Real Numbers: Definition & Composition

Real numbers encompass all numbers that can be represented on a number line. They are broadly classified into two main categories: rational numbers and irrational numbers. This set is denoted by the symbol $\mathbb{R}$. Understanding the distinction between rational and irrational numbers is crucial for many mathematical concepts.

Formulae

There aren’t specific ‘formulae’ in the traditional sense for the *set* of real numbers. However, we can express the set of real numbers conceptually:

$\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$ (Real numbers are the union of rational and irrational numbers.)
$\mathbb{Q}$: The set of rational numbers. These can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers, and $q \ne 0$.
$\mathbb{I}$: The set of irrational numbers. These cannot be expressed as a simple fraction.

Examples

Here are some examples to illustrate real numbers:

Example-1: Consider the numbers 2, -5, $\frac{3}{4}$, 0.75, $\sqrt{9}$, $\pi$, and $\sqrt{2}$.

2, -5, $\frac{3}{4}$, and 0.75 are all real numbers, and they are also rational numbers.
$\sqrt{9}$ is also a real number since it simplifies to 3 (a rational number).
$\pi$ is a real number, but it is irrational. Its decimal representation goes on forever without repeating.
$\sqrt{2}$ is a real number, but it is irrational. Its decimal representation goes on forever without repeating.

Example-2: Demonstrate how we classify a number.

Consider the number $\frac{22}{7}$.

It is a real number.
It can be written in the form of $\frac{p}{q}$ where $p=22$ and $q=7$, and both are integers. Therefore, $\frac{22}{7}$ is a rational number.
When you calculate the value, it’s approximately $3.142857$, which repeats.

Common mistakes by students

Students often struggle with the following:

Confusing rational and irrational numbers: A common mistake is incorrectly classifying numbers. Remembering that irrational numbers cannot be written as simple fractions is key.
Assuming all roots are irrational: Forgetting that the square root of a perfect square (e.g., $\sqrt{9}=3$) results in a rational number.
Misunderstanding repeating decimals: All repeating decimals are rational numbers, not irrational. They can be converted to fractions.

Real Life Application

Real numbers are fundamental to nearly all aspects of real-world measurements and calculations.

Construction: Architects and engineers use real numbers (measurements like lengths, angles, areas, volumes) to design and build structures.
Finance: Real numbers are used extensively in financial calculations, including interest rates, stock prices, and calculating budgets.
Science: Scientists use real numbers to measure physical quantities like temperature, mass, and time.
Computer Science: Real numbers are used in programming to represent quantities that can have fractional parts, like prices or scientific measurements.

Fun Fact

The concept of irrational numbers was initially met with resistance by some mathematicians! The discovery of irrational numbers, like $\sqrt{2}$, challenged the Pythagorean belief that all numbers could be expressed as a ratio of integers.