers (and the Rauzy Fractal) - Numberphile

Edmund Harriss introduces a very cool tiling and talks about Tribonacci Numbers. More links & stuff in full description below ↓↓↓ Numberphile Podcast: https://www.numberphile.com/podcast Or on YouTube: http://bit.ly/Numberphile_Pod_Playlist More Edmund on Numberphile: http://bit.ly/Ed_Harris

#### Don't Know (the Van Eck Sequence) - Numberphile

Neil Sloane on the Van Eck Sequence... Check out Brilliant (get 20% off their premium service): https://brilliant.org/numberphile (sponsor) More links & stuff in full description below ↓↓↓ More Neil Sloane: http://bit.ly/Sloane_Numberphile Van Eck sequence on OEIS: https://oeis.org/A181391

#### How many chess games are possible?

Dr James Grime talking about the Shannon Number and other chess stuff. Squarespace (10% off): http://squarespace.com/numberphile More links & stuff in full description below ↓↓↓ Support us on Patreon: http://www.patreon.com/numberphile NUMBERPHILE Website: http://www.numberphile.com/ Numberp

#### The Discovery That Transformed Pi

For thousands of years, mathematicians were calculating Pi the obvious but numerically inefficient way. Then Newton came along and changed the game. This video is sponsored by Brilliant. The first 314 people to sign up via https://brilliant.org/veritasium get 20% off a yearly subscription. Happy P

#### 357686312646216567629137 - Numberphile

Truncatable Primes with Dr James Grime... Check out Brilliant (and get 20% off their premium service): https://brilliant.org/numberphile (sponsor) More links & stuff in full description below ↓↓↓ Dr James Grime is available for public talks. See his website: http://singingbanana.com More vid

Dr James Grime is back and talking about tortoises. More links & stuff in full description below ↓↓↓ In many ways this video follows on from http://www.youtube.com/watch?v=bFNjA9LOPsg and then http://www.youtube.com/watch?v=CMP9a2J4Bqw James Grime's website is: http://singingbanana.com NUMB

#### Is zero an even number?

Superstorm Sandy had many consequences, some easier to foresee than others. Millions experienced floods and power cuts, the New York marathon was cancelled, and pictures of sharks in the city appeared on the internet. Another outcome was to draw attention to the unique position of the number zero.

#### Fibonacci Mystery - Numberphile

Brady's view on people who write: "FIRST" - http://youtu.be/CmRh9tFYC68 More links & stuff in full description below ↓↓↓ Dr James Grime on the Pisano Period - a seemingly strange property of the Fibonacci Sequence. Available Brown papers: http://periodicvideos.blogspot.co.uk/2013/09/brown.ht

#### The Golden Ratio (why it is so irrational) - Numberphile

Catch a more in-depth interview with Ben Sparks on our Numberphile Podcast: https://youtu.be/-tGni9ObJWk Check out Brilliant (and get 20% off) by clicking https://brilliant.org/numberphile More links & stuff in full description below ↓↓↓ Golden seeds limited edition T-Shirt: https://teespring

#### The Feigenbaum Constant (4.669) - Numberphile

Binge on learning at The Great Courses Plus: http://ow.ly/Z5yR307LfxY The Feigenbaum Constant and Logistic Map - featuring Ben Sparks. Catch a more in-depth interview with Ben on our Numberphile Podcast: https://youtu.be/-tGni9ObJWk Ben Sparks: https://twitter.com/SparksMaths Random numbers: htt

#### Euclid's Big Problem - Numberphile

Trisecting angles and calculating cube roots was a big problem for Euclid and his cohorts. Discussed by Zsuzsanna Dancso at MSRI. More links & stuff in full description below ↓↓↓ TRISECT WITH ORIGAMI: http://youtu.be/SL2lYcggGpc CIRCLE THE SQUARE: http://youtu.be/CMP9a2J4Bqw Support us on Pat

#### The Riemann Hypothesis, Explained

The Riemann hypothesis is the most notorious unsolved problem in all of mathematics. Ever since it was first proposed by Bernhard Riemann in 1859, the conjecture has maintained the status of the "Holy Grail" of mathematics. In fact, the person who solves it will win a \$1 million prize from the Clay

#### A proof that e is irrational - Numberphile

Professor Ed Copeland shows a proof by Joseph "Voldemort" Fourier that e is irrational. Check out episode sponsor http://KiwiCo.com/Numberphile for 50% off your first month of any subscription. The crates are great! More links & stuff in full description below ↓↓↓ Ed Copeland is a physics pro

#### The Dollar Game - Numberphile

Featuring Holly Krieger... Check out Brilliant (and get 20% off their premium service): https://brilliant.org/numberphile (sponsor) More links & stuff in full description below ↓↓↓ With Dr Holly Krieger from Murray Edwards College, University of Cambridge. Check out the monster dollar game s

#### Loop (graph theory)

In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet.

#### The simple maths error that can lead to bankruptcy

As we head into 2021, Worklife is running our best, most insightful and most essential stories from 2020. Read our full list of the year’s top stories here. Fifteen years ago, the people of Italy experienced a strange kind of mass hysteria known as “53 fever”.

#### ANU QRNG – Quantum random numbers

This website offers true random numbers to anyone on the internet. The random numbers are generated in real-time in our lab by measuring the quantum fluctuations of the vacuum. The vacuum is described very differently in the quantum physics and classical physics.

#### Noli turbare circulos meos!

According to Valerius Maximus, the phrase was uttered by the ancient Greek mathematician and astronomer Archimedes. When the Romans conquered the city of Syracuse after the siege of 214–212 BC, the Roman general Marcus Claudius Marcellus gave the order to retrieve Archimedes.

#### Inca Knot Numbers - Numberphile

Alex Bellos discusses how the Incans used knots in string (Quipu) to record numbers. Check out Brilliant (get 20% off their premium service): https://brilliant.org/numberphile (sponsor) More links & stuff in full description below ↓↓↓ Check out the Language Lover's Puzzle Book) on Amazon: htt

#### How modern mathematics emerged from a lost Islamic library

The House of Wisdom sounds a bit like make believe: no trace remains of this ancient library, destroyed in the 13th Century, so we cannot be sure exactly where it was located or what it looked like.

#### Euler's identity

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics.

#### Path-based strong component algorithm

In graph theory, the strongly connected components of a directed graph may be found using an algorithm that uses depth-first search in combination with two stacks, one to keep track of the vertices in the current component and the second to keep track of the current search path.

#### Strongly connected component

In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected.

#### Directed graph

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges often called arcs.

#### Cycle (graph theory)

In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal.

#### Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines).

#### Tarjan's strongly connected components algorithm

Tarjan's strongly connected components algorithm is an algorithm in graph theory for finding the strongly connected components (SCCs) of a directed graph.

#### The violent attack that turned a man into a maths genius

This article was inspired by an episode of The Outlook Podcast, where you can hear more about Jason Padgett's experience in his own words. Jason Padgett sees maths everywhere.

#### From The MIT Press Reader

One of the key findings over the past decades is that our number faculty is deeply rooted in our biological ancestry, and not based on our ability to use language. Considering the multitude of situations in which we humans use numerical information, life without numbers is inconceivable.

#### The maths problem that could bring the world to a halt

It’s not easy to accurately predict what humans want and when they will want it. We’re demanding creatures, expecting the world to deliver speedy solutions to our increasingly complex and diverse modern-day problems.

#### The myth of being 'bad' at maths

Are you a parent who dreads having to help with maths homework? In a restaurant, do you hate having to calculate the tip on a bill? Does understanding your mortgage interest payments seem like an unsurmountable task? If so, you’re definitely not alone.

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