Transcendental Numbers are Uncountable

Theorem

The set of transcendental real numbers is uncountable.

Proof

By definition, a transcendental number (in this context) is a real number which is not an algebraic number.

Recall that the Real Numbers are Uncountable.

Also recall that the Algebraic Numbers are Countable.

The result follows from Uncountable Set less Countable Set is Uncountable.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 11000 10000 00000 00000 00010 00000 00000 00000 0 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00000 \, 00000 \, 00000 \, 0 \ldots$