It's an interesting idea to try to define ln(-1) as j, an entirely new number system. But if you try and follow basic rules, then ln(-1)+ln(-1)=ln(1)=0 j+j=0 j=0 But if you're proposing this is an entirely new system that doesn't follow standard arithmetic rules, then does this solution to x/x=-1 work? It seems to be using j with standard arithmetic rules, which leads to contradictions. If you don't want contradictions and want to create an entirely new system, it might be better off defining what you're allowed to do and what you can't do with j.
@@log_menus_1 Ok, that's interesting. So you propose 2j=0 doesn't mean you can divide both sides by 2? Then it wouldn't follow that 2ipi=0, because in this case, you CAN divide by 2, giving ipi=0, which isn't true. This means that you might need to try redefining what it means to "multiply", so that the definition allows you to deal with both normal numbers like pi or 2 or i, and your new number j, and even k. For example, if 2j=0 but j isn't 0, what does the 2 mean? Without the 2, j=ln(-1), but with it, 2j=0. Can you find some rule to figure out what 3j is, then what 4j is, etc. without any contradictions arising? Are you able to do the same with K? How about numbers like j^k? Right now, your theory is interesting, but it isn't standing on many foundations. People are pointing out "there's contradictions" or "there's inconsistencies", because the rules for what you can and can't do with j and k are unclear. If you can figure out what these rules are, and prove they're perfectly consistent, then you'll have a theory that some experts might start considering, and something maybe no one has done before. If you fail, maybe it wasn't possible, but it would be one of the greatest learning experiences, and you'd have a much deeper understanding of why you can't divide by 0 than many other people who just read textbooks. Learning from actually researching and proving things yourself is incredibly valuable and something not many people can do. And if you're curious about how you're even supposed to build these rules for j and k, you might want to look into areas of math called "foundations of mathematics", "mathematical logic" or "proof theory". These areas deal with axiomatic systems, or those rules you place, and how to prove they're reliable rules.
There is a solution to your equation in infinitely many fields. Since x/x = 1 in any field and for any non-zero value of x, if you want x/x = -1 then you must have -1 =1. That is true in any field with characteristic 2: in other words, 1 + 1 = 0. This is true on the residue classes modulo 2, often written as ℤ₂ or ℤ/[2]. In this field, the only values are 0 and 1, with 1+1 = 0 and 1*1 = 1. There are many other such fields; for example there is one with a size of any power of 2, and there are infinite-sized such fields. I do not know the "virtual number" system that you use, but it seems not to be a field (a number system with the usual, base properties such as closure, commutativity, and associativity of addition and multiplication as well as the distributive property of multiplication over addition).
@@log_menus_1 I watched that video after posting my comment. I found the video hard to understand, since you use some properties of the Real and Complex numbers but ignore others, and it is not clear which properties you are keeping, which ones you are not, and which new properties you are adding. (Those things are clear for a few properties but not for all.) Is there a reference that gives the axioms for the Virtual Numbers?
I understand how the use of familiar properties from real and complex numbers, along with new concepts like Virtual Numbers, can create some confusion. The goal is to establish a new framework that extends beyond the traditional number systems. I'll be sure to clarify the axioms of Virtual Numbers in future videos and provide more concrete examples.
No, all numbers. Every number is complex. Complex numbers can be expressed in polar or standard form, so we’ll go with standard: a+bi 6=6+0i has a real and imaginary part and is therefore complex. You must think that a and/or b can’t be zero, but even in that case, x/x is always 1 Plus, that wasn’t even the question
One must be careful not to dismiss these numbers --- I know they sound absurd . All you have to do is go back to the history of mathematics . When negative numbers and imaginary numbers were first introduced , they were both dismissed ; but , they were both found to be very useful .
Unfortunately brother your division does not associate with multiplication, Since j/j = -1 it should follow that j(j/j) = -j. If it associates we would have (j*j)/j = -j so 0/j = -1 (for now no problem). However if we try to compute now (0j)/j we have (0j)/j = 0/j = -j 0(j/j) = 0(-1) = 0, so it would follow j = 0 which is not true in your number system.
Indeed the deeper reason why this doesn't work is because your algebra of virtual numbers (which is just the dual numbers, they are isomorphic) has j as a zero divisor, so you can't multiply by it
The question was to find a number that satisfies the equation x/x = -1. Nowhere was it stated that j/j = -1. Instead, the value satisfying the equation was calculated step by step.
Please try again to understand more clearly. This equation was solved using singularity numbers, not virtual numbers. I only mentioned virtual numbers because they vanish when divided by 0/j, so then I solved it using singularity numbers. Don’t confuse the two- they are different systems.
