Euler’s Identity is an Equation about constants π and e. Both are “Transcendental” quantities; in decimal form, their digits unspool into Infinity. And both are ubiquitous in scientific laws. But they seem to come from different realms: π (3.14159 …) governs the perfect Symmetry and closure of the Circle; it’s in Planetary Orbits, the endless up and down of light waves. e (2.71828 …) is the foundation of exponential growth, that accelerating trajectory of escape inherent to compound interest, nuclear fission, Moore’s law. It’s used to model everything that grows.
What Euler showed is that π and e are deeply related, connected in a dimension perpendicular to the world of real things - a place measured in units of i, the square root of -1, which of course doesn’t … exist. Mathematicians call it an imaginary number. These diagrams are visual metaphors. Imagine a graph with real numbers on the horizontal axis and imaginary ones on the vertical. Exponential function, f(x) = ex, ordinarily it graphs as an upward swooping curve - the very paradigm of progress. But put i in there, Euler showed, and eix instead traces a circle around the origin - an endless wheel of Samsara intercepting Reality at –1 and +1. Add another axis for Time and it’s a helix winding into the Future; viewed from the side, that helix is an oscillating sine wave.The rest is easy: Take that function f(x) = eix, set x = π, and you get eiπ = -1. Rearrange terms and you have the famous identity: eiπ + 1 = 0.
That’s the essence of Euler’s alchemy: By venturing off the real number line into this empyrean dimension, he showed that disruptive, exponential change (the land of e) reduces to infinite repetition (π). These diagrams combine the five most fundamental numbers in math - 0, 1, e, i, and π - in a relation of irreducible simplicity. e and π are infinitely long decimals with seemingly nothing in common, et they fit together perfectly - not to a few places, or a hundred, or a million, but all the way to forever.
You can take this farther, too. If you write that function above in a more general but still simple form as f(x) = e(zx), where z = (a + bi), what you get is no longer a circle but a logarithmic spiral, combining rotation and growth - now both at the same time- These graceful spirals are also found everywhere in Nature, from the whorls in a nautilus shell to the sweeping arms of Galaxies. And they’re related, in turn, to the Golden Ratio (yet another infinite decimal, 1.61803 …) and the Fibonacci Sequence of Numbers (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …).
But the weirdest thing about Euler’s formula - given that it relies on imaginary numbers - is that it’s so immensely useful in the real world. By translating one type of motion into another, it lets engineers convert messy trig problems into more tractable algebra - like a wormhole between separate branches of math. It’s the secret sauce in Fourier transforms used to digitize music, and it tames all manner of wavy things in quantum mechanics, electronics, and signal processing; without it, computers would not exist.
The square root of 2 equals 1.41421356237… Multiply this successively by 1, by 2, by 3, by 4 and so on, and write down the integer part of each result. Beneath this sequence, make a second list of the numbers missing from the first sequence.
Then the differences between the upper and lower numbers in these pairs are 2, 4, 6, 8, 10, 12, 14, 16, 18…
Are you the square root of -1? Because you can't be real.
Nothing is real. There is no proof that anything exists. Nothing separates “fiction” from “reality”. Feelings are intangible. Do they exist? Does something need to be tangible to exist? Does existence exist? Why does anything need to exist? Is the void that will one day rise up and claim all our souls real? If I focus hard enough will my breasts no longer be real? These are the questions of the hour.
You’d think engineering and relationships would be similar.
There’s a task or a problem and you have to find a solution. Some problems are your fault, some aren’t, some are big, and some you just didn’t see coming. You could design the best system you possibly can and it could still fail.
People are irrational. The limit does not exist. Does not compute. What’s the square root of -1? You’re dividing by 0? What?
A couple of years ago the math department decided to get sweatshirts. We decided to get numbers on the sleeve and everyone started fighting over which number to get. Funny enough, none wanted a normal number, there was the square root of three, e, pi, et cetera. there was one woman who wanted the square root of negative 1, because you know, she was all I, I, I. I myself wanted 0! because I am number one. Turns out in the end we couldn’t get numbers on our sweatshirts because it was too expensive; we still spent 4 or 5 days arguing over who got which number.
Precalculus teacher giving a story so that we remember that 0!=1
Coleslaw has 8 letters. It is also a side dish. What else is a side dish? Potatoes (except they are actually good) Also 8 letters, 8 times 8 is 64. 64 divided by 64 is one. 6 plus 4 is 10. 10 minus 1 is 9. And what’s the square root of 9? 3
You know what else involves 3? Triangles. They have three sides.
WaKE UP AMERICA. Coleslaw is illuminati confirmed.
Yesterday we looked
the equation x2-13y2=1
and used the continued fraction of √13 to find solutions.
We did not specify how to construct such a representation of √13.
For certain numbers you can make them, with a bit of perseverance.
Let us consider √13.
The first thing we do is note that 3<√13<4 and work with
√13-3 rather than √13.
Next we note that (√13-3)(√13+3)=4.
We can (re)write this equality as
But then you can replace the √13-3 in the denominator by the
whole right-hand side, and again, and again, …, this leads
to this continued fraction
That is not yesterday’s continued fraction but you can use it too to
find approximations of √13.
If you add 3 to the convergents of this fraction then you get
11/3, 18/5, 119/33, 393/109, 649/180, ….
This is a subsequence of yesterday’s approximations.
The solutions of x2-13y2=1 and
13y2-x2=1 can also be found in this sequence.
Can we use this to make yesterday’s continued fraction?
Yes, all you need is pencil and paper and the first fraction,
4/(6+(√13-3)) that is.
Divide numerator and denominator by 4; the numerator becomes 1
and the denominator will be 6/4+(√13-3)/4;
this you can rewrite to 1+(√13-1)/4.
Next: divide the numerator and denominator of (√13-1)/4
we get 1 and 4/(√13-1), respectively.
But, as above, we observe that (√13-1)(√13+1)=12 and so
12/(√13-1)=(√13+1), or 4/(√13-1)=(√13+1)/3
which we can turn into 1+(√13-2)/3.
The results of these two steps are displayed below:
If you keep doing this you will after two steps more arrive at the fraction on
the left below.
Now we read our initial equality as
(√13-3)/4=1/(6+(√13-3)) and this gives of the fraction
below on the right
But now we can replace (√13-3) by that fraction on the right,
and again, and again, … and this leads to yesterday’s
Exercise Do the same for √2-1 and √3-1 and construct
approximations of √2 and √3 in this way, and thus also
find solutions to the Pell equations
Transcendental numbers are not the roots of any algebraic equation. The existence of transcendental numbers was proven by Joseph Liouville in 1844.
In 1873 Charles Hermite proved that “e” is transcendental.
The German mathematician Ferdinand von Lindemann, in 1882, succeed in proving that π is transcendental. The area of a circle is π*r^2 (r = radius), that of a square is s^2 (s = side); consequently, the side of a square whose ares is equal to that of a circle with radius 1 is square root of π. A construction with straightedge and compass alone can give only lengths that are algebraic numbers, so Lindemann’s proof that π is transcendental was conclusive evidence that the age-old problem of squaring the circle is insolvable.
The transcendentality of e^π was proved in 1929. Yet, even if we know today that the number of transcendental numbers is infinite, there are still many irrational numbers, e.g. π^π, that defy our curiosity whether they are algebraic or transcendental.
6414 – 4354 = 2060 capsule’s 4th album is “Nexus 2060” 4+4+1+3+7+5+4+1+6+5+4+9+1+5+4 = 144 the square root of 144 is 12 Nexus 2060 is capsule’s 4th album 12 ÷ 4 = 3 3 sides too a triangle triangle = Illuminati