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nworm
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Dear Experts.
Do you know some literature on
representation of rational numbers by formulas? TIA.
Do you know some literature on
representation of rational numbers by formulas? TIA.
x-1 introduces no new complexity in B3. Starting with a number and then inverting it in B3 will introduce new complexity. However, maybe there's someway to start with a number y that requires less complexity than it takes to realize x itself, and then do some things to y that end up with x-1, the end result requiring the same complexity as B2. So B3 gives x-1, but does so in a roundabout way that ends up taking the same complexity. However, you were asked to prove that LB2(r) = LB3(r) for any r. So why would you say that we can't create x-1?nworm said:Thank you very much These ideas are very interesting
I am thinking that we can't create x-1 by using B3.
Yes, that's correct.As function (x-1 + y-1)-1 is expressed in B2 as noniterated superposition, then any formula in B3 can be transformed to formula in B2 without changing of complexity. This implies that LB3(r) >= LB2(r).
I see, very nice. So we are using induction. G (in B3) has the same complexity as F (in B2) as long as the corollary formulas can be made in B3 with the same complexity. But they can since each corollary formula Fi is again just in the form (Fi1-1 + ... + Fin-1)-1 where formulas Fij are corollary to Fi, and hence have even smaller complexity. So using the exact same method, Gi can be made with the same complexity assuming each Gij can be made, and we subsequently look at corollaries of smaller and smaller complexity, until we inevitably reach our base case for our induction. In this case, we're using strong induction.If a corollary formula of a formula in B2 is given by F-1, then this corollary formula is represented as (F1-1+F2-1+...+Fn-1)-1, where Fi are corollary formulas of less complexity or constants 1. Write a new formula G.
G=((...((F1-1+F2-1)-1)-1+F3-1)-1)-1+...+Fn-1)-1 = g(g(...g(g(F1,F2),F3)...),Fn), where g(x,y)=(x-1+y-1)-1.
The formula F and G are equivalent. The complexity of G is the same as the complexity of F. By iterating this transformation we can find a formula H in B3. The complexity of H is the same as the complexity of F. Therefore LB3(r) <= LB2(r). Therefore LB3(r) = LB2(r).
Sorry, I've never seen anything like this.I am repeating my question. Do you know some literature about this (about rational numbers)
I didn’t read a lot of books in the field of number theory (I read “A Course in Number Theory and Cryptography” by Neal Koblitz only).
The representation of rational numbers by formulas refers to the use of mathematical expressions or equations to represent or describe a rational number. This allows for a more concise and systematic way of representing rational numbers, making it easier for scientists to work with them in various calculations and analyses.
Having a representation of rational numbers by formulas is important because it allows for a standardized and universal way of representing these numbers. This makes it easier to communicate and share mathematical ideas and concepts across different fields of study.
There are several types of formulas used to represent rational numbers, such as fractions, decimals, and percentages. Other common representations include ratios, proportions, and mixed numbers.
Scientists use formulas to represent rational numbers in various ways, depending on their specific field of study. For example, in physics, scientists may use formulas to represent measurements and calculations involving rational numbers, while in chemistry, formulas are used to represent the ratios of different elements in a compound.
No, irrational numbers cannot be represented by formulas because they cannot be expressed as a ratio of two integers. For this reason, they are typically represented as decimal numbers or in other non-formulaic ways such as square roots or pi.