@@log_menus_1 sure but if x/x = -1 is solved by j then j/j =-1. It is possible to give other interpretations however, which seems to be what you've done
Why do we introduce complex numbers when x² is always positive, and why do we assume x² could be negative in these contexts? In mathematics, there are no absolutes-every equation and operation has its limits and boundaries
I understand that the concepts of Virtual Numbers and Singularity Numbers might seem unconventional at first, especially when compared to more familiar ideas like imaginary numbers. The goal here is to explore new ways of thinking about numbers and challenge traditional frameworks.
@log_menus_1 Traditional framework is just fine and it's the base for everything. Imaginary, singularity, etc numbers belong to marvel universe and serve other purposes.
Замечательно! Напоминает старого доброго почтальона Печкина (новый мне совсем не зашёл): я вам принёс посылку, но я вам её не дам! е у нас в степени, но эта степень - результат деления на ноль, а на ноль делить нельзя, так что в степень возводить тоже нельзя. Т. е. решение есть, но его как бы и нет.
Знаете, в математике и реальном мире нет ничего, что было бы по-настоящему неопределённым. Всё имеет своё значение, даже если это выходит за пределы привычного. Например, мнимая единица 'i' встречается в уравнении Шрёдингера и играет важную роль в физике. Хотя для некоторых комплексные числа могут казаться 'неопределёнными', они имеют реальную ценность и помогают нам лучше понять вселенную
@@log_menus_1 в комплексных числах есть смысл когда какая-то практическая польза от них. Например, сложить ε^j c π, помножить на √3, извлечь из всего этого ещё корень i-той степени и дальше подставить в уравнение с е. На выходе получаем алгоритм работы нейросети-автоответчика, ракету, холодный ядерный синтез, лекарство от рака, гипердвигатель, наносортировочную машину какую-нибудь... А держать просто числа ради чисел - бессмыслица.
Хотя сейчас может показаться, что комплексные числа не имеют практического применения, каждая новая изобретение или открытие в конечном итоге находит своё применение в будущем. То, что что-то не имеет немедленного применения, не означает, что оно не будет полезно в долгосрочной перспективе. Например, мы пока не можем приземлиться на Марсе или путешествовать за пределы галактики с текущей математикой, но более продвинутые математические концепции именно те, что приведут к этому прогрессу. Мы не можем игнорировать потенциал сингулярных чисел, поскольку они могут быть ключом к новым прорывам в науке и технологиях.
@@log_menus_1 и всё же, возвращаясь, так сказать, к напечатанному, я не считаю, что e^j так уж далеко ушло от ответа "решений нет" (или *nil* языком программирования, если угодно) - это фактически одно и то же. Даже для 1^x=2 есть более-менее осмысленное число е^(пi+1) или как-то так. А тут не пойми что. Если бы е^j давало какие-то действительные числа после преобразований - я ещё понял, потому что выражения с i достаточно легко в таковые конвертируются.
No, sorry. Just like multiplication is repetitive addition, division is repetitive subtraction. A divided by B is the number of times you can subtract B from A until you reach 0. So X divided by X is the number of times x is subtracted from x to equal 0. It is 1 and only 1. It can never be -1. Basically you're asking us to solve an equation that is incorrect. Not a valid question.
If I had asked you to solve the equation x² = -1 before the discovery of complex numbers, you would have said the question is invalid because no number satisfies that equation. Similarly, my question is: find x if x/x = -1. This is analogous to asking for x when x² = -1 before complex numbers were introduced. The point is, my question is valid-you just don’t have an answer for it in the complex numbers framework you're using.
@@log_menus_1 My man, complex numbers are a model that works mathematically and is consistent with basic arithmetic and geometric rules. You just made up a random ass number, said it works, and then broke math. You can always introduce new definitions in mathematics, but you MUST make sure that they are consistent or at least put some sort of boundaries. When using definitions that have some errors, results, at best, need some huge asterisks. However this is complete nonsense. Also are you an AI? Is this a social experiment?
I didn’t break math at all; virtual numbers are consistent and have their own framework where all operations are done, similar to how complex numbers and real numbers each have their own mathematical system. In my introduction to virtual numbers, I showed how they work when we extend the domain of functions from real to virtual. For example, 0! = 1 makes sense if we consider n! = n * (n-1) * (n-2) * ... * 1. When we put n = 0, we get 0! = 0 * (-1) * (-2) * ... * (-n). This results in 0! = 0. Would you say I’m wrong here as well? Also, negative numbers are undefined in the for factorial, but when we extend the domain of integers to complex numbers, we get different results. For instance, (-1/2)! = √π. The results with virtual numbers may seem counterintuitive, but they are correct due to the extension of the real domain to the virtual numbers system. I hope this makes sense.
