Almost All Real Numbers are Transcendental

Theorem

Almost all real numbers are transcendental.

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.

Thus the subset of all real numbers which are not transcendental is countable.

The result follows by the definition of almost all.

$\blacksquare$

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): $0 \cdotp 11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00000 \, 00000 \, 00000 \, 0 \ldots$