The ExtraordinarySums of Leonhard Euler Chapter. 9

Presented by Airen Washington

Leonhard Euler (pronounced oiler)

“His legacy

Surpassed in the long

history of Mathemati

cs” pg 207 1707-1783

“Collected works fill over 70 large

volumes” pg 207

Master of all Mathematica

l Tradespg 207

Leonhard EulerBIRTHPLACE Basel Switzerland

WHEN 1707

Leonhard EulerBORN TO A Calvinist Preacher

GENIUS CHILD He showed signs of genius

STUDIED WITHJohann Bernoulli (Swiss mathematician from Ch.8 Brother to Jacob)

He worked during the week & asked questions on Saturday

Leonhard Euler Fun Facts

“Phenomenal Memory”Memorized the first 100 prime numbers their cubes their fourth, fifth and sixth powers Enjoyed telling stories to his 13

children

Enjoyed growing vegetables

He began loosing his eye sight the mid 1730s

He lost mosthis visionIn 1771

Generous Man, good natured (not crazy)

Leonhard Euler Euler published mathematical papers of high quality

At the tender age of 19, he won a prize from the French Academy

for his analysis of the optimum placement of masts on a chip

1727 he was appointed to the St. Petersburg Academy in Russia, for which he became chair in 1733

Published Intoductio Analysin Infinitorium in 1728

he cleaned up Proofs, added differential calculus and integral calculus

Notable Contributions • Opera Omnia ~ 73 volumes of collected

Papers (his complete works)• Euler triangle in geometry• Euler character istic in topology • Euler circuit in graph theory • Euler constant• Euler polynomials • Euler integrals • Euler identity • Euler’s identity function

Great theorem EVALUATING +…Johann Bernoulli and Leibniz knew it was a number less than 2EULER found a formula that depended on and

3! Means 3 x 2 x 1 = 6 5! Means 5 x 4 x 3 x 2 x 1 = 120 the expression for sin x will continue foreverthe powers are the sequence of odd integersthe denominators are the associated factorials& the signs alternate between positive and negative

Taylor series expansion for the sine function suggested an endless polynomial so Euler examined a finite polynomial…

Taylor Series 𝐹 (𝑥)=∑

𝑘=0

𝑛 𝑓 (𝑘 ) (𝑎)𝑛!

(𝑥−𝑎)𝑘

+

P(x) is a polynomial of degree n having as its n roots

𝒙=𝒂 , 𝒙=𝒃 ,𝒙=𝒄 , . .. ,𝒂𝒏𝒅 𝒙=𝒅

𝑷 (𝒂)=𝑷 (𝒃)=𝑷 (𝒄)= .. .=𝑷 (𝒅 )=𝑶

Define …

We want to show that

Direct Substitution … (first factor is just )

… (second factor is just )

𝑷 (𝒂)=𝑷 (𝒃)=𝑷 (𝒄)= .. .=𝑷 (𝒅 )=𝑶

Further : …

…Further : is a polynomial

meeting the required conditions

Pierre de Fermat(1601-1665)

Made strides toward developing differential calculus, specifically in the area of number theory In his personal copy of Arithmetica ( by Diophantus) he scribbled in the margin …. Proposition 11.8 A result about expressing a perfect square as the sum of two other Perfect squares ( ei. )But it is impossible to divide a cub into two cubes, or a fourth power into two fourth powers

Fermat did so without a Proof….

(Modern Interpretation)Whole Numbers a, b, c

Euler was able to prove n = 3 and n = 4showed that a cube cannot be written as the sum of two cubes, or a fourth power as the sum of two fourth powers

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