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This result allowed him to produce the first proven examples of transcendental numbers.

With an infinity of transcendental numbers, they would never know for sure which number to examine back on Earth.

If there was content inside a transcendental number, it could only have been built into the geometry of the universe from the beginning.

They had not inserted the message into the transcendental number, and could not even read it.

This allowed Liouville, in 1844 to produce the first explicit transcendental number.

This is already enough to demonstrate the existence of transcendental numbers.

He was the first to prove that e, the base of natural logarithms, is a transcendental number.

The existence of transcendental numbers was first established by Liouville (1844, 1851).

At the other extreme, if a is a transcendental number, then T(a) has no equidissection.

He also gave a new method for constructing transcendental numbers.