![]() One type is Proth primes which you make by multiplying a number by a power of 2 then adding 1 (k x 2 n + 1). Proth Primes - Help prove something! There are all sorts of prime types, such as Mersenne primes. Did you know that your mobile phone uses quaternions to work out what way up it is? Dr James Grime steps into the second dimension and beyond to introduce complex numbers (2D), quaternions (4D), octonions (8D) and sedenions (16D). Quaternions - So we have integers, fractions and even irrational numbers, but all of these fit onto the real line. The next one (only a couple of thousand larger) has 64 factors. It is the smallest number with that many factors, so it is called a Highly Composite Number. Highly Composite Numbers - 5040 has 60 factors (1, 2, 3, 4, 5.). Primes and Other NumbersĪll the Numbers - You have heard of rational numbers but what about constructable, computable, algebraic or trancendental numbers? Matt Parker explains that "the maths for these numbers is insane" and we have only found a tiny percentage (essentially 0%) of all the real numbers that exist. Discover how Gödel used primes to write statements like a 2 + b 2 = c 2 as a single number, turning proof into arithmetic. Maybe Riemann's Hypothesis is one of these. Gödel's Incompleteness Theorem - Professor Marcus du Sautoy explains how Gödel proved that mathematics will always have statements that are true, that we know are true - but we won't be able to prove they are true. A good example of why chip making is hard. It very very nearly works - only two glitches. The 10,000 Domino Computer - Being truly insane, Matt Parker and his Domino Computer team (not to be confused with the Domino pizza team) uses dominoes crazy (about 10,000 dominoes) to build a four bit adder circuit. As he says, though, you would have to be "truly insane" to try building a domino computer. He uses 200 of them to create logic gates and combines the gates to create a half adder circuit. Logic and Computationĭomino Addition - Matt Parker uses dominoes to explain why computers use binary. Uses the classic proofs including Cantor's diagonalisation argument. Infinity is bigger than you think - James Grime shows that there are same infinite number of fractions as there are whole numbers - but the real numbers are infinitely bigger than either. Infinitesimals - More about these very very small hyperreal numbers. They are not real numbers, they are called hyperreal numbers. The Opposite of Infinity - Infinitesimals are numbers which are smaller than any positive number but larger than zero. Zero Factorial - Why is the factorial of zero equal to one? ![]() The team also introduce the complex plane here. ![]() ![]() A very good example of mathematical proof by counterexample. Problems with Zero - What is one over zero? What about zero to the power of zero? Find out why these questions cannot be answered. Can it always be done in 6? Includes links to articles and research papers. In the second video he explains how to use it to decide on a seating plan and how many colours a map on a Möbius strip needs.Ī Colorful Unsolved Problem - Dr James Grime describes the Hadwiger–Nelson problem: What are the fewest floor colours needed so that each step changes the colour you are on? Using hexagons you can show you need at most 7 colours, using Moser spindles you can show you need at least 5. The Four Color Map Theorem - Dr James Grime explains the Four Color Map Theorem, its history and its computer-aided solution using network graphs. Recommended if you are planning a weekend party, you need help solving Sudoku puzzles or if you want to find out what graph theory is. The counterexample he found is very very large (drawing it would require all the particles in 10 4000 universes). Graphs and Networksįinding a Counterexample - After 50 years a mathematician has finally disproved Hedetniemi's Conjecture. ![]() Includes links to a book on how to 3D print your own volume filling curves. To do this you need an infinitely long line and a well-designed curve. Space-Filling Curves - How can a one dimensional line (with length but zero width) completely fill a square? All it has to do is to hit every single point in the square. Football fields shaped like an hourglass, basketballs that are only visible using a powerful telescope - until they hit you in the face. Playing Sports in Hyperbolic Space - Dick Canary explains why ball sports really would not work in hyperbolic space. He explains how this links to Fermat Primes.ĭitching the Fifth Axiom - Dr Caleb Ashley explains how deleting Euclid's Parallel Postulate gives access to two whole new worlds: the world of hyperbolic geometry and the world of spherical geometry. Constructing a Heptadecagon - Professor David Eisenbud shows how to construct the 17-sided regular polygon, the heptadecagon, using just a straight edge and compass. ![]()
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