Cheat Sheet: The Joy of X

A Guided Tour of Mathematics from One to Infinity.

Daniel Lanciana
14 min readMar 13, 2023




  • 69 is the only number where the square (4761) and cube (328509) use all digits 0–9
  • The right abstraction leads to new insights, and new power.
  • Maths always involves both invention and discovery: we invent concepts but discover their consequences.


  • Arithmetic deals with the study of numbers and their properties, such as addition, subtraction, multiplication, division, exponentiation, and extraction.
  • The word calculate comes from Latin calculus, meaning a pebble used for counting. Einstein translates to “one stone.”
  • Think of numbers as collections of rocks. Squaring a number actually makes a square shape.
  • Composite numbers are square or rectangular in shape, containing rows. Odd numbers have protuberances (i.e. L-shaped) — which when added (i.e. stacked) — produce squares.
  • Adding all consecutive odd numbers starting from 1 (1+3 = 4; 1+3+5 = 9, …) always produces perfect squares.
  • Commutative law is that the order in which two real numbers are added or multiplied does not affect the result.
  • Decimal from Latin “ten.”
  • Fractions are ratios of integers: rational numbers. The divide symbol (/) is a visual reminder that something is being sliced. Fractions always yield decimals that terminate or eventually repeat periodically. Since decimals can’t be equal to the ratio of any whole number they are irrational numbers. Irrationality is typical — almost all decimals are irrational.
  • The decimal representation of 1/7 repeats every six digits (0.142857142857…). Tripling decimal of 1/3 (0.0333…) you’re forced to conclude that 1 must equal 0.9999…!
  • Roman numerals were difficult to work with for large numbers. Introduction of bar on top of existing symbols to indicate multiplication by a thousand.
  • Babylonians used a base 60 numerical system — the smallest number divisible evenly by 1, 2, 3, 4, 5, and 6 (also 10, 12, 15, 20, and 30). More congenial for calculations that require cutting things into even parts. Minutes, seconds, and degrees in a circle are examples of base 60.
  • Our biology is embedded in counting — two hands of five fingers, base 10 counting using the Hindu-Arabic digits (Latin for “finger” or “toe”). The great invention is there is no symbol for ten; it’s marked by a position — the tens place — as part of a place-value system.


  • Developed by Islamic mathematicians around 800AD, which helped calculate inheritance according to Islamic law. Muhammad ibn Musa al-Khwarizmi coined al-jabr (Arabic for “restoring”), which later morphed into algebra. al-Khwarizmi morphed into algorithm.
  • Helps to visualise algebraic formulas (e.g. (50 + x)² = 2500) as a square divided into areas. Sets the stage for the “completing the square” process.
  • Identities are a type of formula such as factoring or multiplying polynomials.
  • In 600BC, India temple builders computed square roots to construct ritual altars.
  • Until the 1700s, mathematicians believed the square root of negative numbers couldn’t exist. The square root of -1 still goes by the demeaning name of i, for imaginary.
  • Complex numbers are when two types of numbers (real and imaginary) are bonded together to form a complex (or hybrid number); does not mean complicated. Better than real numbers because they always have roots. The pinnacle of number systems; universe of numbers completed.
  • The fundamental theorem states that the roots of any polynomial are always complex numbers.
  • Multiplying a number by i produces a rotation. Useful for voltages and electric/magnetic fields.
  • Quadratic equations include the square of the unknown. From Latin quadratus for “square.” Practical applications include radio tuning, swaying of skyscrapers, animal populations, and the arc of a basketball shot.
  • Today both positive and negative solutions are equally valid, but in the time of al-Khwarizmi the negative solution was ignored as it is geometrically meaningless.
  • Linear algebra is the study of vectors and matrices. It can detect patterns in large data sets, or perform computations involving millions of variables (e.g. Google’s PageRank, face recognition, voting patterns).


  • Functions transforms things — and are often referred to as transformations.
  • Power functions include parabolas (squaring functions), linear functions, and constants. Also inverse square functions, which describe how forces attenuate as they spread out in three dimensions (e.g. how sound dissipates).
  • Exponential growth is the miracle of compounding interest. Logarithms are the inverse; great compressors. Humans perceive music pitch logarithmically — notes rise by equal multiples. Other examples are the Richter and pH scales. Calling a salary “six figures” is roughly the logarithm of salaries.


  • Geometry (geo is “earth” in Latin; metry is “measurement”) created to solve land management issues.
  • Pythagorean theorem to find out how long the diagonal (obtusely called the hypotenuse, which is Greek for “[side] subtending the right angle”) of a triangle is, a² + b² = c².
  • Euclid’s Elements is the most reprinted textbook of all time. Isaac Newton used the logical reasoning structure in his masterwork The Mathematical Principles of Natural Philosophy; so did Spinoza’s Ethics Demonstrated in Geometrical Order. When Thomas Jefferson wrote “we hold these truths to be self-evident” in the Declaration of Independence he was mimicking the style of Elements. Unassailable logic that can make radical conclusions seem inevitable.
  • Ellipses have two points that act as foci — a ray emanating from point F1 in any direction will always find point F2. Defined as a set of points the sum of whose distances from two given points is constant. A whispering wall focuses all sound between two points. To draw an ellipses, put two pins in a wall connected by a piece of string — then use a pen to draw the shape by pulling the string taught; the foci are the two pins.
  • Parabolic curves focus at a single point, F. Defined as a set of all points equidistant from a given point and given line. Applications include amplification for audio recording (nature recording, live sports) and radio waves (satellite dishes). Rays entering will all focus on F, while if you place a light bulb at F it will create a directional beam.
  • Ellipses and parabolas are both cross-sections of the surface of a cone. If the cone is cut level, the intersection is a circle; if sliced with a gentle bias, intersection is an ellipse; if the slice matches the slope of the cone, a parabola; if sliced on a bias grearer than the slope of the cone, two pieces that form a hyperbola.
  • Circles, ellipses and parabolas are collectively known as conic sections — curves obtained by cutting the surface of a cone with a plane. In algebra, they are graphs of second-degree equations. In calculus, they are trajectories of objects (e.g. planets move in elliptical orbits) under gravity.


