Math Files

Life is nonlinear.
So handle it using Math.

[Math Files @Math_files]

In a 1926 interview with Collier's magazine, Nikola Tesla envisioned a world connected through wireless communication.

He described instantaneous global contact, seeing and hearing distant people, and communication devices small enough to fit in a vest pocket.

A striking anticipation of mobile communication and video calling, nearly a century before their widespread use.

Tesla envisioned the possibilities, not the detailed technology of today's smartphones.

[Math Files @Math_files]

In 1806, Sophie Germain feared that Carl Friedrich Gauss might be killed by Napoleon's invading army.

But there was a problem: Gauss did not know who Sophie Germain was.

He knew her as "Monsieur LeBlanc."

Germain, a self-taught French mathematician, had been corresponding with Gauss since 1804 under a man's name, fearing that her mathematics would be dismissed because she was a woman.

Then Napoleon's forces occupied Brunswick, where Gauss lived.

Remembering how Archimedes had been killed by a soldier during the Roman conquest of Syracuse, Germain became alarmed.

She contacted General Pernety, a family acquaintance in Napoleon's army, and asked him to ensure Gauss's safety.

An officer was sent to Brunswick.

He found Gauss alive and unharmed.

But when he mentioned Sophie Germain, Gauss explained that he did not know her.

The mystery was solved in February 1807.

Germain wrote to Gauss, revealing that she was the mathematician who had been sending him mathematical work as Monsieur LeBlanc.

Gauss's reply, dated 30 April 1807, expressed astonishment and admiration for her mathematical talent and determination.

What makes this story especially interesting is that the original correspondence survives.

It is a documented episode in which a mathematician's concern for a colleague's safety unexpectedly revealed her identity.

Walter Gratzer recounts the incident in Eurekas and Euphorias.

The episode also reveals something about the barriers women faced in nineteenth-century mathematics: Germain had to conceal her identity to be confident her work would receive serious consideration.

[Math Files @Math_files]

How long would a spacecraft take to approach the speed of light at a constant acceleration of 1g?

Special relativity gives:

v = c tanh(gτ/c)

After 1 year aboard: v ≈ 0.77c
After 2 years aboard: v ≈ 0.97c
After 3 years aboard: v ≈ 0.996c

Here τ is the time measured aboard the spacecraft.

The speed approaches c asymptotically but never reaches it.

[Math Files @Math_files]

A function can be continuous everywhere yet differentiable nowhere.

W(x) = Σₙ₌₀∞ 2⁻ⁿ cos(13ⁿπx)

The amplitudes shrink, ensuring continuity. But the oscillations become too rapid for a derivative to exist at any point.

[Math Files @Math_files]

Start with 82 and concatenate every integer down to 1.

82, 81, 80, 79, …, 3, 2, 1

Remove the separators. The result is a 155-digit prime number!

There are 73 two-digit integers and 9 single-digit integers, giving 73 × 2 + 9 = 155 digits.

Interestingly, 82 is the smallest starting integer that produces a prime this way.

[Math Files @Math_files]

Why 1/7 is so special

[Math Files @Math_files]

In order to be born, you needed:

2 parents
4 grandparents
8 great-grandparents
16 second great-grandparents
32 third great-grandparents
64 fourth great-grandparents
128 fifth great-grandparents
256 sixth great-grandparents
512 seventh great-grandparents
1,024 eighth great grandparents
2,048 ninth great-grandparents

For you to be born today from 12 previous generations, you needed a total of 4,094 ancestors over the last 400 years.

Think for a moment:
How many struggles?
How many battles?
How many difficulties?
How much sadness?
How much happiness?
How many love stories?
How many expressions of hope for the future? – did your ancestors have to undergo for you to exist in this present moment...

[Math Files @Math_files]

Every time you differentiate, one ball becomes three.

[Math Files @Math_files]

In 1712, Gottfried Wilhelm Leibniz appealed to the Royal Society to settle one of the bitterest disputes in the history of mathematics.

Who had invented calculus: Isaac Newton or Leibniz?

Newton had developed his method of fluxions during the 1660s. Leibniz had independently developed his own calculus during the 1670s and published it in 1684.

