1. The first amazing math fact is that there are infinitely many prime numbers. This was proven by Euclid around 300 BC. The surprising part is that no matter how far you go along the number line, you will always find more primes. The proof uses a clever contradiction argument: assume a finite list of primes, multiply them all together and add 1, then this new number is either prime or divisible by a prime not in the list, contradicting the assumption. 2. The second amazing fact is Euler's identity: $$e^{i\pi} + 1 = 0$$. This formula links five fundamental mathematical constants: the base of natural logarithms $e$, the imaginary unit $i$, the number $\pi$, 1, and 0. It is surprising because it connects seemingly unrelated areas of math: exponential functions, complex numbers, and trigonometry, in a simple and elegant equation. These facts are amazing because they reveal deep and unexpected connections in mathematics, showing the beauty and unity of the subject.