How do we know that there are infinitely many transcendental numbers?

Question

How do we know that there are infinitely many transcendental numbers?

In: 4

Answer 1

If you double a transcendental number, you get a transcendental number. This is because if you half an algebraic number (not transcendental), you get an algebraic number. Why? An algebraic number x satisfies a polynomial equation with rational coefficients ax^n +bx^n-1 +…=0, so x/2 satisfies the polynomial (2^n a)(x/2)^n +(2^n-1 b)(x/2)^n-1 +…=0, so it also satisfies a polynomial with rational coefficients.

So keep multiplying a transcendental number by 2 to get infinitely many transcendental number.