Geometry Biography — Leonhard Paul Euler

Leonhard Paul Euler (April 15, 1707 – September 18, 1783) was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, such as the notion of a mathematical function. He is also renowned for his work in mechanics, optics, and astronomy. Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time.

Euler was born in Switzerland, in the town of Basel to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor’s daughter. Paul Euler’s friend Johann Bernoulli, who was regarded as Europe’s foremost mathematician, would eventually be the most important influence on young Leonhard. At the age of thirteen Euler attended the University of Basel, and in 1723, received a master’s degree with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil’s incredible talent for mathematics.

Euler was at this point studying theology, Greek, and Hebrew in order to become a pastor. Johann Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed his Ph.D. In 1727, he entered the Paris Academy Prize Problem competition, where the problem that year was to find the best way to place the masts on a ship. He won second place, losing to Pierre Bouguer, who is now known as “the father of naval architecture”. Euler subsequently won this coveted annual prize twelve times in his career.

Euler arrived in the Russian capital on May 17, 1727. He was promoted to a position in the mathematics department at the Imperial Russian Academy of Sciences in St Petersburg. Euler mastered Russian and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy. The Academy at St. Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. It was especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy so as to lessen the faculty’s teaching burden, and the academy emphasized research and offered its faculty the time and the freedom to pursue scientific questions.

On January 7, 1734, Euler married Katharina Gsell. The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. SAD. Concerned about turmoil in Russia, Euler left St. Petersburg on June 19, 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick’s niece. Euler wrote over 200 letters to her, which was later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler’s exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler’s personality and religious beliefs. This book became more widely read than any of his mathematical works, and it was published across Europe and in the United States. Its popularity testifies to Euler’s ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.

Despite Euler’s immense contribution to the Academy’s prestige, he was eventually forced to leave Berlin. This was partly because of a personality conflict with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in Frederick’s employ, and the Frenchman enjoyed a prominent position in the king’s social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. [François-Marie Arouet (1694 - 1778), better known by the pen name Voltaire, was a French Enlightenment writer and philosopher known for his wit and defense of civil liberties including freedom of religion. He was an outspoken supporter of social reform. He frequently use of his works to criticize Catholic Church dogma and the French institutions of his day. Voltaire was one of several Enlightenment figures (along with John Locke - not from LOST - and Thomas Hobbes) whose works and ideas influenced important thinkers of both the American and French Revolutions.]

Euler’s eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy. Euler’s sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as “Cyclops”. Euler later suffered a cataract in his good left eye, rendering him almost totally blind. Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory. Euler’s powers of memory and concentration were legendary. For example, Euler could repeat the Aeneid of Virgil. The Aeneid is a Latin epic poem written in the 1st century BC (between 29 and 19 BC) that tells the legendary story of Aeneas, a Trojan who traveled to Italy, where he became the ancestor of the Romans. It is written in dactylic hexameter. The first six of the poem’s twelve books tell the story of Aeneas’ wanderings from Troy to Italy; second half tells of the Trojans’ ultimately victorious war upon the Latins. With the aid of his scribes, Euler’s productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

The situation in Russia had improved greatly, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. His second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife of 40 years. Three years after his wife’s death Euler married her half sister. This marriage would last until his death.

On September 18, 1783, Euler passed away in St. Petersburg after suffering a brain hemorrhage. His eulogy was written by the French mathematician and philosopher Marquis de Condorcet. Condorcet commented, “…il cessa de calculer et de vivre – … he ceased to calculate and to live.” Condorcet told the story of two of Euler’s students who had independently summed seventeen terms of a complicated infinite series, only to disagree in the fiftieth decimal place; Euler settled the dispute by recomputing the sum in his head. Outstanding

Geometry Notes — The Euler Line

In the 18th century, the Swiss mathematician Leonhard Euler noticed that three of the many centers of a triangle are always collinear, that is, they always lie on a straight line. This line has come to be named after him – the Euler line. The three centers that have this surprising property are the triangle’s centroid , circumcenter and orthocenter.

In the figure, the centroid is the black middle point on the line. The centroid is the point where the three medians converge.

The circumcenter is the purple point. The circumcenter is the point where the perpendicular bisectors of the triangle’s sides converge.

The orthocenter is the red point. The orthocenter is the point where the three altitudes of the triangle converge.

The Euler Line

Geometry Notes — Segments of Triangles, The Centers of Triangles – Actually 4 of the 9

Median of a Triangle

A median of a triangle is a segment that joins the vertex of angle to the midpoint of the opposite side.

Because there are three vertices, there are three possible medians. One of the fascinating things about them is that no matter what shape the triangle, all three always intersect at a single point. This point is called the centroid of the triangle. This intersection of three lines at a single point is called concurrent. Two or more lines are said to be concurrent if they intersect in a single point.

There are some fascinating properties of the medians of a triangle:
1. The fact that the three medians always meet at a single point is interesting in its own right
2. Each median divides the triangle into two smaller triangles which have the same area
3. The centroid (point where they meet) is the center of gravity of the triangle.

Centroid facts
1. The centroid is always inside the triangle
2. The centroid is exactly two-thirds the way along each median. Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.

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Perpendicular bisector

A perpendicular bisector is a segment that is perpendicular to another segment at its midpoint.

This section we are more interested in the perpendicular bisectors of a triangle.

The circumcenter is the point where the perpendicular bisectors of a triangle intersect. The circumcenter is also the center of the triangle’s circumcircle – the circle that passes through all three of the triangle’s vertices.

