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Real numbers are the cornerstone of most mathematical concepts that deal with continuous quantities. They form an extensive set that includes both rational and irrational numbers. Rational numbers are those which can be expressed as a fraction or ratio of two integers, where the denominator isn't zero. Irrational numbers, on the other hand, cannot be expressed in such a way.
When talking about real numbers, we often refer to the number line. Every point on this line corresponds to a real number, and this includes both rational numbers like 2/3 or -5, and irrational numbers like \( \pi \) or \( \sqrt{3} \).
This is what makes real numbers such a vital part of mathematics; they cover all possible magnitudes and all forms of numbers within their spectrum, thereby providing a complete and unified way to represent quantities in mathematics.

Inequalities provide a way to express the relative size of two values. These expressions use symbols such as \(<\), \(>\), \(\leq\), and \(\geq\) to indicate that one number is less than, greater than, or equal to another number. In our exercise, the inequality \(a < b\) tells us that \(a\) is less than \(b\).
Understanding inequalities is essential as they offer a means to compare quantities and describe ranges. When dealing with inequalities in problems, it is crucial to remember that any operation performed on the inequality should be done consistently to maintain the relationship, such as adding the same number to both sides.
In the context of our exercise, the understanding of inequalities allows us to logically and confidently find a number that lies in between \(a\) and \(b\), ensuring it follows the specified conditions we are exploring.

Decimal expansion refers to writing numbers using the base-10 number system. For rational numbers, this usually results in either a terminating or repeating decimal. For example, the number 1/4 can be expressed as 0.25, and 2/3 as 0.666..., where the digits repeat.
Irrational numbers, however, have non-terminating and non-repeating decimal expansions. This is a key characteristic that differentiates them from rational numbers. For example, \( \sqrt{2} \) is approximately 1.41421356... and continues infinitely without repeating.
Understanding the nature of decimal expansions helps in grasping why certain numbers are considered irrational. It visually and conceptually represents why they can't be expressed as a neat fraction, instead spiraling into a unique and never-ending sequence of digits.