Fun with Math – Interesting Facts About Numbers and Maths Discoveries

Did you know that “zero” is one of the youngest ideas in mathematics? Ancient civilisations counted, measured, and even did algebra for thousands of years without a true zero. It was finally formalised in India around 628 CE by the mathematician Brahmagupta, in his book Brahmasphutasiddhanta. The word “zero” itself descends from the Sanskrit śūnya, meaning “empty,” which travelled through Arabic (śifr) and Latin (zephirum) before becoming the English word we use today.

Brahmagupta didn’t just use zero as a placeholder — he gave it real mathematical rules, treating it as a number in its own right for the first time in history:

He even attempted rules for dividing by zero, though his answer (that x ÷ 0 = 0) turned out to be wrong — a problem so genuinely tricky that it took mathematicians another thousand years to fully understand why it has no answer at all (see the box below).

From India, the idea spread west. Around 825 CE, the Persian scholar Al-Khwarizmi — whose name gives us the word “algorithm” — described the Indian number system, including zero, in a book that was later translated into Latin. It reached Europe properly in 1202, when the Italian mathematician Fibonacci included it in his book Liber Abaci. Even then, many European merchants and even some governments distrusted the new symbol for over two centuries, and continued doing business in clumsy Roman numerals.

The value of π (3.14159…) — the ratio of a circle’s circumference to its diameter — has fascinated mathematicians for over 4,000 years. The Babylonians approximated it as 3⅓ (3.125), while the Egyptian Rhind Papyrus (c. 1650 BCE) implies a value of about 3.16049. Both are surprisingly close for calculations done entirely by hand.

The first person to calculate π with real mathematical rigour was the Greek mathematician Archimedes of Syracuse, around 250 BCE. Rather than measuring circles physically (which is never very accurate), he trapped π between two numbers using pure geometry, proving that it must lie between 223/71 (≈ 3.1408) and 22/7 (≈ 3.1429) — a method described in the box below.

Centuries later, the Chinese mathematician Zu Chongzhi (429–500 CE) pushed this even further, calculating π as 355/113 — accurate to six decimal places. Astonishingly, no one beat this record for over 800 years.

Today, supercomputers have calculated π to more than 100 trillion digits, yet we already know they’ll never find a pattern: π is irrational (its digits never repeat) and even transcendental — a fact proven by Ferdinand von Lindemann in 1882, which also finally settled a 2,000-year-old open question by proving that it is mathematically impossible to “square the circle” using only a compass and straightedge.

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34…) is built from one of the simplest rules in mathematics: each number is the sum of the two before it. It’s named after the Italian mathematician Leonardo of Pisa, nicknamed “Fibonacci,” who introduced it to Europe in 1202 using a puzzle about how fast rabbits multiply. In fact, Indian scholars — including the poet-mathematician Pingala, writing around 200 BCE — had already discovered the same sequence centuries earlier, while studying the rhythms of Sanskrit poetry.

What makes the sequence genuinely famous, though, is how often it turns up in the natural world. Sunflower seed heads typically show 34 spirals curving one way and 55 the other — both Fibonacci numbers. Pinecones and pineapples show similar spiral counts, and many flowers have a Fibonacci number of petals: lilies have 3, buttercups have 5, and daisies often have 34, 55, or even 89.

As the sequence grows, the ratio between consecutive terms (like 34/21 or 55/34) settles closer and closer to a special number: φ (phi) ≈ 1.6180339887…, known as the golden ratio. This same ratio shows up in the proportions of nautilus shells, hurricane spirals, and even the spacing of leaves around a stem — an arrangement that happens to be a remarkably efficient way to catch sunlight.

The famous Pythagorean Theorem — a² + b² = c² for any right-angled triangle — is credited to the Greek philosopher Pythagoras (c. 570–495 BCE), who founded a secretive brotherhood in the city of Croton that treated numbers as almost sacred. But the relationship itself was already known, and used in practice, over a thousand years earlier: the Babylonian clay tablet Plimpton 322 (c. 1800 BCE) lists Pythagorean triples, and Egyptian “rope stretchers” used loops of rope knotted into a 3-4-5 triangle to lay out perfect right angles when building the pyramids.

What Pythagoras and his followers are credited with is something more powerful than a formula — the first known general proof that the relationship holds for every right triangle, not just the ones people had happened to measure.

The theorem’s discovery also led to a genuine mathematical crisis. Using it on a right triangle with both shorter sides equal to 1 gives a hypotenuse of √2 — and one of Pythagoras’s own students, Hippasus, is said to have proven that √2 cannot be written as a fraction at all. This directly contradicted the school’s core belief that “all is number” (meaning whole numbers and simple ratios). Legend holds that the discovery was so unsettling that Hippasus was thrown overboard at sea for revealing it.

Prime numbers (2, 3, 5, 7, 11, 13…) are often called the “atoms” of mathematics, because of the Fundamental Theorem of Arithmetic: every whole number greater than 1 can be broken down into prime factors in exactly one way. For example, 60 = 2² × 3 × 5 — and no other combination of primes multiplies to 60.

Despite being so fundamental, primes remain full of open mysteries. The Twin Prime Conjecture guesses that there are infinitely many pairs of primes exactly 2 apart (like 11 & 13, or 17 & 19) — still unproven, though in 2013 mathematician Yitang Zhang made history by proving that infinitely many prime pairs exist within some bounded gap (originally 70 million, since narrowed down to 246 by other mathematicians).