If I had asked you to solve x² = -1 before the discovery of complex numbers, would you still ask me to specify the domain? My question is straightforward: find x if x / x = -1.
@@log_menus_1 If you do not specify the domain, then symbol "/" has no meaning and the problem is nonsense. And yes, to pose problem x^2=-1 you have to either specify the domain, or to explicitly request to invent new domain in which symbols -1 and ^2 have meaning. Again: your formulation is dishonest and intends to confuse those who do not have serious mathematical education.
The question of whether mathematics is discovered or invented is indeed a long-standing debate. In the case of complex numbers, I believe they were discovered, not invented. While the formalization of complex numbers as we know them today involved human ingenuity, the concept itself already existed, much like negative or irrational numbers. These mathematical ideas were always part of the structure of the universe; we simply needed to recognize and understand them. So, complex numbers weren’t invented-they were uncovered.
@@xgx899 The question is simple: find x if x / x = -1. You can use any domain or even invent a new one, as long as it satisfies the equation, just like how we extended the domain from real numbers to complex numbers.
It's an interesting idea to try to define ln(-1) as j, an entirely new number system.
But if you try and follow basic rules, then
ln(-1)+ln(-1)=ln(1)=0
j+j=0
j=0
But if you're proposing this is an entirely new system that doesn't follow standard arithmetic rules, then does this solution to x/x=-1 work? It seems to be using j with standard arithmetic rules, which leads to contradictions. If you don't want contradictions and want to create an entirely new system, it might be better off defining what you're allowed to do and what you can't do with j.
j is actually i*pi, if you do j+j, or 2*j, you'll get 2*i*pi
Actually it follows the basic arithmetic as well
j + j = 2j = 2ln(-1) = ln(-1)²= ln(1) = 0
Here 2j = 0 not 2 or j individually 0
Also j+j = 0 is true
j = -j
Ln(-1) = - ln(-1)
Ln(-1) = ln(-1)
j= j
Actually -j = j
So it follows basic arithmetic
2j = 0 so it means 2i*pi = 0?
@@log_menus_1 Ok, that's interesting. So you propose 2j=0 doesn't mean you can divide both sides by 2? Then it wouldn't follow that 2ipi=0, because in this case, you CAN divide by 2, giving ipi=0, which isn't true. This means that you might need to try redefining what it means to "multiply", so that the definition allows you to deal with both normal numbers like pi or 2 or i, and your new number j, and even k. For example, if 2j=0 but j isn't 0, what does the 2 mean? Without the 2, j=ln(-1), but with it, 2j=0. Can you find some rule to figure out what 3j is, then what 4j is, etc. without any contradictions arising? Are you able to do the same with K? How about numbers like j^k? Right now, your theory is interesting, but it isn't standing on many foundations. People are pointing out "there's contradictions" or "there's inconsistencies", because the rules for what you can and can't do with j and k are unclear. If you can figure out what these rules are, and prove they're perfectly consistent, then you'll have a theory that some experts might start considering, and something maybe no one has done before. If you fail, maybe it wasn't possible, but it would be one of the greatest learning experiences, and you'd have a much deeper understanding of why you can't divide by 0 than many other people who just read textbooks. Learning from actually researching and proving things yourself is incredibly valuable and something not many people can do.
And if you're curious about how you're even supposed to build these rules for j and k, you might want to look into areas of math called "foundations of mathematics", "mathematical logic" or "proof theory". These areas deal with axiomatic systems, or those rules you place, and how to prove they're reliable rules.
this is a great exercise to test your ability to detect false logic
My answer: x = 0, because 0 divided itself can be any numbers.
Any numbers? 😂😂😂 Can it be 2?
If any number then this will not satisfy the equation but I think only singularity numbers satisfy the equation
Watching this feels like an acid trip.
There is a solution to your equation in infinitely many fields. Since x/x = 1 in any field and for any non-zero value of x, if you want x/x = -1 then you must have -1 =1. That is true in any field with characteristic 2: in other words, 1 + 1 = 0. This is true on the residue classes modulo 2, often written as ℤ₂ or ℤ/[2]. In this field, the only values are 0 and 1, with 1+1 = 0 and 1*1 = 1. There are many other such fields; for example there is one with a size of any power of 2, and there are infinite-sized such fields.
I do not know the "virtual number" system that you use, but it seems not to be a field (a number system with the usual, base properties such as closure, commutativity, and associativity of addition and multiplication as well as the distributive property of multiplication over addition).
