Irrational Numbers Examples for Students Explained

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Mathematics organizes numbers into distinct categories, each serving a unique purpose. Rational and irrational numbers are two such categories of numbers that are fundamentally different from each other. Rational numbers often expresses itself as a form of fraction of two integers and as decimals too. They can either terminate (1/4 = 0.25) or they go on to repeat forever (1/3 = 0.333…). This predictability makes it easier to represent them on a number line. No matter how complex a rational number appears, mathematicians can always capture these numbers exactly by writing them in a simple ratio. Irrational numbers defy these patterns. Their decimal expansions will stretch on forever without terminating, and no sequence of these number will ever repeat; not even their fractions or integers. Although they are elaborate, they do not represent the numbers precisely.

“Irrational numbers exist in the gaps between rational numbers on a number line, they are infinitely close, yet fundamentally different”.

Despite their unusual nature, irrational numbers carry great significance. Without these numbers, we can not accurately model the evolving world. The ancient Greeks, were the first ones to discover these numbers and have in fact found them deeply troubling . Moving forward, we now embrace these numbers as an essential tools for all mathematical calculations. What do these irrational numbers look like and why do these numbers fall into this category in mathematics? We will now study four famous examples, exploring what makes each of these numbers irrational, where it appears in the real world, why it has significantly fascinated the mathematicians and how irrational numbers can be better explained to students.

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What are the Examples of Irrational Numbers?

Fig 1: Euler’s Number (e) is often used calculate the compound interest

Now that we have understood what irrational numbers are, let’s look at few examples of irrational numbers explained for students, with examples. Pi (π) is an irrational number which approximately equals to 3.14159…, though it’s digits will continue infinitely. Many people use 22/7 or 3.14 as convenient approximations for everyday geometric calculations.

Moving on to the Euler’s number, this number will approximately equal to 2.71828… and, like Pi, it will continue forever without repeating its numerical sequence. Mathematicians classify it as an irrational and transcendental number. In banks and corporates, finance professionals use ‘e’ constantly for compound interest calculations. As you start dividing the Fibonacci numbers, you will start progressing towards 1.61803, which is the golden ratio. There is no fraction to express it in it’s exact numbers. Artists and architects have used this golden ratio for centuries, and it has appeared throughout nature in the form of spiral shells, flower petals, and branching patterns of trees.

The square root of 2 will equal approximately to 1.41421… and its digits never terminate or repeat. We cannot express it as a rational number. Unlike Pi and e, mathematicians classify √2 as algebraic rather than transcendental. Geometrically, √2 represents the diagonal length across a square which has sides of one unit, thereby highlighting its significance in the Pythagorean theorem. To sum it up, these are few examples of irrational numbers for students explained, that are used in day to day mathematical calculations.

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The Discovery of These Irrational Numbers

Fig 2: Illustration of the Fibonacci Series

The discovery of irrational numbers, is in fact an outstanding breakthrough for the numbering system in mathematics. Pi (π) has spanned nearly four thousand years, starting from the beginning of the Babylonian Era, estimated to be around 1900 BCE, to the current era of advanced computing where researchers use it to calculate trillions of digits. The discovery of Pi (π) is indeed a valuable contribution towards enabling numerous calculations, be it in architecture or in computer programming.

“In the 300 BCE, the discovery of the golden ratio by Euclid was a boon, paving the way for professionals to design breathtaking architecture”.

The Euler’s number is believed to have originated from Jacob Bernoulli’s inquiry about compound interest in the year 1683. However, Euler is the one who popularized the letter (e) to represent it and proved that it is an irrational number with an infinite, non-repeating decimal value (approximately 2.71828). Euler discovered the profound nature and properties of (e) through his foundational work with logarithms and calculus. Hippasus of Metapontum discovered that square root of 2 is an irrational number. He also proved that square root of 2 cannot be expressed as a fraction, thereby shattering the core philosophical foundation of the Pythagorean school.

These four examples reveal how mathematics evolved from practical calculations to abstract inquiry, thus proving that even simple geometric relationships could be transcending from rational expression.

Pi (π):

The Babylonians were the first to record the value of Pi on clay tablets as 3.1605. Later on, the Egyptians approximated this value to be the same. Then, Archimedes refined this value to as 3.1418 using his very own polygon method.Subsequently, the Chinese mathematicians named Liu Hui and Zu Chongzhi pushed this accuracy further, calculating the values 3.1415 and 3.1415926 respectively. During the Renaissance period, Ludolph Van Ceulen devoted his entire life for computing Pi, therefore deriving the value of pi to 35 decimal places.

Moving on to 1940s, John von Neumann’s team harnessed computers to accelerate these calculations. Then in the year 2019, Emma Haruka Iwao shattered records of these discoveries of Pi by reaching to 31.4 trillion digits in her calculations.

Euler’s number (e):

The disocvery of Euler’s number started when Jacob Bernoulli posed a simple question,

“How much would one dollar grow to, if a bank compounded interest at increasingly frequent intervals”?

To answer this question, he analyzed the returns, compounding them monthly, then daily, and finally to hourly intervals, ultimately pushing them towards the continuous compounding in finance. Remarkably, his calculations have revealed something surprising. As the compounding frequency approaches infinity, the returns do not grow without any bound. Instead, they converge to a precise limit, that is approximately 2.71828. This unexpected discovery fascinated many mathematicians. Add to that, Leonhard Euler later transformed this curiosity into a cornerstone of mathematics. First, he popularized the constant and introduced the symbol e. Additionally, he demonstrated its linkage to exponential functions and calculus.

