Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture.
Bertrand Russell
Welcome to the blog Math1089 – Mathematics for All.
Welcome to a captivating journey through the realm of mathematics! Prepare to be amazed as we unveil 40 Mathematical Marvels that will challenge your perceptions, spark your curiosity, and leave you in awe of the beauty and complexity of numbers, shapes, and patterns. From mind-bending paradoxes to facts, from mysteries to cutting-edge discoveries, this collection showcases the endless wonders of mathematics. Whether you’re a seasoned mathematician or simply curious about the power of numbers, join us on this fascinating exploration of mathematical marvels that will inspire, intrigue, and delight.
The symbols + and –, referring to addition and subtraction, first appeared in 1456 in an unpublished manuscript by the mathematician Johann Regiomontanus. The plus symbol, as an abbreviation for the Latin word ‘et’ (meaning ‘and’), was found earlier in a manuscript dated 1417; however, the downward stroke was not quite vertical.
In 1631, the multiplication symbol × was introduced by the English mathematician William Oughtred (1574–1660) in his book Keys to Mathematics.
Consider the function f(n) = n2 + n + 41, as discovered by Euler, which produces distinct primes for the 40 consecutive integers n = 0 to 39. There are many other prime-generating functions as well.
As simple as the playing board is, players can place their Xs and Os on the tic-tac-toe board in 9! = 362,880 ways.
Sometime around 2400 B.C., the ancient Sumerians noticed the apparent circular track of the Sun’s annual path across the sky and knew that it took about 360 days to complete the journey. Thus, it was reasonable for them to divide the circular path into 360 degrees to track the Sun’s daily movement. This eventually led to our modern 360-degree circle.
In 1631, the multiplication symbol × was introduced by the English mathematician William Oughtred (1574–1660) in his book Keys to Mathematics. He also introduced the abbreviations sin and cos for the sine and cosine functions.
The symbol ≥ (greater than or equal to) was first introduced by the French scientist Pierre Bouguer in 1734.
If the walls of a museum are all regular (or even a convex) polygons, one guard is sufficient to observe all the walls of a museum. In 1973, Victor Klee asked this question. In general, A museum with n walls can be guarded by n/3 guards.
In 1844, the French mathematician Joseph Liouville (1809–1882) considered the following interesting number
L=0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001. . .
known today as the Liouville constant. Can you guess what rule he used to create it?
The Copeland-Erdös constant is the (irrational) number 0.23571113171923 . . ., created by concatenating the prime numbers 2, 3, 5, 7, 11, . . .
Euler is responsible for our common, modern-day use of many famous mathematical notations—for example, f(x) for a function, e for the base of natural logs, i for the square root of –1, π for pi, and Σ for summation.
The comparison between doing nothing and making small, consistent efforts throughout the year is strikingly illustrated by the mathematical example provided.
(0.99)365 ≈ 0.025
(1.00)365 ≈ 1.000
(1.01)365 ≈ 37.783
The Austrian mathematician Christoff Rudolff was the first to use the square root symbol √ in print; it was published in 1525 in Die Coss.
Parallel lines represent soulmates who are never meant to meet each other.
Asymptotes are always getting closer but will never be together.
Successive discounts of 50%, 35%, and 15% don’t mean the item will come to you for free; rather, it is the same as a single discount of 72.375%!
The following equation is certainly eye catching. Notice that the sums on each side of the = sign total 365—the number of days in a year.
102 + 112 + 122 = 132 + 142.
The English mathematician John Wallis (1616–1703) introduced the mathematical symbol for infinity (∞) in 1655 in his Arithmetica Infinitorum.
73939133 is the largest number known such that all of its digits produce prime numbers as they are stripped away from the right!
73939133
7393913
739391
73939
7393
739
73
7
The difference between consecutive prime numbers is always even, except for two particular prime numbers. What are they?
The first person known to have proved that there are an infinite number of primes was Euclid (third century B.C.).
Consider the beautiful formula
relating five mathematical constants, e, π, i, and Φ , the golden ratio and √5. Do you know of any other such type of relation?
In 1637, the philosopher René Descartes was the first person to use the superscript notation for raising numbers and variables to powers—for example, as in x2.
Chairs commonly have four legs because it provides stability and balance. With four legs placed at each corner of the seat, the weight distribution is even, reducing the likelihood of the chair tipping over. This design also offers ample support for people sitting on the chair, ensuring comfort and safety. Additionally, four legs make it easier to level the chair on uneven surfaces compared to chairs with fewer legs. Overall, the four-legged design has become a standard for chairs due to its practicality and functionality.
If we assume that each of the variables A, B, C, D, E, F, and G represents a single digit, the only positive integer solution to the equation A × B × C = C × D × E = E × F × G is 8 × 1 × 9 = 9 × 2 × 4 = 4 × 6 × 3.
Ibn Yahya al-Maghribi Al-Samawal in 1175 was the first to publish x0 = 1. In other words, he realized and published the idea that any number raised to the power of 0 is 1.
A Reuleaux triangle is a curved triangle with constant width. It can be constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. They are named after Franz Reuleaux, a German engineer.
The greater-than and less-than symbols (> and <) were introduced by the British mathematician Thomas Harriot in his Artis Analyticae Praxis, published in 1631.
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton. Multiplication of quaternions is non-commutative and are generally represented in the form a + bi + cj + dk where the coefficients a, b, c, d are real numbers, and 1, i, j, k are the basis vectors.
In abstract algebra, the biquaternions are the numbers w + xi + yj + zk, where w, x, y, and z are complex numbers, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients.
Twelve fifty-nine (12:59) is the largest prime time of day on a 12-hour clock in hours and minutes. If you include seconds, it’s 12:59:59.
The Sierpinski triangle is a fractal named after the Polish mathematician Waclaw Sierpinski. It is formed by dividing an equilateral triangle into four smaller triangles and removing the middle one. The process is then repeated on the remaining triangles.
The fraction 1/998999 contains a number of obvious instances of Fibonacci numbers, 1, 1, 2, 3, 5, 8 . . . , in which each successive number is the sum of the previous two. I’ve underlined the Fibonacci numbers to make them easy to find:
1/998999 = 0.000001001002003005008013021034055089 . . .
A golygon (or serial isogon of 90°) is any rectilinear polygon with all right angles, whose sides are consecutive integer lengths. To create a golygon, start at one point, and take a first step one unit to the north or the south. The second step is two units to the east or the west. The third is three units to the north or the south. Continue until your path closes on itself and reaches the starting point. No crossing or backtracking is allowed.
If you concatenate positive integers, 1, 2, 3, 4, . . . , and lead with a decimal point, you get Champernowne’s number (C10),
C10 = 0.1234567891011121314 . . .
Like π, e, and Liouville’s number, Champernowne’s number is transcendental.
Can you get a bike with square wheels to roll smoothly on a road? Round wheel bikes are common, but a road made up of inverted catenaries will do the trick! See the following image.
The number 0.23571113171923 . . . is the famous Copeland-Erdös constant, created by concatenating the prime numbers 2, 3, 5, 7, 11, . . . in order. The constant is irrational.
Christian Kramp (1760–1826) introduced the ! as the factorial symbol in 1808 as a convenience to the printer.
The number 7777277227777772327777772222332222772333533327723555532772352532772355553277233353332772222332222777777232777777227727777 is the smallest star-congruent prime containing all four prime digits—and no other digits. This means that the 121-digit number can be arranged in the form of a six-pointed star:
The points orthocentre, circumcentre, centroid, and centre of nine-point circle are always collinear and they lie on the Euler line.
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