Math Files
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Life is nonlinear.
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This formula may look a bit strange to you, but that is the beauty of mathematics—sometimes you get a mathematical result in an astonishing way.
The second law of thermodynamics says that the total entropy of a closed system cannot decrease. Entropy is a measure of how many microscopic states are possible in a system.
At first, black holes might seem to have very little entropy because they are described by only a few properties, such as their mass, charge, and angular momentum.
However, in 1972, physicist Jacob Bekenstein proposed that black holes should have entropy.
His argument was based on the second law of thermodynamics. If an object with entropy falls into a black hole, its entropy would no longer be directly accessible outside the black hole. If the black hole had no entropy of its own, this could lead to a decrease in the total entropy of the universe, which would conflict with the second law.
So, how much entropy does a black hole have?
In 1974, Stephen Hawking showed that black holes have a temperature and emit thermal radiation, now called Hawking radiation. This means that black holes have thermodynamic properties.
Using the relationship between energy, temperature, and entropy, physicists found that the entropy of a black hole is proportional to the area of its event horizon.
This result is known as the Bekenstein–Hawking formula:
S = A/4ℓₚ²
where S is the black hole's entropy, A is the area of its event horizon, and ℓₚ is the Planck length.
The important point is that black holes do not violate the second law of thermodynamics. Instead, they can contain an extremely large amount of entropy. For a given region of space, a black hole represents the maximum possible entropy allowed by known physics.
So, although a black hole looks simple from the outside, its entropy can be extremely large.
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Muhammad ibn Musa al-Khwarizmi was a Persian scholar who worked in 9th-century Baghdad. His work had a major influence on mathematics, especially algebra and the idea of algorithms.
His book The Compendious Book on Calculation by Completion and Balancing, written around 820 CE, introduced systematic methods for solving equations. The Arabic word al-jabr in its title later became the English word algebra.
Al-Khwarizmi was born around 780 CE in Khwarazm, in present-day Uzbekistan. He later worked in Baghdad at the House of Wisdom, a major center of scholarship during the Islamic Golden Age.
His approach changed how mathematical problems could be solved. Instead of relying mainly on geometric figures, he described rules for manipulating equations, combining terms, and finding unknown quantities. These methods became an important part of later mathematics.
Al-Khwarizmi also helped spread the Hindu-Arabic numeral system, including zero and place-value notation. His work on Indian numerals was later translated into Latin. The Latin form of his name, Algoritmi, gave rise to the word algorithm.
His methods were based on clear, ordered steps: perform an operation, check a condition, and continue according to specific rules. This is closely related to the modern idea of an algorithm—a defined procedure for solving a problem.
Al-Khwarizmi died around 850 CE, centuries before modern computers existed. Yet the words algebra and algorithm, derived from his work and name, remain part of mathematics and computer science today.
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According to Gauss, “There have been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein.” Why, then, have so few people heard of the latter? Perhaps because Ferdinand Eisenstein died at the age of just 29.
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George Green was a baker and grain miller who became an important mathematician largely through self-study.
He was born in England in 1793 and spent most of his life working in his family business while studying mathematics in his spare time.
He had very little formal education, so it is not fully known how he learned advanced mathematics. Despite this, he developed ideas that later became important in mathematical physics.
In 1828, he published an essay that introduced what we now know as Green’s theorem. In simple terms, Green’s theorem connects what happens along the boundary of a closed region with what happens throughout the area inside it. Instead of calculating certain quantities only along the boundary, we can calculate them by considering the corresponding quantities over the entire region. Green’s theorem is closely related to the divergence theorem and Stokes’ theorem.
Green’s life also shows that important mathematical work can come from people working outside traditional academic settings.
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Best time to teach calculus
THE FIBONACCI SEQUENCE
In the Liber Abaci, Fibonacci posed the following problem about the number of offspring produced by a pair of rabbits:
"A man put one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year, if the nature of these rabbits is such that every month each pair bears a new pair which, from the second month on, becomes productive?"
Assuming that none of the rabbits dies, a pair is born during the first month, so there are two pairs present. During the second month, the original pair produces another pair. One month later, both the original pair and the firstborn pair produce new pairs, so that there are three adult pairs and two young pairs. This process continues in the same way.
The important point is that each month the young pairs mature and become adult pairs. Therefore, the new number of adult pairs is equal to the previous number of adult pairs plus the previous number of young pairs. Also, each pair that was adult in the previous month produces one young pair, so the new number of young pairs equals the previous number of adult pairs.
When continued indefinitely, the sequence that arises from the rabbit problem is
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …
This is called the Fibonacci sequence, and its terms are called Fibonacci numbers.
The position of each number in the sequence is traditionally indicated by a subscript. Thus,
u₁ = 1, u₂ = 1, u₃ = 2,
and so on, where uₙ denotes the nth Fibonacci number.