Moreover, Euler also proved that e appears naturally in problems to study radioactivity decay and population growth. Today researchers have used this value across multiple fields, ranging from biochemistry to bioanalytical techniques, to study radioactivity decay.

The golden ratio (Φ):

The golden ratio connects deeply to the Fibonacci sequence,0, 1, 1, 2, 3, 5, 8, 13… .As the numbers grow larger, dividing consecutive terms produces ratios that progressively approach Φ. Hence the, Fibonacci patterns and the golden ratio appear together throughout nature. Remarkably, this ratio emerges everywhere in the natural world. For instance, it governs the spiral arrangement of sunflower seeds and the curve of nautilus shells

Using Leonardo da Vinci’s pictures, Luca Pacioli praised this proportion in his book “De Divina Proportione” at the beginning of the 16th century. In 1835, Martin Ohm first used the phrase “golden ratio.” Mark Barr appears to have used this symbol of phi (Φ) for the first time around the beginning of the 20th century in honor of the Greek sculptor Phidias (c. 490–430 BC), who many art historians believe used the golden ratio widely in his masterpieces.

The square root of 2 (√2):

In 1800 BCE, the Babylonians estimated √2 to 1.41421296, which was remarkably accurate and within a millionth of the actual amount. The Sulbasutras, ancient Indian mathematical books, provides another early approximation which asserts that an increase in the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth. This approximation is the seventh in a line of progressively accurate approximations based on the sequence of Pell numbers, which may be calculated from the continued fraction expansion of √2. Despite having a lower denominator, it is only slightly less accurate than the Babylonian approximation. For practical reasons, early mathematicians needed √2, mainly to calculate a square’s diagonal. Therefore through these examples, the irrational numbers for students explains well.

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The Use of These Numbers in Daily Life

These four irrational numbers play essential roles across numerous fields, shaping everything from ancient architecture to modern technology. In geometry, mathematicians and engineers use Pi (π) to calculate the dimensions of circles and spheres. Specifically, Pi determines the relationship between a circle’s diameter and its circumference. Also, architects rely on Pi when designing domes, arches, and curved structures. Additionally, astronomers apply Pi to calculate planetary orbits and the vast distances between celestial bodies.

The golden ratio (Φ) finds applications in finance, accounting, design, and art. For instance, graphic designers use Φ to create visually balanced compositions. Similarly, architects incorporate this ratio into building facades and floor plans. Moreover, financial analysts apply Fibonacci retracements that is closely linked to Φ,when predicting stock market movements. Consequently, this ancient ratio remains relevant in modern economic analysis. Scientists use Euler’s number (e) extensively for calculating radioactive decay and half-lives of isotopes. Beyond physics, biologists employ e to model population growth and the spread of diseases. Additionally, bankers and economists rely on e for continuous compound interest calculations. Therefore, this constant bridges natural sciences and financial mathematics.

Engineers, designers, and physicists widely use the square root of 2 (√2) in their work. Notably, the international paper size system (A4, A3, A2) uses √2 proportions, allowing sheets to maintain the same ratio when folded in half. Electrical engineers apply √2 when calculating root mean square of voltages. In physics, √2 appears naturally in problems involving right triangles and diagonal measurements.

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Conclusion

Irrational numbers stretch on forever without terminating and never repeat the same sequence of digits. No fraction can capture them exactly, yet they appear everywhere in our world.

Throughout history, mathematicians discovered several irrational numbers that transformed our understanding of mathematics. Pi (π) gave us the key to understanding circles and spheres. The golden ratio (Φ) revealed hidden proportions in nature, art, and architecture. Euler’s number (e) unlocked the mathematics of growth, decay, and continuous change. The square root of 2 (√2) demonstrated that even simple geometry produces numbers beyond rational expression. Hence, the irrational numbers examples for students is explained in great detail.

Each irrational number carries a rich history of discovery. Ancient civilizations first approximated these values through practical necessity. Further, brilliant mathematicians proved their irrationality, sometimes facing resistance from those who found such numbers philosophically troubling. Today, we celebrate these constants as fundamental tools that connect abstract mathematics to the physical world. Understanding irrational numbers makes mathematics more engaging and accessible for students. Hence, these examples demonstrate that numbers are not merely abstract symbols but powerful tools that shape our understanding of reality. From ancient Babylon to modern computing, irrational numbers continue to inspire curiosity and drive mathematical progress.

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Frequently asked questions (FAQs)

References

1. Agarwal, R. P., & Agarwal, H. (2021). Origin of irrational numbers and their approximations. Computation, 9(3), 29. https://doi.org/10.3390/computation9030029
2. De Spinadel, V., & Paz, J. (1999). A New Family of Irrational Numbers with Curious Properties. Humanistic Mathematics Network Journal, 1(19), 33–37. https://doi.org/10.5642/hmnj.199901.19.14
3. Obersteiner, A., & Hofreiter, V. (2017). Do we have a sense for irrational numbers? Journal of Numerical Cognition, 2(3), 170–189. https://doi.org/10.5964/jnc.v2i3.43
4. Rosangliana, D. (2024). HISTORY AND APPLICATIONS OF PI (π). Zenodo (CERN European Organization for Nuclear Research). https://doi.org/10.5281/zenodo.15612843

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Ms. Adrita Roy is a dual master’s holder specializing in Biological Sciences and Sustainable Development. She completed her Master’s in Biological Sciences from Bangalore University in 2025, followed by a Master’s in Sustainable Development from the University of Sussex in 2026.

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