Watch virtual numbers video to understand
@@log_menus_1 I watched that video after posting my comment. I found the video hard to understand, since you use some properties of the Real and Complex numbers but ignore others, and it is not clear which properties you are keeping, which ones you are not, and which new properties you are adding. (Those things are clear for a few properties but not for all.) Is there a reference that gives the axioms for the Virtual Numbers?
I understand how the use of familiar properties from real and complex numbers, along with new concepts like Virtual Numbers, can create some confusion. The goal is to establish a new framework that extends beyond the traditional number systems. I'll be sure to clarify the axioms of Virtual Numbers in future videos and provide more concrete examples.
x/x=-1 then x=0
0=-1*0
Because 0/0 is set of all number
We can write 0/0 = -1 as 0= 0×-1
0/0 is not 1..
It's 0k in singularity numbers
solutions for
x/x=1 are all non-complex numbers
No, all numbers.
Every number is complex. Complex numbers can be expressed in polar or standard form, so we’ll go with standard: a+bi
6=6+0i has a real and imaginary part and is therefore complex. You must think that a and/or b can’t be zero, but even in that case, x/x is always 1
Plus, that wasn’t even the question
But question was x/x = -1
I know always x/x = 1 ,
My question was find x if x/x = -1
@@log_menus_1 you’re actually stupid
thanks
When one divides by 0, then anything is true.
No 👎
@@log_menus_1 At 4:50, that's a division by 0.
1/0 is singularity unit which is k
x/x = -1 already in F2
Not sure whats going on here, but it seems suspicious.
One must be careful not to dismiss these numbers --- I know they sound absurd . All you have to do is go back to the history of mathematics . When negative numbers and imaginary numbers were first introduced , they were both dismissed ; but , they were both found to be very useful .
Everytime i see these weird equations i just guess it has something to do with i
I solved it with k
Unfortunately brother your division does not associate with multiplication,
Since j/j = -1 it should follow that j(j/j) = -j. If it associates we would have (j*j)/j = -j so 0/j = -1 (for now no problem).
However if we try to compute now (0j)/j we have
(0j)/j = 0/j = -j
0(j/j) = 0(-1) = 0, so it would follow j = 0 which is not true in your number system.
Indeed the deeper reason why this doesn't work is because your algebra of virtual numbers (which is just the dual numbers, they are isomorphic) has j as a zero divisor, so you can't multiply by it
The question was to find a number that satisfies the equation x/x = -1. Nowhere was it stated that j/j = -1. Instead, the value satisfying the equation was calculated step by step.
Please try again to understand more clearly. This equation was solved using singularity numbers, not virtual numbers. I only mentioned virtual numbers because they vanish when divided by 0/j, so then I solved it using singularity numbers. Don’t confuse the two- they are different systems.
@@log_menus_1 sure but if x/x = -1 is solved by j then j/j =-1. It is possible to give other interpretations however, which seems to be what you've done
@@TC159 No! x/x = -1 is an equation, what I want is to find x to satisfy this equation, I don't think j is a solution to his equation
X,2×+5=8
??
this makes no sense x/x is always equals to 1 why would introducing new variables change anything
Why do we introduce complex numbers when x² is always positive, and why do we assume x² could be negative in these contexts? In mathematics, there are no absolutes-every equation and operation has its limits and boundaries
Pointless, useless, totally absurd. 😂😂😂
If my method is wrong, feel free to try your own method. Find .
@log_menus_1 What method??? As if it wasn't enough absurd with the imaginary numbers now we get the singularity numbers!
I understand that the concepts of Virtual Numbers and Singularity Numbers might seem unconventional at first, especially when compared to more familiar ideas like imaginary numbers. The goal here is to explore new ways of thinking about numbers and challenge traditional frameworks.
@log_menus_1 Traditional framework is just fine and it's the base for everything. Imaginary, singularity, etc numbers belong to marvel universe and serve other purposes.
Замечательно! Напоминает старого доброго почтальона Печкина (новый мне совсем не зашёл): я вам принёс посылку, но я вам её не дам! е у нас в степени, но эта степень - результат деления на ноль, а на ноль делить нельзя, так что в степень возводить тоже нельзя. Т. е. решение есть, но его как бы и нет.
