1. Definition
An irrational number is a real number that cannot be expressed as \(\dfrac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\). In other words, it cannot be written as a ratio of two integers.
2. Symbol and Representation
The set of irrational numbers does not have a single standard letter symbol like \(\mathbb{Q}\) for rationals, but irrationals are usually described as \(\mathbb{R} \setminus \mathbb{Q}\) — the real numbers that are not rational.
Graphically, irrational numbers appear all along the real number line, filling the gaps between rational points.
3. Examples of Irrational Numbers
Here are common examples of irrational numbers and brief explanations:
- \(\sqrt{2}\): The square root of 2 is not expressible as a fraction of integers; it is the length of the diagonal of a unit square.
- \(\pi\): The ratio of a circle's circumference to its diameter. Its decimal expansion never terminates or repeats.
- \(e\): The base of natural logarithms; appears in limits and continuous growth problems.
- The golden ratio \(\varphi = \dfrac{1+\sqrt{5}}{2}\): An irrational number arising in geometry and recursion.
- Non-perfect-root expressions, e.g. \(\sqrt{3},\ \sqrt{5}\): Square roots of non-perfect squares are irrational.
4. Decimal Representation
Irrational numbers have decimals that neither terminate nor repeat. Their decimal expansions go on forever without a fixed repeating pattern.
4.1. Non-terminating Non-repeating Decimals
An irrational number's decimal expansion is called non-terminating non-repeating. For example, \(\pi = 3.1415926535\ldots\) and \(\sqrt{2} = 1.41421356\ldots\) continue infinitely with no repetitive block.
4.2. Why Decimals Differ from Rationals
Rational numbers produce either terminating or repeating decimals. Since irrationals are not rational, their decimal expansions cannot fall into those two categories.
4.3. Examples Table
| Number | Decimal (start) | Type |
|---|---|---|
| \(\sqrt{2}\) | 1.41421356\(\ldots\) | Non-terminating, non-repeating |
| \(\pi\) | 3.14159265\(\ldots\) | Non-terminating, non-repeating |
| \(e\) | 2.71828182\(\ldots\) | Non-terminating, non-repeating |
5. Properties of Irrational Numbers
Irrational numbers have several important properties. Some are similar to rational numbers because both are part of the real numbers, while others are quite different.
5.1. Positive, Negative, and Zero
Irrational numbers can be positive or negative just like rational numbers. However, zero is not irrational because it can be written as a ratio of integers.
5.1.1. Examples
- Positive irrational number: \( \sqrt{5} \approx 2.236... \)
- Negative irrational number: \( -\pi \approx -3.14159... \)
- Why 0 is rational: \(0 = \dfrac{0}{1}\), which fits the form \(\dfrac{p}{q}\).
5.2. Closure Under Arithmetic
Unlike rational numbers, irrational numbers are not closed under addition, subtraction, multiplication, or division. The result can be either rational or irrational.
5.2.1. Addition and Subtraction
- Irrational + irrational can be rational: \(\sqrt{2} + (2 - \sqrt{2}) = 2\)
- Irrational + irrational can be irrational: \(\sqrt{2} + \sqrt{3}\)
- Irrational + rational is always irrational: \(\pi + 1\)
5.2.2. Multiplication and Division
- Irrational × irrational can be rational: \(\sqrt{2} \times \sqrt{2} = 2\)
- Irrational × irrational can be irrational: \(\sqrt{2} \cdot \pi\)
- Irrational ÷ rational can be irrational: \(\dfrac{\pi}{2}\)
5.3. Density Property
Irrational numbers are dense on the real number line. This means that between any two real numbers, there is always an irrational number.
5.3.1. Examples
- Between 1 and 2: \(\sqrt{2} \approx 1.414...\)
- Between 3 and 4: \(\sqrt{10} \approx 3.162...\)
- General method: For any a and b (a < b), the number \(a + \dfrac{b-a}{\sqrt{2}}\) is always irrational.
5.4. Uncountability
The set of irrational numbers is uncountable, meaning that there are infinitely more irrationals than rationals. This comes from Cantor's diagonal argument.
5.4.1. Examples
- Rationals can be listed (countable): 0, 1, -1, 1/2, -2/3, ...
- Irrationals cannot be listed: no sequence can capture all irrationals.
- Example sets of irrationals: All numbers with non-repeating infinite decimals.
5.5. Relation with Real Numbers
Real numbers are made up of all rational and irrational numbers. Irrational numbers fill in the "gaps" between rational numbers.
5.5.1. Examples
- Real number set: \(\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})\)
- Rational example: \( \dfrac{3}{4} \)
- Irrational example: \( \pi \)
- Mixed real interval: Between any two rationals there are irrational numbers and vice versa.
5.6. Approximation by Rationals
Although irrational numbers cannot be written exactly as fractions, they can be approximated by rationals with any desired accuracy.
5.6.1. Examples
- Approximating \(\pi\): \(3.14, 3.141, 3.14159, ...\)
- Approximating \(\sqrt{2}\): \(1.4, 1.41, 1.414, ...\)
- Continued fractions: \(\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \cdots}}\)
- Fractions getting closer: \( \dfrac{22}{7}, \dfrac{333}{106} \) approximate \(\pi\).