0 Members and 1 Guest are viewing this topic.

We can represent any rational numbers by combining two integers as numerator and denominator. Some irrational numbers can be stated as a rational number powered by another rational number. Let's call them power numbers. How many more expansion procedures like that are required to cover the whole real numbers?

Quote from: hamdani yusuf on 02/07/2022 15:41:12To be clear,Looks like another never ending, merry-go-round thread by Hamdani to put on ignore. Bye and have fun.

To be clear,

I honestly don't know. It depends what you consider to be an "expansion procedure like that".

For example, would you allow taking limits? If {Xn} is a sequence of numbers you already admit in your set, then why not also include the Real number X whenever the sequence Xn → X. That seems like a reasonable expansion procedure. Anyway, that expansion procedure gets you straight from Q to R in one step.

Is there a set of numbers which contains more than algebraic numbers but less than real numbers?

∞^{∞} = ∞

∞−∞ and ∞/∞) are not generally well-defined.

https://en.m.wikipedia.org/wiki/Transcendental_numberIn mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are π and e.[1][2]Though only a few classes of transcendental numbers are known — partly because it can be extremely difficult to show that a given number is transcendental — transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers comprise a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic.[3][4][5][6] The converse is not true: not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping rational, algebraic non-rational and transcendental real numbers.[3] For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0. The golden ratio (denoted {\displaystyle \varphi }\varphi or {\displaystyle \phi }\phi ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x2 − x − 1 = 0. The quality of a number being transcendental is called transcendence.

https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry)The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable.The set of all periods is countable, and all periods are computable,[6] and in particular definable.Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods".

https://en.m.wikipedia.org/wiki/Transcendental_functionIn mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.[1][2] In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and root extraction.[3]

https://en.m.wikipedia.org/wiki/Computable_numberIn mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers[1] or the computable reals or recursive reals.

https://en.m.wikipedia.org/wiki/Continued_fractionIn mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.[1] In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers {\displaystyle a_{i}}a_{i} are called the coefficients or terms of the continued fraction.[2]

https://en.m.wikipedia.org/wiki/Transcendental_numberIn 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is a^{b} necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).[18]

If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is ab necessarily transcendental?

Here are what I got after go down the rabbit hole.

However, 2^{∞} > ∞(At least for the counting numbers), so this one seems suspect.

If you could divide infinitely, we might ask how would anything get anywhere at a fundamental length

Hi.Quote from: hamdani yusuf on 03/07/2022 13:12:23If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is ab necessarily transcendental?That has a significant copy-and-paste error. ab should be a^b = a^{b}Best Wishes.

Quote∞^{∞} = ∞However, 2^{∞} > ∞(At least for the counting numbers), so this one seems suspect.Quote from: Wikipedia∞−∞ and ∞/∞) are not generally well-defined.In practice, this means that you need more information to determine the answer.For example, L'Hopital's rule allows you to calculate Limit of a/b as a→∞ and b→∞in those scenarios where you know the derivative of a and b (and both aren't infinite).

Time stamps:0:00 - Fake sphere proof1:39 - Fake pi = 4 proof5:16 - Fake proof that all triangles are isosceles9:54 - Sphere "proof" explanation15:09 - pi = 4 "proof" explanation16:57 - Triangle "proof" explanation and conclusion

If the numerator and denominator are allowed to be infinite, then any real number can be expressed as ratio between two integers with infinite precision. The same thing can't be done for complex numbers or hypercomplex numbers.

For a start,

When expressed as ratio, 1/2, 1/3, and 1/4 don't seem to need much different amount of information. But when expressed in decimal, 1/3 needs infinite number of digits. Although the decimal digits are repetitive, which make the information compressible.

If the numerator and denominator are allowed to be infinite, then any real number can be expressed as ratio between two integers with infinite precision.

Quote from: hamdani yusuf on 07/07/2022 09:26:35When expressed as ratio, 1/2, 1/3, and 1/4 don't seem to need much different amount of information. But when expressed in decimal, 1/3 needs infinite number of digits. Although the decimal digits are repetitive, which make the information compressible.Use base 12 rather than 10.The problem goes away.