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What is a Liouville number?
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Liouville number. In number theory, a Liouville number is a real number with the property that, for every positive integer , there exists a pair of integers with such that. Liouville numbers are "almost rational ", and can thus be approximated "quite closely" by sequences of rational numbers.
5 days ago · A Liouville number is a transcendental number which has very close rational number approximations. An irrational number beta is called a Liouville number if, for each n, there exist integers p>0 and q>1 such that 0<|beta-p/q|<1/(q^n).
Liouville's constant, sometimes also called Liouville's number, is the real number defined by L=sum_(n=1)^infty10^(-n!)=0.110001000000000000000001... (OEIS A012245). Liouville's constant is a decimal fraction with a 1 in each decimal place corresponding to a factorial n!, and zeros everywhere else.
Liouville number, in algebra, an irrational number α such that for each positive integer n there exists a rational number p/q for which p/q < |α − (p/q)| < 1/qn. All Liouville numbers are transcendental numbers—that is, numbers that cannot be expressed as the solution (root) of a polynomial.
- William L. Hosch
- Transcendental Number
- Algebraic Number
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- More Transcendental Numbers
- Transcendental Numbers Are Common
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A Transcendental Number is any number that is not an Algebraic Number Examples of transcendental numbers include π (Pi) and e (Euler's number).
What then is an Algebraic Number? We can imagine all kinds of polynomials: 1. x − 1 = 0 has x = 1, 2. x + 1 = 0 has x = −1, 3. 2x − 1 = 0 has x = ½, 4. x2 − 2 = 0 has x = √2, 5. and so on All integers, all rational numbers, some irrational numbers(such as √2) are Algebraic. In fact it is hard to think of a number that is notAlgebraic. But they do e...
Back in 1844, Joseph Liouvillecame up with this number: It is a very interesting number because: 1. it is irrational, and 2. it is not the root of any polynomial equation and so is not algebraic. In fact, Joseph Liouville had successfully made the first provable Transcendental Number. That number is now known as the Liouville Constant. and is in th...
It took until 1873 for the first "non-constructed" number to be proved as transcendental when Charles Hermite proved that e (Euler's number) is transcendental. Then in 1882, Ferdinand von Lindemann proved that π (pi) is transcendental. In fact, proving that a number is Transcendental is quite difficult, even though they are known to be very common ...
Most real numbers are transcendental. The argument for this is: 1. The Algebraic Numbers are "countable" (put simply, the list of whole numbersis "countable", and we can arrange the algebraic numbers in a 1-to-1 manner with whole numbers, so they are also countable.) 2. But the Real numbers are "Uncountable". 3. And since a Real number is either Al...
In a similar way that a Transcendental Number is "not algebraic", so a Transcendental Function is also "not algebraic". More formally, a transcendental function is a function that cannot be constructed in a finite number of steps from the elementary functions and their inverses. An example of a Transcendental Function is the sine function sin(x).
Mar 24, 2023 · The fact that a Liouville number is transcendental (cf. Transcendental number) follows from the Liouville theorem (cf. Liouville theorems ). These numbers were studied by J. Liouville [1] . Examples of Liouville numbers are: $$\alpha_1=\sum_ {n=1}^\infty2^ {-n!},$$. $$\alpha_2=\sum_ {n=1}^\infty (-1)^n2^ {-3^n},$$.
Transcendental Numbers. Benjamin Church. December 3, 2022. Contents. 1 Introduction. 2 Algebraic Numbers and Cantor’s Theorem. 3 Diophantine Approximation. 4 Irrationality Measure. 5 Liouville Numbers. 2. 3. 5. 6. 1 Introduction. The rational numbers (Q) are incomplete in two diferent ways.
