Final answer:
The smallest positive integer is 1, while the smallest positive rational and irrational numbers cannot be definitively determined as they can be made increasingly smaller without reaching an absolute smallest value.
Step-by-step explanation:
The smallest positive integer is 1. A positive integer is a whole number greater than zero, and 1 is the first such number in the sequence of natural numbers.
The smallest positive rational number cannot be definitively stated because there is no 'smallest' positive rational number. Positive rational numbers can be made as small as desired by choosing a large enough denominator, for example, 1/2, 1/100, 1/1000, and so forth. However, as long as the numerator and denominator are both positive, the number will be positive, but we can always find a smaller number, and hence cannot define the smallest.
Similarly, there is no 'smallest' positive irrational number. Irrational numbers are numbers that cannot be expressed as a simple fraction, such as π or the square root of 2 (√2), and they can be made arbitrarily close to zero while remaining positive, meaning there is no smallest value.