Maths With R.Khan

Pascal Triangle

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Ants Can Count Steps.

Ants are social insects that evolved from vespoid Wasp in the mid cretaceous period, about 150 millions years ago. After the rise of flowering plants, about 100 millions years ago, ants diversified into numerous species.
The Saharan desert ant, Cataglyphis fortis, travels immense distances over sandy terrain, often completely devoid of landmarks, as it searches for food.These creatures are able to return to their nest using a direct route rather retracing their outbound path. Not only do they judge directions, using light from the sky for orientation but they also appear to have a a built-in " computer" that functions like a pedometer that counts their steps and allows them to measure exact distances. An ant may travel as far as 160 feet( about 50 meters) until it encounters a dead insect, wheareupon it tears a piece to carry directly back to its nest.

By manipulating the leg lengths of ants to give them longer and shorter strides, a research team of German and Swiss scientists discovered that the ants "count" steps to judge distance. For example, after ants had reached their destination, the legs were lengthened by adding stilts or shortened by partial amputation. The researchers then returned the ants so that the ants could start their journey back to the nest. Ants with the stilts traveled too far and passed the nest entrance while those with the amputated legs did not reach it. However, if the ant started their journey from their nest with the modified legs, they were able to compute the appropriate distances. This suggests that stride length is the crucial factor. Moreover, the highly sophisticated computer in the ant's brain enables the ant to compute the quantity related to the horizontal projection of its path so that it does not become lost even if the sandy landscape develops hills and valleys during its journey.

(20th President of U.S ) who Proved .

James A. Garfield (1831-1881) was the twentieth president of the United States. After he graduated from Williams College in 1856, he taught Greek, Latin, mathematics, history, philosophy, and rhetoric at Western Reserve Eclectic Institute, now Hiram College, in Hiram, Ohio, a private liberal arts institute. In addition to teaching, he also practiced law, was a brigadier general in the Civil War, served as Western Reserve’s president, and was elected to the U.S. Congress.

Garfield contributed an original proof of the Pythagorean Theorem to the hundreds that have been recorded over the centuries. For a repository of original proofs of the Pythagorean Theorem, see Elisha Loomis’ The Pythagorean Proposition. Garfield developed his proof in 1876 while a member of Congress; that was the year Alexander Graham Bell developed the telephone. This “very pretty proof of the Pythagorean Theorem,” as Howard Eves described it, was published in the April 1, 1876 issue of the New-England Journal of Education. Evidently the editor of the journal erroneously (or perhaps in political jest) called the theorem Pons Asinorum or “Bridge of Asses,” actually a nickname for the isosceles triangle theorem (Euclid's Elements, Book I, Proposition 5). The latter theorem may have earned its name because many medieval students had difficulty understanding the proof, for which the diagram somewhat resembles a bridge, and therefore could not cross over the bridge to subsequent proofs in Euclid's Elements.
Garfield's proof of the Pythagorean Theorem essentially consists of a diagram of a trapezoid with bases a and b and height a+b. He looked at the area of the diagram in two different ways: as that of a trapezoid and as that of three right triangles, two of which are congruent.

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Today marks the death anniversary of Maryam Mirzakhani, a mathematician and professor of mathematics at Stanford University. Mirzakhani established herself as a prominent figure in her subject area, having done her research on topics including Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. It was due to her research that in 2005, she was honored in Popular Science's fourth annual "Brilliant 10" in which she was acknowledged as one of the top ten young minds who have pushed their fields in innovative directions.

In 2014, Maryam was awarded the Fields Medal - the most prestigious award in mathematics. She is both the first, and to date, the only woman and the first Iranian to be honored with the award.

The Mysterious Connection between Honey Bees and Mathematics ““…that vast book which stands forever opened before our eyes, I mean the universe, … cannot be read until we have learned the language……

Differentiation Rules

X^2+1=0
This was the equation which lead to the discovery of Complex numbers ,the most important branch of mathematics which has countless applications in mathematics and physics,

a)"Sum of the angles in any triangle is 180 degree"
This is true when the Geometry is Euclidean or planner,in general this is not true,in Riemanian Geometry (also known as Differential Geometry) the sum could b less than 180 or greater than 180

b)" a triangle could have at most one right angle(90 degree)"
This is true for Euclidean geometry ,in general a triangle may have more than 1 right angles,2 or at most 3,.....