Rational numbers and irrational numbers together form the real numbers. After Cantor discovered that there could be as many rational numbers as positive integers, which are a subset of rational numbers, he naturally wondered: Could there be as many real numbers as irrational numbers, which are a subset of real numbers? After in-depth research, Cantor reached a negative conclusion. To illustrate Cantor's conclusion, let's first note a fact: The ray emitted from point A makes the points on the semicircle correspond to the points on the real axis one by one, and the perpendicular line makes the points on the interval (0, 1) correspond to the points on the semicircle one by one. Therefore, it can be concluded that the real numbers on the interval (0, 1) are as numerous as all real numbers.
Now, Cantor only needs to prove that the real numbers on the interval (0, 1) cannot be arranged into an infinite sequence like rational numbers. To this end, he gave a very ingenious proof by using the method of reductio ad absurdum: Suppose the real numbers on the interval (0, 1) can be arranged into a sequence, denoted as x₁, x₂, x₃, …. We know that every real number on the interval (0, 1) can be uniquely represented by an infinite decimal, and each xᵢ can be expressed as xᵢ=0.aᵢ₁aᵢ₂aᵢ₃… (i=1, 2, …).
Now, construct a new number x=0.b₁b₂b₃…, such that b₁≠a₁₁, b₂≠a₂₂, b₃≠a₃₃, …, and so on. Of course, the number x constructed in this way is a real number on the interval (0, 1), but it cannot be any of the numbers in the sequence x₁, x₂, x₃, ….
This shows that no matter how the real numbers on the interval (0, 1) are arranged, there will always be at least one number that cannot be arranged into this sequence. Therefore, although there are infinitely many real numbers on the interval (0, 1) and rational numbers, the infinity of the real numbers (that is, the infinity of all real numbers) is greater than the infinity of all rational numbers (that is, the infinity of all natural numbers). Since the real numbers minus the rational numbers are irrational numbers, it is clear that all irrational numbers cannot be arranged into a sequence (because two sequences can always be combined into one sequence), which shows that irrational numbers are more numerous than rational numbers.