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A page from Richard Feynman's notebook
Hardy and Ramanujan’s most famous joint paper was in the theory of partitions, the ways of representing a given whole number n as the sum of positive whole numbers. The number 5, for example, can be “partitioned” seven ways:
5 = 5
5 = 4 + 1
5 = 3 + 2
5 = 3 + 1 + 1
5 = 2 + 2 + 1
5 = 2 + 1 + 1 + 1
5 = 1 + 1 + 1 + 1 + 1
As n grows, the number of partitions balloons.
For n = 10, there are 42.
For n = 50, there are 204,226.
For n = 100, there are 190,569,292.
And for n = 200 there are 3,972,999,029,588.
In 1918, in a forty-page paper on partition theory, Hardy and Ramanujan offered a surprisingly accurate asymptotic formula for the number of partitions of an integer n.
In 1942, Erdős was able to show that Hardy and Ramanujan didn’t need to use heavy machinery to deduce the first term of their formula, that the term could be found by “elementary” methods. Elementary techniques are not necessarily simpler. In this context, elementary means that the proof of the formula relies on a restricted set of numbers, the so-called real.
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The sum of the reciprocals of the primes i.e. 1/2 + 1/3 + 1/5 + ... is infinite, but the sum of the reciprocals of the known primes is less than 5 and will always be so!
Maryam Mirzakhani, the Fields Medalist who died of breast cancer at age 40, made important contributions to mathematics.
There are probably 100 mathematicians right now working full time on something she left behind.
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Visualization of the concept of integration by parts
On the left is Niels Bohr, one of the pioneers of quantum mechanics. On the right is Albert Einstein, known for his theory of relativity. This photograph was taken in December 1925 at the home of physicist Paul Ehrenfest in Leiden.
Einstein disagreed with the probabilistic nature of quantum theory and famously said, “God does not play dice.” Bohr responded, “Einstein, stop telling God what to do.”
Their disagreement became one of the most important debates in the history of physics. Although they strongly disagreed, they respected each other and continued to engage seriously with each other’s ideas.
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Plato: God ever geometrizes!
Jacobi: God ever arithmetizes!
Kronecker: God created the natural numbers, all else is the work of man!
When Henry Briggs (1561–1630) died, his epitaph claimed that ‘his soul still astronomizes and his body geometrizes’.
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This iconic photograph captures the legendary Danish theoretical physicist Niels Bohr (right) working on quantum equations at a blackboard with his son, Aage Bohr (left), looking on.
Together, they form one of the only four father-and-son duos in history to have both received Nobel Prizes in Physics.
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This formula, developed by Lennart Berggren, Jonathan and Peter Borwein in 1997, calculates π correctly to about 42 billion digits. It is discussed in more detail in a 2004 paper by Professor Thomas Osler.
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Is mathematics invented or discovered?
Tesla is often quoted.
“If you only knew the magnificence of the 3, 6 and 9, then you would have a key to the universe.”
The exact words are disputed but it doesn’t matter. The pattern is real.
The key tool is the digital root
keep adding a number’s digits until only one remains.
(Example: 38 → 3+8=11 → 1+1=2)
It’s just a fast way to find the remainder when divided by 9 (multiples of 9 give 9 instead of 0).
Now start with 1 and keep doubling, taking the digital root each time.
1 → 2 → 4 → 8 → 7 → 5 → 1…
An endless six-step cycle. 3, 6 and 9 never appear.
They form a fixed “spine” while the others rotate around them like a vortex.
Mathematically this happens modulo 9.
The rotating digits are exactly the numbers that share no common factors with 9.
Doubling visits every one of them and only them because 2 is a special generator (a primitive root) modulo 9.
That is the precise content of Tesla’s vortex.
The property isn’t automatic. It works for 9, 11, 13, 19… but fails for many others.
When it holds for a prime power, the rotating “shell” is always (p−1) times the fixed spine.
The ratio equals 2:1 if and only if the prime is 3.
So the only places that give Tesla’s exact geometry a nontrivial spine with a perfect 2:1 shell are the pure powers of 3.
9, 27, 81, 243, 729…
and that tower continues forever.
Base 10 made the pattern easy to notice, but the reason it’s special has nothing to do with counting in tens.
It’s special because 3 is the smallest odd prime true in every base. Roughly 37% of all primes allow a complete vortex like this.
Larger primes can never land on the spine at all.
The arithmetic simply makes the geometric intuition precise and shows that 9 is the smallest, cleanest place where it appears.
Tesla Was Right: The 3-6-9 Vortex as a Primitive-Root Law Prime Distribution, Spine Geometry, and the Artin Density of the Doubling
The Prime Lattice Coherence Framework: A Unified Master Document
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Leo Moser, quoted in Howard Eves, Mathematical Circles Adieu, 1977:
"It has been claimed that only three people read the three volumes of Russell and Whitehead’s monumental Principia Mathematica in its entirety, namely Russell, Whitehead, and the proofreader. There has been some scepticism, however, about Russell and Whitehead belonging to the list."
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