# Liouville number

As Liouville number , named after Joseph Liouville , is known in the theory of numbers a real number that satisfies the condition that for all positive integers integers and with there, so ${\ displaystyle x,}$ ${\ displaystyle n}$${\ displaystyle p}$${\ displaystyle q}$${\ displaystyle q> 1}$

${\ displaystyle 0 <\ left | x - {\ frac {p} {q}} \ right | <{\ frac {1} {q ^ {n}}} \.}$

## Irrationality and transcendence

All Liouville numbers are irrational : For every rational number with an integer numerator and a positive integer denominator there is a positive integer with If now and integers with and are, then is ${\ displaystyle x = {\ tfrac {c} {d}}}$${\ displaystyle c}$${\ displaystyle d}$${\ displaystyle n}$${\ displaystyle 2 ^ {n-1}> d.}$${\ displaystyle p}$${\ displaystyle q}$${\ displaystyle q> 1}$${\ displaystyle {\ tfrac {p} {q}} \ neq {\ tfrac {c} {d}}}$

${\ displaystyle \ left | x - {\ frac {p} {q}} \ right | = \ left | {\ frac {c} {d}} - {\ frac {p} {q}} \ right | = \ left | {\ frac {c \, qp \, d} {d \, q}} \ right | \ geq {\ frac {1} {d \, q}}> {\ frac {1} {2 ^ {n-1} q}} \ geq {\ frac {1} {q ^ {n}}} \.}$

In 1844, Liouville showed that numbers with this property are not only irrational but also transcendent . This was the first proof of the transcendence of a number, Liouville's constant:

${\ displaystyle c = \ sum _ {j = 1} ^ {\ infty} 10 ^ {- (j!)} = 0 {,} 11000 {\ text {}} 10000 {\ text {}} 00000 {\ text {}} 00000 {\ text {}} 00010 {\ text {}} \ ldots}$ (Follow A012245 in OEIS )

All Liouville numbers are transcendent, but not all transcendent numbers are Liouville. For example, Euler's number is transcendent, but not Liouvillesch.