I'm an expert in the field of mathematics, particularly with a focus on number theory. Fractions, as you might already know, are a fundamental part of our numerical system. They are a way to express a part of a whole, and they are represented as a ratio of two integers, where the numerator is the part and the denominator is the whole.

In mathematics, a rational number is defined as any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) where p and q are integers and q is not zero. This definition includes all integers because any integer can be written as a fraction with a denominator of 1. For example, the integer 5 can be expressed as \( \frac{5}{1} \). It also includes finite decimals and repeating decimals, as they can be converted into fractions. For instance, the repeating decimal 0.262626... can be expressed as \( \frac{26}{99} \), and the finite decimal 0.241 can be expressed as \( \frac{241}{1000} \).

On the other hand, an irrational number is a real number that cannot be expressed as a simple fraction. This means it cannot be written as a ratio of two integers. Irrational numbers are non-repeating, non-terminating decimals that do not have a pattern that repeats indefinitely. Examples of irrational numbers include the mathematical constants π (pi), e (Euler's number), and the square root of any non-perfect square, such as \( \sqrt{2} \) or \( \sqrt{3} \).

It is important to note that the distinction between rational and irrational numbers lies in their representation. While rational numbers can be represented as a ratio of two integers, irrational numbers cannot. This is a fundamental concept in mathematics and is crucial for understanding the properties of numbers and their operations.

In summary, fractions are rational numbers because they can be expressed as a ratio of two integers. Rational numbers encompass a wide range of numbers, including integers, finite decimals, and repeating decimals. Irrational numbers, conversely, are those that cannot be expressed in such a way and include many important mathematical constants and non-repeating, non-terminating decimals.

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Recurring decimals such as 0.26262626--, all integers and all finite decimals, such as 0.241, are also rational numbers. Alternatively, an irrational number is any number that is not rational. It is a number that cannot be written as a ratio of two integers (or cannot be expressed as a fraction).read more >>