But Newton's supporters began accusing Leibniz of borrowing his ideas.

Furious at the accusation, Leibniz asked the Royal Society to investigate and defend his reputation.

There was one extraordinary problem.

The president of the Royal Society was Isaac Newton himself.

A committee was appointed in 1712 to examine the historical correspondence and determine who deserved priority.

Its report, published in 1713 under the title Commercium Epistolicum, strongly supported Newton.

But the investigation had a remarkable hidden feature.

Newton had personally selected evidence and substantially prepared the report that appeared under the Society's authority.

In effect, one of the two disputants had helped write the judgment in his own case.

And the story did not end there.

In 1715, an anonymous review appeared defending the Royal Society's report and Newton's claims.

The author?

Isaac Newton.

He had helped prepare the official account and then anonymously defended it.

The dispute continued until Leibniz's death in 1716, leaving a bitter chapter in the history of science.

Today, historians recognize that Newton and Leibniz independently developed calculus. Newton worked on it earlier, while Leibniz published first and introduced notation that remains fundamental.

Hal Hellman recounts this extraordinary rivalry in Great Feuds in Mathematics.

The episode reminds us that even the greatest mathematical minds were capable of personal bitterness, institutional maneuvering, and an intense desire for recognition.

Calculus changed mathematics forever. The battle over who deserved credit revealed a much more human side of its inventors.

[Math Files @Math_files]

A volunteer cubes a secret two-digit number and reveals 314,432.

You can recover 68.

Since 6³ < 314 < 7³, the root starts with 6. Among digit cubes, only 8³ ends in 2, so it ends in 8. The trick works for every perfect cube of a two-digit number.

[Math Files @Math_files]

This number is prime

[Math Files @Math_files]

Not every great mathematician was a child prodigy.

Lagrange turned seriously to mathematics only in his late teens.

Hilbert later said he was not an unusually brilliant child.

Grothendieck’s most extraordinary mathematical development came later,

while Minkowski was closer to a genuine early prodigy.

[Math Files @Math_files]

In 1907, while working at the Swiss Patent Office in Bern, Albert Einstein experienced what he later called "the happiest thought of my life."

He imagined a person falling freely from a rooftop.

During the fall, the person would feel weightless. Any object released alongside them would appear stationary relative to their body, neglecting air resistance and tidal effects.

Einstein recognized that a freely falling observer could locally eliminate the effects of gravity.

This insight became the foundation of the equivalence principle, connecting gravitation with acceleration.

Eight years later, in 1915, Einstein completed general relativity, replacing Newton's gravitational force with a geometric description of spacetime.

A thought experiment about falling became the starting point for a new theory of gravity.

[Math Files @Math_files]

A 22-year-old problem in matrix analysis was solved in 2026.

Shanmu Jin, a neurosurgery resident, used AI to help prove Crouzeix's conjecture:

‖p(A)‖₂ ≤ 2 max_{z∈W(A)} |p(z)|

Here W(A) is the numerical range of A.

The constant 2 is optimal. Independent proofs followed.

[Math Files @Math_files]

In 1978, John McKay noticed something that looked almost embarrassingly trivial.

But the two sides came from mathematical worlds that seemed unrelated.

196883 is the smallest nontrivial irreducible representation dimension of the Monster group, while 196884 appears as the first nonconstant coefficient of the modular-function.

[Math Files @Math_files]

Can you jump straight to a distant digit of π?

The Bailey–Borwein–Plouffe formula (1995) extracts hexadecimal digits without calculating the preceding ones.

In 2022, Simon Plouffe proposed a Bernoulli-number approach to decimal digits.

[Math Files @Math_files]

In 1902, a puzzle showed that an equilateral triangle could be cut into just four pieces and rearranged into a perfect square.

Could three pieces suffice?

A proof published in 2026 by Demaine, Kamata and Uehara established that four is the minimum for polygonal pieces rearranged without flipping.

A 124-year-old geometric question, settled.

[Math Files @Math_files]

Add 2 to the sum of the first 2, 4, 6, ... primes. The first TEN results are all prime: 7, 19, 43, 79, 131, 199, 283, 383, 503, 641. Then the 11th result breaks the pattern: 793 = 13 × 61.