Знаете, в математике и реальном мире нет ничего, что было бы по-настоящему неопределённым. Всё имеет своё значение, даже если это выходит за пределы привычного. Например, мнимая единица 'i' встречается в уравнении Шрёдингера и играет важную роль в физике. Хотя для некоторых комплексные числа могут казаться 'неопределёнными', они имеют реальную ценность и помогают нам лучше понять вселенную
@@log_menus_1 в комплексных числах есть смысл когда какая-то практическая польза от них. Например, сложить ε^j c π, помножить на √3, извлечь из всего этого ещё корень i-той степени и дальше подставить в уравнение с е. На выходе получаем алгоритм работы нейросети-автоответчика, ракету, холодный ядерный синтез, лекарство от рака, гипердвигатель, наносортировочную машину какую-нибудь... А держать просто числа ради чисел - бессмыслица.
Хотя сейчас может показаться, что комплексные числа не имеют практического применения, каждая новая изобретение или открытие в конечном итоге находит своё применение в будущем. То, что что-то не имеет немедленного применения, не означает, что оно не будет полезно в долгосрочной перспективе. Например, мы пока не можем приземлиться на Марсе или путешествовать за пределы галактики с текущей математикой, но более продвинутые математические концепции именно те, что приведут к этому прогрессу. Мы не можем игнорировать потенциал сингулярных чисел, поскольку они могут быть ключом к новым прорывам в науке и технологиях.
@@log_menus_1 и всё же, возвращаясь, так сказать, к напечатанному, я не считаю, что e^j так уж далеко ушло от ответа "решений нет" (или *nil* языком программирования, если угодно) - это фактически одно и то же. Даже для 1^x=2 есть более-менее осмысленное число е^(пi+1) или как-то так. А тут не пойми что. Если бы е^j давало какие-то действительные числа после преобразований - я ещё понял, потому что выражения с i достаточно легко в таковые конвертируются.
No, sorry. Just like multiplication is repetitive addition, division is repetitive subtraction. A divided by B is the number of times you can subtract B from A until you reach 0. So X divided by X is the number of times x is subtracted from x to equal 0. It is 1 and only 1. It can never be -1. Basically you're asking us to solve an equation that is incorrect. Not a valid question.
If I had asked you to solve the equation x² = -1 before the discovery of complex numbers, you would have said the question is invalid because no number satisfies that equation.
Similarly, my question is: find x if x/x = -1.
This is analogous to asking for x when x² = -1 before complex numbers were introduced.
The point is, my question is valid-you just don’t have an answer for it in the complex numbers framework you're using.
@@log_menus_1 My man, complex numbers are a model that works mathematically and is consistent with basic arithmetic and geometric rules. You just made up a random ass number, said it works, and then broke math. You can always introduce new definitions in mathematics, but you MUST make sure that they are consistent or at least put some sort of boundaries. When using definitions that have some errors, results, at best, need some huge asterisks. However this is complete nonsense. Also are you an AI? Is this a social experiment?
I didn’t break math at all; virtual numbers are consistent and have their own framework where all operations are done, similar to how complex numbers and real numbers each have their own mathematical system. In my introduction to virtual numbers, I showed how they work when we extend the domain of functions from real to virtual.
For example, 0! = 1 makes sense if we consider n! = n * (n-1) * (n-2) * ... * 1. When we put n = 0, we get 0! = 0 * (-1) * (-2) * ... * (-n). This results in 0! = 0.
Would you say I’m wrong here as well? Also, negative numbers are undefined in the for factorial, but when we extend the domain of integers to complex numbers, we get different results. For instance, (-1/2)! = √π.
The results with virtual numbers may seem counterintuitive, but they are correct due to the extension of the real domain to the virtual numbers system. I hope this makes sense.
Total fraud. Before solving any equation, domain must be specified.
If I had asked you to solve x² = -1 before the discovery of complex numbers, would you still ask me to specify the domain?
My question is straightforward: find x if x / x = -1.
@@log_menus_1 If you do not specify the domain, then symbol "/" has no meaning
and the problem is nonsense.
And yes, to pose problem x^2=-1 you have to either specify the domain, or to explicitly request to invent new domain in which symbols -1 and ^2 have meaning.
Again: your formulation is dishonest and intends to confuse those who do not have serious mathematical education.
@@log_menus_1😂😂😂😂 "discovery of complex numbers"...
The question of whether mathematics is discovered or invented is indeed a long-standing debate. In the case of complex numbers, I believe they were discovered, not invented. While the formalization of complex numbers as we know them today involved human ingenuity, the concept itself already existed, much like negative or irrational numbers. These mathematical ideas were always part of the structure of the universe; we simply needed to recognize and understand them. So, complex numbers weren’t invented-they were uncovered.
@@xgx899 The question is simple: find x if x / x = -1.
You can use any domain or even invent a new one, as long as it satisfies the equation, just like how we extended the domain from real numbers to complex numbers.
You made so many mistakes. It's not even funny
I haven't, but if I had, what would be the solution to the equation?
I used singularity numbers