There are infinitely many irrational numbers between 0 and 1.
While both rational and irrational numbers exist in infinite quantities between 0 and 1, the "size" of infinity representing irrational numbers is larger than the "size" of infinity representing rational numbers. This is because the set of rational numbers is countable, meaning they can be put into a one-to-one correspondence with the natural numbers (1, 2, 3,...). The set of irrational numbers, however, is uncountable.
Examples of Irrational Numbers Between 0 and 1:
- √2 / 2 (approximately 0.707)
- π / 4 (approximately 0.785)
- 1 / √3 (approximately 0.577)
- Any number of the form 0.a₁a₂a₃..., where the decimal expansion neither terminates nor repeats, such as 0.1010010001...
Why are there infinitely many?
Suppose you have an irrational number, x, between 0 and 1. For instance, let's say x = 0.1010010001... (as above). Then x/2 is also an irrational number between 0 and 1. In fact, dividing any irrational number by any integer greater than 1 always results in another irrational number. You can repeat this process indefinitely, each time generating a new irrational number. Also, if you have two different irrational numbers a and b, where a<b, the number (a+b)/2 will always be another irrational number between a and b, and thus also between 0 and 1. Again, this process can be repeated indefinitely.
Rational vs. Irrational
Despite the infinity of rational numbers between 0 and 1, like 1/2, 1/3, 2/3, 1/4, 3/4, and so on, there's a "denser" infinity of irrational numbers. In any interval, no matter how small, you can always find an irrational number. This means that they are more abundant in a certain mathematical sense.
