Table of Contents
Understanding Irrational Numbers
In the broader chapter on number systems, you have already met natural numbers, integers, and rational numbers. Irrational numbers are another type of real number, but with a special property: they can never be written as a ratio (fraction) of two integers.
This chapter focuses on what makes a number irrational, how to recognize such numbers, and why they are important.
What Is an Irrational Number?
A number is irrational if:
- It is a real number (it is a point on the number line),
- It cannot be written as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$,
- Its decimal representation never ends and never repeats in a fixed pattern.
By contrast, a rational number is any number that can be written as a fraction of integers. Rational numbers have decimal forms that either:
- terminate (end), like $0.5$ or $3.125$, or
- repeat a pattern, like $0.3333\ldots$ or $2.145145145\ldots$
Irrational numbers do neither: their decimal digits go on forever without settling into a repeating block.
Examples of irrational numbers:
- $\sqrt{2} \approx 1.4142135623\ldots$
- $\pi \approx 3.1415926535\ldots$
- $e \approx 2.7182818284\ldots$
None of these can be written as $\dfrac{p}{q}$ with integers $p$ and $q$.
Decimal Expansions: Non-Terminating and Non-Repeating
The most practical way for beginners to recognize an irrational number is by looking at its decimal expansion.
- If the decimal stops, like $4.75$, the number is rational.
- If the decimal repeats a pattern, like $0.27272727\ldots$, the number is rational.
- If the decimal goes on forever without any repeating pattern, the number is irrational.
For example:
- $\sqrt{2} = 1.4142135623730950\ldots$
The digits appear random; no pattern repeats forever. - $\pi = 3.1415926535897932\ldots$
Again, no repeating block of digits.
It is important to understand that writing an irrational number with just a few decimal places (like $3.14$ for $\pi$) is always an approximation. The exact value has infinitely many non-repeating decimal places.
Famous Irrational Numbers
Several irrational numbers appear very often in mathematics and in applications.
The Number $\sqrt{2}$
The number $\sqrt{2}$ is the length of the diagonal of a square with side length $1$. It cannot be expressed as a fraction of integers.
A classic result in mathematics shows that no matter what integers $p$ and $q$ you pick, the fraction $\dfrac{p}{q}$ will never equal $\sqrt{2}$ exactly. This was one of the earliest discovered irrational numbers.
The Number $\pi$
The number $\pi$ is the ratio of the circumference of a circle to its diameter:
$$
\pi = \frac{\text{circumference of a circle}}{\text{diameter of the circle}}.
$$
For every circle, this ratio is the same number $\pi$, and it is irrational. That means:
- You can approximate $\pi$ as $3.14$ or $\dfrac{22}{7}$,
- But no fraction $\dfrac{p}{q}$ equals $\pi$ exactly.
The Number $e$
The number $e$ appears in many contexts involving growth and change (such as interest, population models, and calculus). It is also irrational. Its approximate value is:
$$
e \approx 2.718281828\ldots
$$
While its deeper meaning is explored in more advanced chapters, here you only need to know that it is a real, irrational number widely used in mathematics.
Approximations vs Exact Values
Because irrational numbers have infinitely many non-repeating decimal places, we can never write out the whole number. Instead, we use:
- Symbols like $\sqrt{2}$, $\pi$, $e$ to represent exact irrational values.
- Decimal approximations like $1.41$, $3.14$, or $2.72$ when we only need an approximate value.
For example:
- $\sqrt{2}$ is exact; $1.41$ is an approximation.
- $\pi$ is exact; $3.14$ and $\dfrac{22}{7}$ are approximations.
Approximations are good enough for many practical calculations, as long as you are aware that they are not exact.
How Irrational Numbers Relate to Rational Numbers
Although irrational numbers cannot be written as fractions, they still fit on the same number line as rational numbers. Between any two distinct real numbers, no matter how close, there are:
- Infinitely many rational numbers, and
- Infinitely many irrational numbers.
Two key ideas:
- Rational and irrational numbers are mixed together on the number line; they are not separated into different regions.
- The set of irrational numbers is not “rare” or small; in a precise mathematical sense, there are “more” irrational numbers than rational ones.
While a full discussion of the structure of the real number line comes later, here the main point is: irrational numbers are an essential part of filling in all the points on the line, not just special exceptions.
Examples and Non-Examples
To build intuition, here are some common expressions and whether they are irrational or not. The justifications here rely only on simple observations and the definitions given.
- $0.101001000100001\ldots$ where the number of zeros between the $1$s keeps increasing:
This decimal never repeats a fixed pattern, so it represents an irrational number. - $0.123123123\ldots$
The block 3$ repeats forever. This is rational, not irrational. - $\sqrt{4}$
$\sqrt{4} = 2$, which is an integer (and therefore rational). So this is not irrational. - $\sqrt{3}$
It cannot be expressed as a fraction of integers (similar reasoning to $\sqrt{2}$). It is irrational. - $\sqrt{9} = 3$
This is an integer, so it is rational, not irrational. - $0.5$
Decimal terminates, so rational. - $0.\overline{7} = 0.7777\ldots$
Decimal repeats the digit $, so rational.
The key check is always: can it be written as $\dfrac{p}{q}$ with integers $p$ and $q$? If yes, it is rational. If no, and it is real, it is irrational.
Why Irrational Numbers Matter
Even for beginners, it is useful to know why irrational numbers are not just a curiosity:
- Geometry: Distances like $\sqrt{2}$ (the diagonal of a unit square) or $\pi$ (in circles) naturally appear.
- Measurement: Many exact lengths, areas, and angles lead to irrational numbers.
- Completeness of the number line: To have a number for every possible length or continuous measurement along the line, you must include irrational numbers.
Without irrational numbers, many simple geometric and physical quantities could not be represented exactly by numbers.
Summary
- An irrational number is a real number that cannot be written as a fraction of integers.
- Its decimal expansion never ends and never repeats in a fixed pattern.
- Important examples include $\sqrt{2}$, $\sqrt{3}$, $\pi$, and $e$.
- We use special symbols for exact irrational numbers and decimals or fractions for approximations.
- Irrational numbers are mixed with rational numbers all along the number line and play a central role in geometry and measurement.