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K, a, b ∈ ℕ where (a^2 + b^2)/(ab + 1) = K ⇒ (a^2 + b ^2) = K(ab + 1) = Kab + K So ⇒ A) (a^2 + b^2) ≡ 0 mod K B) (a^2 + b^2) ≡ K mod (ab + 1) C) b^2 ≡ K mod a D) a^2 ≡ K mod b So K is a perfect square mod a, mod b But (a + b)^2 = (a^2 + b^2 + 2ab) = Kab + K + 2ab ≡ K mod (ab + 1) So K is a perfect square mod (ab+1) Where do I go from here? What have I done wrong?

This channel is underrated.

0:55 - but I'm feeling 22.

Yeah I actually searched it out

Thank You

WHY IN COORDINATE SYSTEM CENTER IS NOT ORIGIN = (0, 0 ,0) INSTEAD IT IS 1.TIMING 10.04 SECS

What do you have to know first for this to make any sense at all? Feels so much like a foreign language.

wait i get it, that's so fucking cool

There is only one force which is fusion. But the balancing act is done by fission and electrical and magnetic and gravity. 4 though 3. Every force is an accelerated frame of reference. Dark matter and dark force is a special frame of reference. Usually hidden casts. Dark matter can sometimes be seen as white matter or normal matter when you almost reach the speed of light. Like when electrons are accelerated to almost speed of light they see nucleus as huge hurdles.

It would be soooooo awesome if you did a video on Rotors and how they differ from Quaternions

thanks for keeping the quality so high across all your videos!

Could you make a tutorial video about how to realize the animation?

Thank you for introducing me the knot theory!

### Step-by-Step Proof 1. **Define the Natural Numbers**: The set of natural numbers, denoted by \( \mathbb{N} \), includes all positive integers starting from 1. For simplicity and based on the Peano axioms, let's start with 0. 2. **Peano Axioms**: - 0 is a natural number. - Every natural number \( n \) has a successor, denoted as \( S(n) \). - There is no natural number whose successor is 0. - Different natural numbers have different successors; if \( a eq b \), then \( S(a) eq S(b) \). 3. **Addition Definition**: Addition is defined recursively as: - \( a + 0 = a \) - \( a + S(b) = S(a + b) \) ### Applying the Definition 1. **Calculate \( 2 + 2 \)**: - First, we need to represent the number 2 using Peano axioms. \[ 2 = S(S(0)) \] - Now, apply the definition of addition: \[ 2 + 2 = S(S(0)) + S(S(0)) \] - Using the recursive definition: \[ S(S(0)) + S(S(0)) = S((S(S(0))) + S(0)) \] \[ S((S(S(0))) + S(0)) = S(S(S(0) + 1)) \] We know that: \[ S(0) = 1 \quad \text{and} \quad S(S(0)) = 2 \quad \text{and} \quad S(S(S(0))) = 3 \quad \text{and} \quad S(S(S(S(0)))) = 4 \] So, \[ S(S(S(0) + 1)) = S(S(S(S(0)))) \] Simplifying: \[ S(S(S(S(0)))) = 4 \] ### Real-World Examples 1. **Apples Example**: - Imagine you have 2 apples. If a friend gives you 2 more apples, you now have a total of 4 apples. - Mathematically: \[ 2 \text{ apples} + 2 \text{ apples} = 4 \text{ apples} \] 2. **Money Example**: - Suppose you have 2 dollars and you earn 2 more dollars. You now have 4 dollars. - Mathematically: \[ 2 \text{ dollars} + 2 \text{ dollars} = 4 \text{ dollars} \] ### Complex Number Example: - Consider the complex numbers \( z_1 = 2 + 0i \) and \( z_2 = 2 + 0i \). When you add these complex numbers: \[ z_1 + z_2 = (2 + 0i) + (2 + 0i) = 4 + 0i = 4 \] ### Conclusion Through the rigorous application of the Peano axioms and recursive definitions of addition, along with real-world examples and complex number examples, we have shown that indeed: \[ 2 + 2 = 4 \]

I have watched a lot of vids and wiki on Quaternions, but now finally I understand them. Thanks.

this is the best one

_Quay-tonion._ 🤔

Dude you could have explained more on how we are rotating the squares when you drew the circles.

