The World of Numbers · Lesson 5 of 7
Irrational Numbers
“Some numbers refuse to end, repeat, or fit into a fraction—decimal rebels.”
• Understand why irrational numbers are needed. • Derive √2 from the diagonal of a unit square. • Follow the proof by contradiction that √2 is irrational. • Construct irrational lengths such as √2 on the number line. • Understand π as an irrational number and explore Madhava’s infinite series.
Rational numbers are spread throughout the number line so closely that between any two rational numbers, we can always find another rational number. This property is called density. However, being densely packed does not mean that rational numbers include every possible number on the number line.
Geometry shows us that some lengths cannot be written as a fraction of the form (p/q), where (p) and (q) are integers . For example, certain diagonal lengths naturally lead to numbers such as (sqrt{2}) and (sqrt{3}), which cannot be expressed exactly as fractions. Numbers of this kind are called irrational numbers. Together, rational and irrational numbers fill the entire number line.
A number that cannot be expressed in the form p/q where p and q are integers and q ≠ 0.
The Diagonal of a Unit Square
Consider a square of side 1 unit. Its diagonal forms the hypotenuse of a right triangle with perpendicular sides 1 and 1. By the Baudhāyana–Pythagoras theorem, the square of the diagonal equals 1² + 1² = 2.
The important question is not whether √2 exists—it clearly exists as a geometric length—but whether it can be written as a fraction of two integers. The answer is no.
Proof by Contradiction
A method in which we assume the opposite of the statement we want to prove and then show that the assumption leads to an impossibility.
Proof that √2 Is Irrational
Assume, for the sake of contradiction, that √2 is rational. Then it can be written in lowest terms as p/q, where p and q are integers, q ≠ 0, and p and q have no common factor other than 1.
Square both sides.
Since p² is twice an integer, p² is even. If the square of an integer is even, the integer itself must be even. Therefore p is even. Write p = 2k.
Now q² is also even, so q is even. Therefore both p and q are divisible by 2. But we began by assuming p/q was already in lowest terms. That is the contradiction.
The assumption that √2 is rational leads to p and q having a common factor of 2. Therefore the assumption is false and √2 is irrational.
The Same Idea for Other Square Roots
A similar contradiction argument can be used for square roots such as √3, √5, √7 and √10. When the integer under the square root is not a perfect square, its square root is often irrational.
Constructing √2 on the Number Line
Irrational does not mean 'cannot be located'. Geometry allows us to construct irrational lengths exactly. To construct √2, begin with OA = 1 on the number line. At A, draw a perpendicular AB = 1. Join O to B. Then OB = √2 by the Pythagorean theorem. Using a compass centred at O with radius OB, mark the same length on the number line.
Extending the Construction to √n
By repeatedly adding a perpendicular segment of length 1 to a previously constructed hypotenuse, one can build the square-root spiral and obtain lengths √2, √3, √4, √5 and so on.
The Story of π
Another famous irrational number is π, the ratio of a circle’s circumference to its diameter. Āryabhaṭa gave a highly accurate approximation 3927/1250 = 3.1416 and indicated that it was an approximation rather than an exact fractional value.
π is the constant ratio of a circle's circumference to its diameter.
Madhava’s Infinite Series
Because π cannot be represented exactly by a single fraction, mathematicians developed infinite processes that approach it more and more closely. The chapter highlights Madhava of Sangamagrama and the Kerala School, and gives the infinite series
An infinite series does not mean physically adding every term one by one until the end, because there is no final term. Instead, we study the value approached by the partial sums as more and more terms are included.
Practice Problems
- Explain why the diagonal of a unit square has length √2.
- Reproduce the proof by contradiction that √2 is irrational in your own words.
- Use the same contradiction structure to prove that √5 is irrational.
- Describe the ruler-and-compass construction used to locate √2 on the number line.
- Extend the square-root construction idea to describe how √3 can be obtained.
- Explain why π cannot be represented exactly by a single fraction if it is irrational.
Key Takeaways
• Irrational numbers cannot be expressed as p/q. • √2 appears naturally as the diagonal of a unit square. • Proof by contradiction shows that √2 cannot be rational. • Irrational lengths can still be constructed and located on the number line. • π is irrational and can be approximated through fractions or infinite series. • Rational and irrational numbers together are needed to fill the real number line.
Next, we unite rational and irrational numbers into the real numbers and learn how decimal expansions reveal which type a number is.