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@numbermathics
Parabola and the Vertex
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Geometry speaks without words 😍👌
Animated diagram illustrating the definitions of the cosine and sine functions as the x coordinate and y coordinate of a point on the unit circle making an angle teta with the x axis
Finding n roots of unity
This leads to Quantization of θ because there are only specific values that become possible for θ when we impose that after n rotations it has to return back to its same starting point. **
And the following are those values:
The problem of finding the values of θ for given values of n is more generally known as the N roots of unity.
We will leave it as an exercise, but the following animation plots all the possible the values of θ for integer values of n=2,3,4,.. from the above equation that we found:
Source
And those are the roots of the equation z^n = z i.e if you start at these points on the unit circle and make n rotations you will get back to the same point that you started with.
Have a good one!
** A more physical way to think about this is matching boundary conditions. For more insight on how boundary conditions lead to quantization, take a look at this post.
*** Finding out the n roots of unity (Video)
Get it?
The myth: You can't fold a paper in half more than eight times.* The reality: Given a paper large enough—and enough energy—you can fold it as many times as you want. The problem: If you fold it 103 times, the thickness of your paper will be larger than the observable Universe: 93 billion light-years. Seriously.
Koch Snowflake: Finite area, Infinite perimeter.
The Koch Snowflake has finite area but infinite perimeter… yeah that happens with fractals. This abstract curve requires an infinite process (depicted in the gif) to construct and is an example of a fractal–a mathematical set (usually a curve or geometric figure) which exhibits a repeating pattern that displays at every scale. (More about fractals here https://en.wikipedia.org/wiki/Fractal)
But How? It seems clear that the area would be finite since the figure encloses a finite amount of space. To grasp why the Koch Snowflake has infinite perimeter, notice how as the iterations progress, the edges become more and more intricate. Now imagine trying to draw the edges with a pen. Since the construction of the snowflake continues indefinitely, the edges become infinitely intricate and you could never finish detailing these intricacies with your pen (that is the intuitive argument at least. I’ll leave the precise calculations up to you).
Fractals may seem so abstract and impractical but they actually have many useful real-world applications. For example, Benoit Mandelbrot (considered the “father of fractals”) found that stock market prices could be modeled with a factual curve. Check the wiki page for a long and diverse list of applications.
Fractal geometry may seem more abstract than traditional geometry but Mandelbrot argues that fractals are “the geometry of nature”. Objects in nature have random irregularities and are seemingly infinite in their intricacies. Attempting to incorporate this in drawings or animations is extremely difficult. Movie special effects and CGI often use fractals to make objects appear more natural looking. Since fractals can be made with mathematical formulae they are easy to generate with a computer. The first Star Wars movies were renowned for their special effects and were some of the first to use fractals to generate life-like explosions and landscapes of other worlds.
MAGIC SQUARE OF 6x6 - Magic Sums of 111 and Solar Symbol The Magic Square of 6x6, having the sum of all its numbers (Sigma 36) adding to 666 has always been a Solar Glyph, a veritable and secret pin number or access code to enter another solar system or dimension or galaxy, via the Triplet 111 Harmonic. This is one reason why 666 was demonized, because it is really a representation of the light, to create fear and not love, to scare you from enter the Higher Worlds, or from accessing your DNA. On a deeper level, when you draw a line from 1 to 2 to 3 etc to the last number 36, you arrive at a special Yantra or pattern, symbolically you are looking for Order amidst the apparent Chaos, you are translating Number Into Art, but when you superimpose this pattern upon itself 4 times, at angles of 90 degrees you get the famous, most anointed, holy symbol of the Swastika! And this is just the beginning of The Million Year War. Jain 108
Meet the First Woman to Win Math’s Most Prestigious Prize
“As an 8-year-old, Maryam Mirzakhani used to tell herself stories about the exploits of a remarkable girl. Every night at bedtime, her heroine would become mayor, travel the world or fulfill some other grand destiny.
Today, Mirzakhani — a 37-year-old mathematics professor at Stanford University — still writes elaborate stories in her mind. The high ambitions haven’t changed, but the protagonists have: They are hyperbolic surfaces, moduli spaces and dynamical systems. In a way, she said, mathematics research feels like writing a novel. “There are different characters, and you are getting to know them better,” she said. “Things evolve, and then you look back at a character, and it’s completely different from your first impression.”
Learn more about Maryam Mirzakhani at wired.
Draw three circles, each intersecting the other two in two points, and connect their points of intersection. The lengths of this line segments connecting the intersection points then satisfy a⋅c⋅e = b⋅d⋅f.
This neat fact was discovered by Hiroshi Haruki.
