Leonhard Euler was an 18th century Swiss mathematician who made revolutionary contributions to many areas of mathematics. He introduced modern mathematical notation including the use of f(x) to denote a function, popularized the letter e as the base of natural logarithms, and used Σ for summations. Some of his most important theorems include Euler's formula relating the number of faces, vertices and edges of a polyhedron; his formula connecting trigonometric functions and complex exponentials; his theorem for homogeneous functions; and Euler's totient theorem relating co-prime integers. Euler worked as a mathematician his entire life until suffering a brain hemorrhage and passing away in St. Petersburg in 1783.

1. Leonhard
Euler
15 April 1707 - 18
September 1783

2. His life
Leonhard Euler was one of the giants of 18th Century mathematics. He was born
in Basel, Switzerland, and he studied for a while under Johann Bernoulli at Basel
University. He is said to have produced on average one mathematical paper every
week. He was a revolutionary thinker in the fields of geometry, trigonometry,
calculus, differential equations, number theory and notational systems. In 1738,
he became almost blind in his right eye.
Euler, working on the day of his passing, suffered from a brain hemorrhage and
died during the night of September 18, 1783, in St. Petersburg.
Euler's legacy has been enormous in terms of shaping the modern playing field of
mathematics and engineering

3. Contributions

4. Mathematical notation
He introduced the concept of
a function and was the first to
write 𝑓(𝑥) to denote the
function 𝑓 applied to the
argument 𝑥. He also
introduced the modern
notation for the trigonometric
functions, the letter 𝑒 for the
base of the natural logarithm,
the letter Σ for summations
and the letter ‘𝑖’ to denote
the imaginary unit.

5. Euler's Formula about Geometry
For any polyhedron that doesn't intersect
itself, the
– Number of Faces
– plus the Number of Vertices (corner points)
– minus the Number of Edges
always equals 2
This can be written: F + V − E = 2
Try it on the cube:
A cube has 6 Faces, 8 Vertices, and
12 Edges,
so: 6 + 8 − 12 = 2

6. EULER'S FORMULA ABOUT COMPLEX NUMBERS
A mathematical formula in complex analysis that establishes the fundamental
relationship between the trigonometric functions and the complex exponential
function.
eix = cos x + i sin x
e = base of the natural logarithm
i = imaginary unit
x = the angle in radians
Example: when x = 1.1
eix = cos x + i sin x
e1.1i = cos 1.1 + i sin
1.1
e1.1i = 0.45 + 0.89 i

7. Euler’s
Theorem for
Homogenous
function
If 𝑓 𝑥, 𝑦 is a homogeneous function
of degree ′𝑛′ then
𝑥
𝜕𝑓
𝜕𝑥
+ 𝑦
𝜕𝑓
𝜕𝑦
= 𝑛𝑓

8. Euler’s Totient Theorem :-
It states that if 𝑥 and 𝑛 are co-prime positive integers, then
→ 𝑥∅ 𝑛 𝑚𝑜𝑑 𝑛 = 1 𝑚𝑜𝑑 𝑛
Where ∅ 𝑛 → euler's totient function
𝑥∅ 𝑛 ≡ 1 𝑚𝑜𝑑 𝑛

9. Thank You