eiπ + 1 = 0
is immensely useful for mathematically modeling phenomena, such as the rhythmic flow of alternating current.
Kant " Sapere aude! " ---Latin for “Dare to know"
e dollars would be the amount that $1 of savings would be worth after a year if the bank compounded its annual 100% interest on it continuously.
German mathematician Hermann Weyl put it with greater gravitas, the goal of mathematics is “the symbolic comprehension of the infinite with human, that is finite, means.”
What’s the chance that not a single guest gets the right hat?
It turns out that this probability gets ever closer to 1 divided by e as the number of guests gets ever larger. Using 2.718 as an approximation for e, my calculator shows that 1/e equals roughly 0.37, which means there’s about a 37% chance that every guest walks out with the wrong hat.
Strangely, the probability is about 37% regardless of whether there are 50 or 50,000 guests.
As Euler brilliantly proved, plugging in such i-containing numbers for the x in ex effectively endows the function with the ability to model another common form of change: the oscillatory change of phenomena, such as alternating current, sound waves, or the back-and-forth motion of a child on a swing
The number π is like e in another important way: it’s an irrational number, which was proved in 1761 by Swiss mathematician Johann Lambert, one of Euler’s contemporaries. In 1882, π was demonstrated to possess a more unusual property by German mathematician Carl Louis Ferdinand von Lindemann: it’s a transcendental number, a type of irrational that’s extra-far-removed from the integers, fractions, and other relatively ordinary quantities encountered in arithmetic and algebra.
(e is also transcendental, which was proved in 1873 by French mathematician Charles Hermite. Mathematicians, including Euler, suggested the existence of transcendental numbers during the seventeenth and eighteenth centuries. But none were actually known to exist until 1844, when French mathematician Joseph Liouville proved that a group of infinitely complicated fractions he’d dreamed up were transcendental.)
A transcendental number is defined as a number that isn’t the solution of any polynomial equation with integer constants times the x’s.
(e raised to the π power) is also known to be transcendental, but no one has been able to determine whether πe, ee, and ππ are transcendental or not. The term transcendental refers to the fact that such numbers lie outside (or transcend) the “algebraic” set of numbers that can be solutions of polynomial equations.
In 1671, for example, Scottish mathematician James Gregory discovered an astonishing equation into which π seemed to have quietly slipped while he was playing around with infinite sums. The equation implied that when fractions with consecutive odd-integer denominators are combined in this straightforward way,
1 − 1/3 + 1/5 − 1/7 + 1/9 − 1/11 +
the grand total equals precisely one-fourth of π, or π/4.
Consider this question: 1 − 1 + 1 − 1 + 1 − 1 + … = what? This thought-twister is called Grandi’s series—it’s a much-contemplated puzzler in mathematics. Euler believed that its sum equals 1/2, as did other mathematicians of his time. Today, it’s considered a “divergent” series
∞ (sometimes called the lazy eight), which was introduced in 1655 by English mathematician John Wallis
Euler when he was in his late 20s:
π2/6 = 1/12 + 1/22 + 1/32 + 1/42 + ….
This formula is even more shocking than the Gregory-Leibniz one—it reveals a startling connection between π and the positive integers, 1, 2, 3,… . Before Euler derived it, several mathematicians, including Leibniz, had tried unsuccessfully to figure out what all those similar fractions add up to; the problem was first posed in 1644 by Italian mathematician Pietro Mengoli
Another Italian mathematician, Rafael Bombelli, discovered a cool but disquieting thing around 1570. Instead of discarding a cubic-equation solution that contained imaginary numbers, he played around with it using standard algebraic techniques and showed that it was actually a camouflaged real number.
cos θ = 1 − θ 2/2! + θ 4/4! − θ 6/6! + θ 8/8! + …
and
sin θ = θ − θ 3/3! + θ 5/5! − θ 7/7! + θ 9/9! +
eθ = 1 + θ + θ 2/2! + θ 3/3! + θ 4/4! + θ 5/5! +
geometric interpretation of complex numbers.
The Norwegian innovator was Caspar Wessel,
Thus, ii = e−π/2, and although e−π/2 has a funny-looking negative exponent, it is a real number—it’s actually about 0.208.