91Ó°ÊÓ

A rational number is any number that can be represented as a fraction. To be specific, it is of the form \(\frac{p}{q}\), where both \(p\) and \(q\) are integers, and \(q\) is not zero. This definition is crucial because it allows us to understand how numbers can be broken down into parts. For example, \(\frac{1}{2}\), 0.75 (which is \(\frac{3}{4}\) in fraction form), and -3 (which can be written as \(\frac{-3}{1}\)) are all rational numbers.
If a number can be written as a simple fraction or a finite or repeating decimal, it qualifies as a rational number.
Numbers like 1.333... (which is \(\frac{4}{3}\)) are also rational because the decimal repeats.

Irrational numbers, unlike rational numbers, cannot be expressed as a simple fraction. This means they cannot be written in the form \(\frac{p}{q}\) where both \(p\) and \(q\) are integers, and \(q\) is not zero. These numbers have decimal expansions that never terminate and never repeat.
Examples of irrational numbers include \( \text{π} \) (pi), which is approximately 3.14159..., and \(\text{e}\) (Euler's number), which is approximately 2.71828.... Another common example is \( \text{√2} \), which is approximately 1.41421...
In essence, irrational numbers fill in the 'gaps' between the rational numbers on the number line. They give us a complete number system when combined with rational numbers.

The main difference between rational and irrational numbers lies in how they can be represented and their decimal expansions.
**Representation:**

• Rational numbers can be expressed as fractions \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\) is not zero.
• Irrational numbers cannot be expressed in this way; they cannot be written as simple fractions.

**Decimal Expansions:**

• Rational numbers have decimal expansions that either terminate (like 0.5) or repeat (like 0.333...).
• Irrational numbers, on the other hand, have decimal expansions that are non-terminating and non-repeating. For example, \(\text{Ï€} \) (pi) and \(\text{√2}\) have endless, non-repetitive decimal sequences.

In summary, rational numbers are predictable and can be expressed in fractional form, while irrational numbers are unpredictable and cannot be neatly expressed as fractions.