The circumcircle always passes through all three vertices of a triangle. Its center is at the point where all the perpendicular bisectors of the triangle’s sides meet.
In the special case of a right triangle, the hypotenuse is a diameter of the circumcircle, and its center is exactly at the midpoint of the hypotenuse.

Circumradius
1. The radius of the circumcircle. The radius is given by the formula abc/[(a+b+c)(a+b-c)(a+c-b)(b+c-a)]^.5 where a,b, and c are the lengths of the sides of the triangle.
2. In the special case of an equilateral triangle, where all three sides (a,b,c) are have the same length, there is a simpler formula: s/3^.5 where s is the length of any side of the triangle.
3. In the special case of a right triangle, the radius is half the length of the hypotenuse.

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Angle bisector

An angle bisector is a ray that divides an angle into two congruent angles.

In this section, we are more interested in the angle bisectors of a triangle.

The incenter is the point where the angle bisectors intersect. The incenter is also the center of the triangle’s incircle – the largest circle that will fit inside the triangle.

Properties of the incenter
1. The incenter is the center of the triangle’s incircle, the largest circle that will fit inside the triangle and touch all three sides.
2. The triangle’s incenter is always inside the triangle.

The incircle of a triangle is also known as “inscribed circle.” Each of the triangle’s three sides is a tangent to the circle.

Incenter – the location of the center of the incircle. The point where the angle bisectors meet.
Inradius – the radius of the incircle. The radius is given by the formula 2a/p where a is the area of the triangle and p is the perimeter of the triangle, the sum of its sides.

For the special case of an equilateral triangle the inradius is also given by the formula s/(2*3^.5) where s is the length of a side.

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Altitude

An altitude is a segment from the vertex of an angle that is perpendicular to the opposite side or to the line containing the opposite side. A triangle has three altitudes.

An interesting fact is that the three altitudes always pass through a common point called the orthocenter of the triangle.

The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside. To make this happen the altitude lines have to be extended so they cross.

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Summary of triangle centers.

There are many types of triangle centers. Below are four of the most common.

Incenter – Located at intersection of the angle bisectors.

Circumcenter – Located at intersection of the perpendicular bisectors of the sides.

Centroid – Located at intersection of medians.

Orthocenter – Located at intersection of the altitudes of the triangle.

In the case of an equilateral triangle, all four of the above centers occur at the same point.

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The Euler line – an interesting fact. It turns out that the orthocenter, centroid, and circumcenter of any triangle are collinear – that is, they always lie on the same straight line called the Euler line, named after its discoverer – Leonhard Paul Euler.

Pre-Calculus Notes — Video — Rational Inequality

Problem 53

Pre-Calculus Notes — Solving Quadratic and Rational Inequalities

Pre-Calculus Assignment — Book 3.6

Statistics Notes — Expected Value

Expected value

At a carnival, you pay $2.00 to play a coin-flipping game with three fair coins. On each coin one side has the number 0 and the other side has the number 1. You flip the three coins at one time and you win $1.00 for every 1 that appears on top. Are your expected earnings equal to the cost to play? We’ll answer this question in several steps.

a. In this game, the random variable of interest counts the number of 1s that show. What is the sample space for the values of this random variable?  The sample space is (0, 1, 2, 3), since any of these numbers of 1s can appear.

b. There are eight equally likely outcomes for throwing three coins. They are 000, 001, 010, 011, 100, 101, 110, and 111.

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d. The expected value is μ = ΣxP(x) = 0 + 0.375 + 0.750 + 0.375 = $1.50

The expected value is $1.50. It cost $2.00 to play the game; the expected value is less than the cost. The carnival is making money. In the long run, the carnival can expect to make an average of about 50 cents per player.

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Another way is to consider the gain of the game. You pay $2.00.
If you get zero 1s, your gain is -2, actually its a loss, not a gain.
If you get one 1, your gain is -1, again its a loss, but play along.
If you get two 1s, your gain is 0.
If you get three 1s, your gain is 1.

The expected value of your gain is μ = ΣxP(x) = -0.250 – 0.375 + 0 + 0.125 = -$0.50

The expected value of your gain is -50 cents. It cost $2.00 to play the game, you can expect to lose 50 cents each time you play. The carnival is making money. Overall, all players should expect to lose 50 cents each time they play, money the carnival will gladly take from them.

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In gambling, an expected value of 0 implies that a game is a fair game — an unlikely occurrence.

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At a raffle, 1000 tickets are sold at $5 each for four prizes of $1000, $200, $100, and $50. You buy one ticket. What is the expected value of each ticket? What is the expected value of your gain?

The expected value is μ = ΣxP(x) = 1 + 0.2 + 0.1 + 0.05 + 0 = $1.35

The expected value is $1.50. It cost $5.00 to play the game; the expected value is less than the cost. The fundraiser is making money. The fundraisers can expect to make an average of $3.65 per player.

The expected value of your gain is μ = ΣxP(x) = 0.995 + 0.195 + 0.095 + 0.045 – 4.98 = -$3.65

The expected value of your gain is -$3.65. It cost $5.00 to buy a raffle ticket, you can expect to lose $3.65 on each ticket you buy. The fundraiser will make money. Overall, all buyers should expect to lose $3.65 each time they buy a raffle ticket, money to buy wonderful equipment for wonderful kids. Thanks, we appreciate you.