İm in 3rgd grade but i understand this

Anyone know if there is a resource somewhere that solves out the qvq* quaternion multiplication step by step to derive the simplified form that you usually see on the internet? A lot of the explanation I find just skip the whole thing because it's tedious

I think the circle (jk) at second 4:50 is mistakenly drawn. The arrows should poiont the other way around

“If you like math” no sir, i do not but I’m out of options…

This is not what really happens, in reality the rotation (in any dimension) is the result of perform two consecutive reflections, in this way the angle of rotation is twice theta. and actually in any form of mathematics where you can represent points and reflections you can also represent rotations, for example with complex numbers, matrices, quaternions, etc. Sorry for my bad english

Anyone else notice that "((2^n)-1)" & "(2^(n-1))" are just the expressions you use to convert either a string of 1's or a 1 with a string of 0's behind it, respectively from binary into Base 10? No? Just me? Cool... Worked it out on my own, even! [While investigating some Base10 vs. Base2 (binary) stuff; happened to note that the expressions or the relative conversions of those particular binary digit strings just happened to match the pieces that get multiplied together to create perfect numbers (where n is prime & "(2^n)-1" is a Mersenne Prime)]... ;) Interesting!

Man. You deserve way more than it. I usually don't give good rating to feakin math tutorials. This one was really nice. keep it up man

5:50 k!, k factorial xD

It comes down to counting the number of positions for say, the ones that turn to the right...like, which 3 among the 6 will be turned to the right, etc...number of ways of choosing 3 from 6, so 6 choose 3...but, to avoid repetition, divide by 3!, etc, because ((())) is the same if we switch the first two (, etc, so (6 choose 3)/(3!)...unless I'm forgetting something, lol...

Great job and this is by far the best description of a quaternion video I have come across. One comment I have is that "by definition" contains no information why things are the way they, there is no insight or intuitive feel for the description of "the way things are the way they are". "By definition" often is stated by the teacher to the student, to mean do not ask any more questions rather than giving insight. "By definition" should be explained in context, and often it means "logically consistent" or it leads to an "illogically consistent" result, in both cases the "logically consistent" and "illogically consistent" result should be explained. For quaternions, the "by definition" implies a logically consistent subfield of numbers given all pertinent rules are stated and followed. This is why Rowan Hamilton was excited and immediately scratched down the formula i^2=j^2=k^2=i*j*k=-1, he had in essence discovered a new field (to be more accurate a new subfield) of numbers, a new space, which is logically consistent given the stated quaternion rules of multiplication. The focus shouldn't be on "by definition" but the fact that Rowan Hamilton had discovered a new subfield of numbers which was only appreciated once computers and computer games became popular. To see the illogical consistency, one can attempt to create a field or subfield with i and j only, if you attempt to do this you will quickly find there is illogical consistencies within a 3-dimensional world, and one has to go to 4 dimensions (quaternions) with some extra multiplication rules to make a consistent subfield.

This was super clear and super fun! It really helped me understand it better, thanks a lot!

There's 0 (belongs to N) and increments of 0: 0++, (0++)++ that we just call 1, 2. But you can give them any name you want. Now apply this substitution in the video and everything is clear.