The Goldbach Conjecture, proposed in 1742, claims every even number greater than 2 is the sum of two primes (8 = 3+5, 20 = 7+13). It has been checked by computer for every even number up to 4 quintillion — and not one exception has ever been found — yet nobody has ever proven it must be true for all even numbers. And looming over all of prime number theory is the Riemann Hypothesis, one of the most important unsolved problems in all of mathematics, with a $1 million prize attached for whoever proves it.

Here’s a trick that feels like real magic the first time you see it:

No matter which starting number you pick — 852, 731, 940, anything — you will always land on exactly 1089. It isn’t magic, and it isn’t a coincidence: it’s guaranteed by algebra, and the full reason why is short enough to fit in the box below.

1. Pick any 3-digit number where the first and last digits are different (e.g., 521).
2. Reverse it (125).
3. Subtract the smaller from the larger (521 − 125 = 396).
4. Reverse that result (693).
5. Add the two numbers together (396 + 693 = 1089).

As of October 2024, the largest known prime number is 2^136,279,841 − 1 — a staggering 41,024,320 digits long. Printed in a standard book, it would run to thousands of pages of digits and nothing else. It was discovered by Luke Durant, a former NVIDIA engineer, using thousands of cloud GPUs donated to the Great Internet Mersenne Prime Search (GIMPS), a volunteer computing project that has found every record-breaking prime since 1996.

Every single record-holder for decades has taken the form 2^p − 1, called a Mersenne prime after the 17th-century French monk Marin Mersenne, who studied numbers of this exact shape. That’s not a coincidence: mathematicians have a remarkably fast way to test whether a Mersenne number is prime (described below), which makes numbers of this specific form the only realistic candidates for record-breaking searches — testing an arbitrary number this large for primality by ordinary methods would take longer than the age of the universe.

Such enormous primes aren’t just for bragging rights, either: prime numbers are the mathematical backbone of the encryption that keeps online banking, passwords, and private messages secure.

The Cake-Cutting Problem asks a question that sounds simple but hides real mathematics: how do you divide a cake among several people so that nobody feels short-changed — without a referee, and without anyone needing to trust anyone else?

For two people, the answer has been known for centuries: “I cut, you choose.” One person cuts the cake into two pieces, and the other picks whichever piece they want. The cutter, knowing this, is motivated to cut as evenly as possible — and whichever piece the chooser picks, they’re guaranteed to feel they got at least half.

For three or more people, it gets surprisingly hard. It took until the 1960s for mathematicians John Selfridge and John Conway to independently find a method (now called the Selfridge–Conway procedure) that guarantees an “envy-free” division for three people — meaning nobody would rather have someone else’s slice — though it can require cutting the cake into as many as five pieces along the way. For any number of people, a fully general envy-free method was only found in 2016, and while it’s mathematically guaranteed to work, the number of cuts it might need to make is so astronomically large it’s completely impractical — a reminder that “solved” in mathematics doesn’t always mean “practical.”

Ancient Egyptian mathematicians were remarkably advanced for their time, using a base-10 system and sophisticated fraction techniques to plan and build monuments like the Great Pyramid of Giza — a structure so precisely aligned that its base is level to within about 2 centimetres over 230 metres.

Their approach to fractions, however, was completely different from ours. The Egyptians only had a symbol for unit fractions — fractions with a numerator of 1, like 1/2, 1/3, or 1/4. Any other fraction had to be rewritten as a sum of different unit fractions. This method, preserved in the Rhind Mathematical Papyrus (c. 1650 BCE, itself a copy of an even older document), was used to solve very practical problems — like fairly dividing loaves of bread or jugs of beer among workers.

A special set of fractions, called the Eye of Horus fractions (1/2, 1/4, 1/8, 1/16, 1/32, 1/64), were used for measuring grain volumes and were linked to an Egyptian myth about the god Horus. Curiously, added together they total 63/64 — not quite a whole — which ancient scribes explained mythologically rather than as a rounding error.

Some numbers are so unusual — or so enormous — that mathematicians have given them their own names.

Is 1 a prime number? It has exactly one divisor, it can’t be split into smaller factors, and for much of the 19th century, many respected mathematics textbooks confidently listed it as prime. Today, every mathematician agrees it is not — and the reason isn’t just a matter of definition-tweaking, but something that would genuinely break mathematics if it were ignored.

The modern definition requires a prime number to have exactly two distinct divisors: 1 and itself. The number 1 only has one divisor (itself), so it fails the definition immediately. Mathematicians instead put 1 into its own special category, called a unit.

Pascal’s Triangle is a triangular grid of numbers where every entry is the sum of the two numbers directly above it. It’s named after the French mathematician Blaise Pascal, who wrote a systematic treatise on it in 1654 — but, much like the Fibonacci sequence, it was discovered independently and far earlier in other cultures: by Indian scholar Pingala around 200 BCE, by Persian mathematicians Al-Karaji and Omar Khayyam around 1000 CE (where it’s still called “Khayyam’s Triangle” in Iran), and by Chinese mathematician Yang Hui in 1261 (where it’s known as “Yang Hui’s Triangle” to this day).

The triangle is bursting with hidden patterns. Each row sums to a power of 2 (row n adds up to 2ⁿ). One diagonal contains the ordinary counting numbers; the next contains the triangular numbers (1, 3, 6, 10…); and if you add up numbers along certain shallow diagonals, the Fibonacci sequence from Fact 3 appears again, hiding in plain sight.

Pascal’s own major breakthrough was using the triangle to help found modern probability theory, in a famous exchange of letters with fellow mathematician Pierre de Fermat about how to fairly split the stakes in an interrupted gambling game.