Skip to main content

How do we know pi is an irrational number?

 

From - Live Science

By  

Edited by - Amal Udawatta

The symbol for pi made from numbers on a black background.
Irrational numbers go on and on. How do we know that pi has no ending? (Image credit: kr7ysztof/Getty Images)
Are there mathematical ways to prove that pi is an irrational number that has no end?

Originally defined as the ratio between the circumference of a circle and its diameter, pi — written as the Greek letter π — appears throughout mathematics, including in areas that are completely unconnected to circles such as chemistry, physical sciences and medicine.

Pi belongs to a huge mathematical group called irrational numbers, which go on forever and cannot be written as fractions. Scientists have calculated pi to 105 trillion digits, although most of us are more familiar with the approximation 3.14. But how do we know that pi is an irrational number?

Rational numbers, which make up the majority of numbers we use in day-to-day life (although less than half of all possible numbers), can be written in the form of one whole number divided by another. Pi, with its complicated string of decimals, certainly doesn't appear to be part of this group at first glance.

0 of 1 minute, 23 secondsVolume 0%
01:17
00:17
01:23
 
PLAY SOUND

"Rationality is the practical property of having access to the number explicitly, i.e. without any approximation … so being able to write the number in a finite amount of symbols," Wadim Zudilin, a mathematician at Radboud University in the Netherlands, told Live Science.

Related: What is the largest known prime number?

However, actually proving that you can't write pi as a fraction is a surprisingly knotty issue. Mathematicians don't have a universal method to show that a particular number is irrational, so they must develop a different proof for each case, explained Keith Conrad, a mathematician at the University of Connecticut. "How do you know a number is not a fraction?" he said. "You're trying to verify a negative property."

Despite this difficulty, over the past 300 years, mathematicians have established different proofs of pi's irrationality, using techniques from across mathematics. Each of these arguments begins with the assumption that pi is rational, written in the form of an equation. Through a series of manipulations and deductions about the properties of the unknown values in this equation, it subsequently becomes clear that the math contradicts this original assertion, leading to the conclusion that pi must be irrational.

The specific math involved is often incredibly complex, typically requiring a university-level understanding of calculus, trigonometry and infinite series. However, each approach relies on this central idea of proof by contradiction.

"There are proofs using calculus and trigonometric functions," Conrad said. "In some of them, π is singled out as the first positive solution to sin(x) = 0. The first proof by Lambert in the 1760s used a piece of mathematics called infinite continued fractions — it's a kind of infinitely nested fraction."

However, rather than proving pi is irrational directly, it's also possible to confirm irrationality using a different property of the number. Pi belongs to another numerical group called transcendental numbers, which are not algebraic and, importantly, cannot be written as the root of a polynomial equation. Because every transcendental number is irrational, any proof showing that pi is transcendental also proves that pi is irrational.

"Using calculus with complex numbers, you can prove π is transcendental," Conrad said. "The proof uses the very famous equation called Euler's identity: e +1 = 0."

Although pi's universal importance may arise from this intangible irrationality, seven or eight decimal places is usually more than sufficient for any real-world applications. Even NASA uses only 16 digits of pi for its calculations.

"We approximate the value for practical purposes, 3.1415926 — that's already a lot of information!" Zudilin said. "But of course in mathematics, it's not satisfactory. We care about the nature of the numbers."


Comments

Popular posts from this blog

Why did Homo sapiens outlast all other human species?

  From - Live Science By  Mindy Weisberger Edited by - Amal Udawatta Reproductions of skulls from a Neanderthal (left), Homo sapiens (middle) and Australopithecus afarensis (right)   (Image credit: WHPics, Paul Campbell, and Attie Gerber via Getty Images; collage by Marilyn Perkins) Modern humans ( Homo sapiens ) are the sole surviving representatives of the  human family tree , but we're the last sentence in an evolutionary story that began approximately 6 million years ago and spawned at least 18 species known collectively as hominins.  There were at least nine  Homo  species — including  H. sapiens  —  distributed around Africa, Europe and Asia by about 300,000 years ago, according to the Smithsonian's  National Museum of Nat ural History  in Washington, D.C. One by one, all except  H. sapiens  disappeared.  Neanderthals  and a  Homo  group known as the  Denisovans  lived alongside...

New Comet SWAN Now Visible in Small Scopes

     From :- Sky & Telescope  By :- Bob King  Edited by :- Amal Udawatta This spectacular image of Comet SWAN (C/2025 F2) was taken on April 6th and shows a bright, condensed coma 5′ across and dual ion tails. The longer one extends for 2° in PA 298° and the other 30′ in PA 303°. Details: 11"/ 2.2 RASA and QHY600 camera. Michael Jaeger Amateur astronomers have done it again — discovered a comet. Not by looking through a telescope but through close study of  publicly released, low-resolution images  taken by the  Solar Wind Anisotropies  (SWAN) camera on the orbiting  Solar and Heliospheric Observatory  (SOHO). On March 29th, Vladimir Bezugly of Ukraine was the first to report a moving object in SWAN photos taken the week prior. Michael Mattiazzo of Victoria, Australia, independently found "a pretty obvious comet" the same day using the same images, noting that the object was about 11th magnitude and appeared to be brightening. R...

The indigenous women saving India's endangered giant yams

  From BBC News   By-  Kamala Thiagarajan   Edited by - Amal Udawatta Sai Krishan, Thirunelly Tribal Special Intervention Programme Lakshmi and Shantha with a species of tuber locally called the Noorang (Credit: Sai Krishan, Thirunelly Tribal Special Intervention Programme) In a tribe in southern India, a group of women are working hard to revive the country's ancient native tubers, and bring them back into everyday culture. Lakshmi spends several hours each day digging out large lumpy and hairy yam tubers, starchy roots that grow below the soil. Some weigh an unwieldy 5kg (11lb) and are 4.5ft-long (1.4m), almost as tall as she is. It's painstaking work, says 58-year-old Lakshmi, who goes by one name. First, she has to cut out the thick shoot above the ground. Then, she uses shovels to dig up the earth around the buried stem and a paddle-like flat chisel to gently pry out the tuber. She uses her hands to dig the tuber out of the ground to avoid damaging its delicate...