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Rational numbers are numbers that can be expressed as a fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \eq 0\). This means you can write any rational number as a ratio of two integers. Here are some examples of rational numbers:

• 1/2 (which equals 0.5)
• -3 (which can be written as -3/1)
• 7 (which is 7/1)

As you can see, rational numbers include both fractions and whole numbers, as long as they can be expressed in the fraction form. Importantly, all rational numbers are also real numbers since they possess a specific location on the number line.

Real numbers are a broad category that includes all the numbers you can think of on the number line. This set includes

• rational numbers
• irrational numbers

Irrational numbers cannot be written as fractions \(\frac{a}{b}\). Examples of irrational numbers include:

• \( \pi (pi) \) which is approximately 3.14159
• \( \sqrt{2} \) which is about 1.414

If we combine both rational and irrational numbers, we cover the entire set of real numbers. In essence, any number that can have a place on the number line is a real number. These numbers can be used for most arithmetic operations we perform in daily life and various fields of science and engineering.

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. An important concept in set theory is 'subset.'
When we say that \( \mathbb{Q} \) is a subset of \( \mathbb{R} \), we mean that all elements of the set \( \mathbb{Q} \) (rational numbers) are also elements of the set \( \mathbb{R} \) (real numbers). In mathematical notation, this relationship is written as \( \mathbb{Q} \subset \mathbb{R} \).
This simple symbol \( \subset \) represents a powerful idea that helps in understanding the structure and relationships between different sets. In real-life applications, these notions are useful in data organization, analyzing groups of objects, and understanding mathematical hierarchies.