Irrational Numbers: Identification & Properties

Irrational numbers are real numbers that cannot be expressed as a simple fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. They cannot be written as a ratio of two integers. This is the defining characteristic.

Identification: Irrational numbers are usually identified by their inability to be expressed as a fraction or through their decimal representation. Common examples include:

Square roots of non-perfect squares (e.g., $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$).
The mathematical constant $\pi$ (pi), which represents the ratio of a circle’s circumference to its diameter.
The mathematical constant $e$ (Euler’s number), the base of the natural logarithm.

Representation on the Number Line: Although they cannot be written exactly, irrational numbers can still be represented on the number line. The process often involves geometric constructions or approximations based on their decimal expansions. For example, $\sqrt{2}$ can be constructed using a right triangle with legs of length 1. $\pi$ and $e$ have specific points on the number line, although their exact location is impossible to pinpoint visually.

Decimal Expansions: Irrational numbers have decimal expansions that are non-terminating (they never end) and non-recurring (they don’t have a repeating pattern of digits). The decimal representation goes on infinitely without any discernible pattern.

Formulae

There are no single “formulae” for *all* irrational numbers as they are a diverse set. However, some related concepts that use them might involve formulae.

The Pythagorean theorem: $a^2 + b^2 = c^2$, where if $a$ and $b$ are rational, $c$ can be irrational (e.g., a=1, b=1, c=$\sqrt{2}$)
Circumference of a circle: $C = 2\pi r$
Area of a circle: $A = \pi r^2$

Examples

Example-1: Consider the number $\sqrt{7}$.

We know that 7 is not a perfect square (1, 4, 9, 16…). Therefore, $\sqrt{7}$ is irrational.

Its approximate decimal value is 2.645751311… Notice that this is non-terminating and non-repeating. It cannot be written as a simple fraction.

Example-2: Consider the number $\frac{\pi}{2}$.

While $\pi$ is irrational, dividing it by 2 also results in an irrational number. Any constant *non-zero* multiple or division of an irrational number also ends up as irrational.

Its approximate decimal value is 1.57079632679… Again, the decimal representation is non-terminating and non-repeating.

Common mistakes by students

Confusing Repeating Decimals with Irrational Numbers: Students sometimes confuse repeating decimals with irrational numbers. Repeating decimals (e.g., 0.333…) are actually rational numbers, because they can be expressed as a fraction ($\frac{1}{3}$ in the example). Irrational numbers *never* have a repeating pattern. For example, 0.1010010001… is irrational as the number of ‘0’s increases between each 1.

Incorrectly Approximating Irrational Numbers as Exact Values: Students sometimes treat approximations of irrational numbers (e.g., using 3.14 for $\pi$) as the exact value, especially in calculations. This will introduce some amount of error in your calculation.

Assuming all Roots are Irrational: Students might think all roots are irrational. However, square root of perfect squares like 4 ($\sqrt{4}$ =2) are rational.

Real Life Application

Construction and Architecture: When designing circular structures or using precise dimensions that involve square roots (e.g., calculating the diagonal of a square), irrational numbers are crucial. $\pi$ is fundamental in circular designs.

Engineering: Calculations involving the strength of materials and wave properties may incorporate irrational numbers.

Computer Science: Irrational numbers are used in algorithm and modelling of any situation that has continuous value change, such as a stock price over time, which is more precisely handled using these numbers.

Fun Fact

There are infinitely many irrational numbers, just like there are infinitely many rational numbers and integers! The set of irrational numbers is considered “uncountable,” meaning there are more irrational numbers than rational numbers.