Pythagorean Pal @pythagoreanpal

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Pythagorean Pal @pythagoreanpal

2 years ago

Synthetic Division. Further information for the book Precalculus for Everybody #mathematics

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Pythagorean Pal @pythagoreanpal

2 years ago

Proof: a,b > 0 a = b ab = b² ab-a² = b²-a² a(b-a) = (b+a)(b-a) a = (b+a) a = a+a a = 2a 1 = 2 This "popular paradox" is actually no paradox at all, it is a mathematical error. In the simplification of step 6 there is a camouflaged division by zero: (b-a) = 0 #Mathematics

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Pythagorean Pal @pythagoreanpal

2 years ago

A frustrating problem for the ancient Greeks was finding a square with the area of a given circle, using only compass and straightedge. In 1882 it was proven to be impossible since the side of the square would be x=r√π, but π is not root of polynomials with rational coefficients

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Pythagorean Pal @pythagoreanpal

2 years ago

Konigsberg was a city joined by 7 bridges over a river with two islands. It was tradition to go through the city trying to cross the bridges only once, until Euler proved this was impossible. His proof came with a diagram (graph), giving rise a new branch of mathematics, Topology

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Pythagorean Pal @pythagoreanpal

2 years ago

There are four non-convex solids, also called Kepler-Poinsot solids. They are not the Platonic solids of Ancient Greece; they were just discovered in the 18th century. One of their applications is to predict the structure and behavior of porous materials, such as rubber or sponge

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Pythagorean Pal @pythagoreanpal

2 years ago

If 3 golden rectangles intersect symmetrically, where each one is perpendicular to the other two, then the vertices of the rectangles will match the 12 corners of a regular icosahedron (20-sided polyhedron) #Mathematics #Geometry

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Pythagorean Pal @pythagoreanpal

2 years ago

In the Islamic religion, prayers are carried out in direction of the sacred places, that is why, in the Islamic golden age, the caliphs promoted research in mathematical geography. As a result, Muslim scholars made incredible contributions in mathematics, physics and astronomy

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Pythagorean Pal @pythagoreanpal

2 years ago

The Transcendental Numbers are a set of numbers (real or complex) that cannot be Algebraic, that is, roots of a non-zero polynomial. These numbers have to be irrational, but not every irrational is a transcendental, for instance, √2 and ϕ (golden mean) are not transcendentals

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Pythagorean Pal @pythagoreanpal

2 years ago

Trigonometric functions are widely used in 3D rendering to calculate positions in 3D scenes. This is done by using the camera's position, orientation, and field of view to calculate the ray that passes through the point and the camera. It also applies to lights and reflections

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Pythagorean Pal @pythagoreanpal

2 years ago

Diophantine Equations are equations whose solution can only include integers. They are named after the Greek mathematician, Diophantus of Alexandria. These equations can be as simple as x+y=5, but they can also be so complex that they are still areas of research #Mathematics

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Pythagorean Pal @pythagoreanpal

2 years ago

This concept of "Parallelism" in Precalculus for Everybody causes some controversy😅. In Euclidean Geometry, two parallel lines never coincide. In Analytic Geometry, lines are parallel if their slopes are equal, that is, both can coincide at all their points (can be the same)

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Pythagorean Pal @pythagoreanpal

2 years ago

This photo of a rainbow seen from the sky went viral, but many thought it was a fake. Actually, a rainbow is not a semicircle, nor a circle; it is a cone, in whose apex are our eyes. Light breaks (refraction) and these rays form the cone as they pass through a sphere (droplets)

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Pythagorean Pal @pythagoreanpal

2 years ago

In the 16th century, the mathematician John Napier used logarithms to develop a system that converts multiplication, division and root extraction operations into addition and subtraction. This is known as Napier's Bones, and was used worldwide before calculators were produced

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Pythagorean Pal @pythagoreanpal

2 years ago

Hyperbolic Geometry is a "non-Euclidian" geometrie, however, they only differ in one postulate, "parallelism". In Euclidean, if you have a line L and a point P (not in L), then there is only one L' parallel to L that passes through P. In Hyperbolic, there are infinitely many L's

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Pythagorean Pal @pythagoreanpal

2 years ago

Pascal's Theorem was a work of Blaise Pascal when he was only 16 years old. It states that if a hexagon is inscribed in a conic section, then the three intersection points of the opposite sides will form a line, also known as Pascal Line #Mathematics

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Pythagorean Pal @pythagoreanpal

2 years ago

If the dimensions of a billiard table are given in whole numbers (e.g. 7 x 5), then a ball hitting one of the walls, at an angle of 45 degrees, will end up in one of the corners after "table length + table width - 2" rebounds. #Mathematics

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Pythagorean Pal @pythagoreanpal

2 years ago

Pascal's Limacon is a curve of degree 4. It is produced by the path of a fixed point on a circle rolling on another of equal radius. It has many applications in mechanics. This curve wasn't discovered by the famous Blaise Pascal, rather by his father, Étienne Pascal #Mathematics