Math
Choose any number and write down its divisors:
14 1 2 7 14
Then write down the number of divisors of each of these divisors:
14 1 2 7 14 1 2 2 4
Now the square of the sum of this last group will always equal the sum of its members’ cubes:
(1 + 2 + 2 + 4)2 = 13 + 23 + 23 + 43
Discovered by Joseph Liouville.
n roots of unity
When one is dealing with complex numbers, it is many a times useful to think of them as transformations. The problem at hand is to find the n roots of unity. i.e
As is common knowledge z = 1 is always a solution.
Multiplication as a transformation
Multiplication in the complex plane is mere rotation and scaling. i.e
Now what does finding the n roots of unity mean?
If you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up at 1.
We just need to find the complex numbers that do this.i.e
This implies that :
And therefore :
Take a circle, slice it into n equal parts and voila you have your n roots of unity.
Okay, but what does this imply ?
Multiplication by 1 is a 360o / 0o rotation.
When you say that you are multiplying a positive real number(say 1) with 1 , we get a number(1) that is on the same positive real axis.
Multiplication by (-1) is a 180o rotation.
When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis
The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.
Multiplication by i is a 90o rotation.
Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation.
This applies to -i as well.
so on and so forth,
Have a great day!
The generalized product rule ( Leibniz Formula )
The generalized product rule at the face of it not intuitive and that’s probably why a lot of people make mistakes when asked to differentiate functions n-times.
If f and g are functions that are continuous and differentiable everywhere, then:
Now, we would like to find out a generalized expression for the n-th derivative of fg.
In order to arrive at that formulation lets calculate a few derivatives to see whether we can find a pattern:
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You must have noticed a pattern in the above expressions. The coefficients seem are the one in the binomial expansion of (x+y)n
Therefore we can write the expression for the n-derivative of fg as the following:
where (i) means to differentiate i-times. This is also known as Leibniz Formula.
Have a good one!
How high can you count on your fingers?
How high can you count on your fingers? It seems like a question with an obvious answer. After all, most of us have ten fingers, or to be more precise, eight fingers and two thumbs. This gives us a total of ten digits on our two hands, which we use to count to ten.
It’s no coincidence that the ten symbols we use in our modern numbering system are called digits as well. But that’s not the only way to count. In some places, it’s customary to go up to twelve on just one hand. How? Well, each finger is divided into three sections, and we have a natural pointer to indicate each one, the thumb. That gives us an easy to way to count to twelve on one hand.
And if we want to count higher, we can use the digits on our other hand to keep track of each time we get to twelve, up to five groups of twelve, or 60.
Better yet, let’s use the sections on the second hand to count twelve groups of twelve, up to 144.
That’s a pretty big improvement, but we can go higher by finding more countable parts on each hand. For example, each finger has three sections and three creases for a total of six things to count. Now we’re up to 24 on each hand.
And using our other hand to mark groups of 24 gets us all the way to 576. Can we go any higher? It looks like we’ve reached the limit of how many different finger parts we can count with any precision. So let’s think of something different.
One of our greatest mathematical inventions is the system of positional notation, where the placement of symbols allows for different magnitudes of value, as in the number 999. Even though the same symbol is used three times, each position indicates a different order of magnitude. So we can use positional value on our fingers to beat our previous record. Let’s forget about finger sections for a moment and look at the simplest case of having just two options per finger, up and down. This won’t allow us to represent powers of ten, but it’s perfect for the counting system that uses powers of two, otherwise known as ‘binary’.
In binary, each position has double the value of the previous one, so we can assign our fingers values of 1, 2, 4, 8…all the way up to 512. And any positive integer, up to a certain limit, can be expressed as a sum of these numbers. For example, the number seven is 4+2+1. So we can represent it by having just these three fingers raised. How high an we go now? That would be the number with all ten fingers raised, or 1,023. Is it possible to go even higher? It depends on how dexterous you feel! If you can bend each finger just halfway, that gives us three different states -down, half bent, and raised. Now, we can count using a base-three positional system, up to 59,048. And if you can bend your fingers into four different states or more, you can get even higher. That limit is up to you, and your own flexibility and ingenuity.
Even with our fingers in just two possible states, we’re already working pretty efficiently. In fact, our computers are based on the same principle. Each microchip consists of tiny electrical switches that can be either on or off, meaning that base-two is the default way they represent numbers. And just as we can use this system to count past 1,000 using only our fingers, computers can perform billions of operations just by counting off 1’s and 0’s.
From the TED-Ed Lesson How high can you count on your fingers? (Spoiler: much higher than 10) - James Tanton
Animation by TED-Ed
feb. 1, 2017 / class notes + study guide from algebra! logs are really fun :P
“If you need to know the best counter to the Queen’s Gambit you ask a Chess Grandmaster. If you need to figure out the average number of steps it would take before a randomly moving knight returns to its starting square, you ask a mathematician. “ And ask, we shall. If you haven’t been watching PBS Infinite Series, you are missing OUT.
PITÁGORAS - Probablemente el mayor troll de la historia http://ift.tt/2hc5N27
