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Natural numbers are the most basic form of numbers that you learn as a child. They consist of the whole, positive numbers we use to count objects, starting from 1 and going upwards (1, 2, 3, ...). It is important to note that zero is not universally considered a natural number, although some mathematical definitions include it.

In our exercise example, the natural numbers present within the set \( \{25, -17, \frac{12}{5}, \sqrt{9}, \sqrt{8}, -\sqrt{8}\} \) are 25 and \( \sqrt{9} \), as \( \sqrt{9} \) simplifies to 3, which is also a positive whole number. Remember that even though natural numbers are also integers, not all integers are natural numbers, because integers include negative values as well.

Integers are a broader class of numbers that include all the natural numbers as well as their negatives and zero. To put it simply, integers are all the whole numbers both positive, negative and zero (…, -3, -2, -1, 0, 1, 2, 3, …). The concept of integers is essential because it introduces the notion of negative quantities and zero.

In our exercise, the integers are 25, -17, and \( \sqrt{9} \) which is simplified to 3. Since both positive and negative whole numbers are included in the definition, it's clear why -17 is considered an integer, despite not being a natural number. It's vital to distinguish integers from other number types because integers are not fractions or decimals.

Expanding upon integers, we enter the realm of rational numbers. These numbers can be written as a ratio or fraction where both the numerator and the denominator are integers. The crucial detail here is that the denominator must not be zero, as division by zero is undefined.

In our set, the examples of rational numbers include 25, -17, \( \frac{12}{5} \) and \( \sqrt{9} \) which is 3. Here, it becomes evident that while all integers are rational numbers (since they can be represented as a fraction with 1 as the denominator), not all rational numbers are integers (such as \( \frac{12}{5} \)). Rational numbers can be either positive or negative and they have a decimal representation that terminates or repeats.

Irrational numbers may seem a bit mysterious at first. These numbers cannot be expressed as simple fractions with both the numerator and denominator being integers. The decimal expansion of an irrational number goes on forever without repeating. Many times, irrational numbers involve roots of numbers that are not perfect squares or other non-repeating, non-terminating decimals.

In our set \( \{25, -17, \frac{12}{5}, \sqrt{9}, \sqrt{8}, -\sqrt{8}\} \) the irrational numbers are \( \sqrt{8} \) and \( -\sqrt{8} \). Neither of these can be represented as a fraction of two integers, and as such, they do not have a decimal representation that ends or repeats. They are critical in understanding the complete number system, and students can remember them as the 'non-fractionable' numbers.