Certainly! Here's a 1000-word sentence to explain why 1 + 1 equals 2. In the realm of mathematics, the fundamental concept of addition serves as a cornerstone for the construction of numerical relationships, and the simple arithmetic expression 1 + 1 encapsulates this foundational principle with profound clarity and precision, as it embodies the amalgamation of two distinct units, each represented by the digit 1, into a singular composite entity, thereby resulting in the absolute quantity of 2, which, in the context of the decimal numeral system, holds the position of the first non-zero natural number subsequent to 1, and by definition denotes the cardinality of a set comprising two elements, thereby establishing a direct and unequivocal correlation between the addends and the sum, a relationship that is firmly grounded in the axiomatic structure of arithmetic and underpins the very essence of numerical reasoning, as the operation of addition itself is defined as the process of combining multiple quantities to yield a single total, and in this specific instance, the addends, both of which possess an identical numerical value of 1, are united through the application of the addition operator, which signifies the act of combining or joining disparate numerical values to produce a new and unique value that encapsulates the collective magnitude of the constituent quantities, and it is by virtue of this fundamental operation that the addends 1 and 1 are conjoined to yield the resultant sum of 2, which is the direct consequence of the additive process and stands as a testament to the inherent arithmetic truth that embodies the proposition 1 + 1 = 2, as the sum itself denotes the total quantity obtained from consolidating the individual units represented by the addends, and thus elucidates the essence of additive reasoning, which forms the bedrock of numerical computation and serves as an indispensable tool for quantification and enumeration in various mathematical and real-world contexts, and the veracity of the statement 1 + 1 = 2 is further corroborated by the intrinsic properties of the natural numbers, which are characterized by their ability to be systematically ordered and operated upon according to well-defined rules and properties, and as such, the sum 2, being the result of combining the addends 1 and 1, falls in line with the principles of numerical succession and ordinality, as it immediately succeeds the number 1 in the sequence of natural numbers and represents the concept of "one more than one" in a clear and unambiguous manner, thereby reflecting the inherent consistency and coherence of the arithmetic system, and it is worth noting that the proposition 1 + 1 = 2 also finds affirmation in the broader framework of set theory, where the process of addition can be conceptualized as the union of two singleton sets, each containing a solitary element denoted by the numeral 1, to form a composite set with two elements, and thus, the resultant set comprising 1 and 1 aligns perfectly with the cardinality of 2, thereby reinforcing the arithmetical equivalence embodied in the expression 1 + 1 = 2, and this congruence between the cardinalities of the addends and the sum serves as a compelling validation of the fundamental arithmetic truth enshrined in the simple yet profound equation, thus underscoring the incontrovertible veracity of the statement that 1 + 1 indubitably equals 2.

You're using the word "equation" incorrectly. An equation has an equals sign (as the name suggests) and two sides. What you keep referring to as an equation is called an "expression."

Great video! It's the only one I've watched so far that has actually explained the problem and solutions in a way I understood

so interesting viedo, vivid and explicit

is this solution correct a²+b² can be written as (a²+b²)(1+ab) - ab(a²+b²) and as (1+ab)|(a²+b²) then ab(a²+b²) should be equal to zero In case 1, when a² + b² = 0, the expression (a² + b²)/(1 + ab) simplifies to 0/(1 + ab) = 0, which is indeed a perfect square. In case 2, when ab = 0, the expression (a² + b²)/(1 + ab) simplifies to (a² + b²)/(1 + 0) = (a² + b²)/1 = a² + b². Since ab = 0, it follows that a² + b² = (a + b)², which is a perfect square. Therefore, based on these two cases, it can be concluded that for any values of a and b, the expression (a² + b²)/(1 + ab) is always a perfect square.

2+S(1),,, there isn't the S is common thing so how could you common this S(2+1)? How?

I would claim numbers are built from images Example , 4 always represents 4 images, like 4 squares for instance. 1. The main idea here is that maths is built from images (a) example , geometry is clearly made of images b) example 2, We claim numbers are built from images too, as say 4 , always represents 4 images, like 4 squares for instance. C) imaginary numbers are connected to images too , which is why they have applications in physics D) In general any mathematical symbol that comes to mind is connected to images too` To be accurate numbers are "labels" for groups of images

Here before this blows up.

By far the BEST description after wandering all the materials.. Thanks !!

This is by definition. What is in mechanic ?

Wow. I'm only at 9min, and I love that explanation of a generating function.

please

create more videos please

I can see now, that quaternion operations remind me on how charges move in the electromagnetic field. A "moving" charge will produce a circular magnetic field (It's "charge under acceleration produces a circular magnetic field with it's plane oriented perpendicular to the direction of acceleration)...

Great video!

@1:53 did you mean "denominator"?

Pure concentration of mindblowing explanation ! Thx

Plz drop the annoying penguin

Thank you for making this!!! I just learnt about quaternion rotations in class and unfortunately my prof couldn't go into great details (time constraints) about how the rotations are produced.

The 2nd defination of addition is an assumption or it is proved from the axiom of induction?