  • Goes beyond the measurement of triangles to include mathematics of cycles.
  • Practical applications in ocean waves, brain waves, electric generators, sound waves, ripples in a pond, ridges in sand dunes, and zebra stripes.
  • Pattern formulation is the emergence of sinusoidal structure from a background of bland uniformity.
  • Sine wave tracks something moving in a circle. sin a is pronounced “sine of a.” Whenever any state of featureless equilibrium loses stability, the first pattern to appear is a sine wave (or combination of them). The atoms of structure; nature’s building blocks.
  • Fourier analysis of sine waves shows unwanted oscillations caused by the Gibbs phenomenon, which can produce blurring or other artifacts in digital imaging. Gibbs artifacts can be spotted and cancelled out.
  • Quantum mechanics describe all matter as packets of sine waves.


  • The mathematics of change. Algebra works when something changes at a constant rate, but doesn’t work for changing change.
  • Practical applications include the spread of epidemics, the flight of a curveball, orbits of planets, circadian rhythms, or finding the best/cheapest/fastest path.
  • Two main concepts are the derivative (how fast something changes) and the integral (how much change is accumulating). Integrals were discovered in Greece around 250BC, while derivatives in England and Germany in the 1600s.
  • The fundamental theorem of calculus forged a powerful link between the two. If you integrate the derivative of a function from one point to another, you get the net change in the function between the two points.
  • Fastest path obeys Snell’s Law, which describes how light rays bend as they pass from air to water. The rays bend to minimise travel time; light behaves as if it were considering all possible paths, then taking the best one! Nature somehow knows calculus.
  • Integral sign is a long-necked S for “summation.” Integral calculus can sum all the atoms between the Earth and the sun — at least the idealised limit — by treating both as solid spheres composed of infinitely many points of continuous matter.
  • Take a can, cut off the top, then core a larger object (e.g. potato) from two mutually perpendicular directions. The result has square cross-sections created from round cylinders. A stack of infinite layers tapers from a large square in the middle. Nature unfolds in slices — virtually all the laws of physicals discovered in the past 300 years have this character. The conditions in each slice of time or space determine what will happen in adjacent slices. The implications were profound.


  • The limiting number approached by the sum — or how something changes through the accumulated effect of many tiny events.
  • Practical applications include radioactive decay or how many people you should date before choosing a mate.
  • If you took $1,000 that increased 50% over two periods it gives $2,250; divided by 100 equal periods of of 1% interest gives $2,704.81; if interest was calculated infinitely often (continuous compounding) the interest would be $1,000 multiplied by e, or $2,71828…

Differential Equations

  • The most powerful tool humanity has created for making sense of the material world. Newton used them to solve the ancient mystery of planetary motion for the two-body problem (e.g. a planet around the sun). The laws of physics are always expressed as differential equations.
  • Newton turned his attention to three-body problems (e.g. the sun, Earth and moon) but couldn’t solve it — and neither could anyone else. It contains the seeds of chaos.
Vector field


  • Latin root vehere meaning “to carry.” A step to a mathematician.
  • Vector algebra and vector calculus (vector fields). Vector fields emerge in magnetic-field lines.
  • In vector calculus, the derivative operator is named del — which uses the opposite symbol to the Greek letter for delta. Practical application includes how bumblebees and hummingbirds fly.
  • James Maxwell discovered that electric and magnetic fields could propagate as symbiotic waves — each pulling the other forward. He calculated the speed of these hypothetical waves as the same as the speed of light measured by Hippolyte Fizeau a decade earlier. Maxwell unified three seemingly unrelated phenomena: electricity, magnetism, and light.
  • These waves are used in radio, television, cell phones and wifi.


  • To find the meaning in the haystack of data.
  • Things that seem random and unpredictable when viewed in isolation often turn out to be orderly and predictable when viewed in aggregate.
Galton board
  • The idealised version of a bell curve is called the normal distribution, but it is not nearly as ubiquitous as it once seemed. Many phenomena look more like an L-curve — when looked through a logarithmic lens.
  • Power-law distributions have heavy tails. The 1987 stock market crash caused a drop of over twenty standard deviations, which is all-but-impossible (10 to 50th power) in traditional bell-curve statistics. The stock market is a heavy-tailed distribution; so are earthquakes, wildfires, and floods.
  • Avoid the complicated Bayes’s theorem by thinking in terms of natural frequencies. People often miscalculate risk.


  • The atoms of arithmetic, primes are atomic — indivisible. Just as everything is composed of atoms, every number is composed of primes.
  • Number 1 is not a prime! It should be, but is left out in modern mathematics to satisfy a theorem that states any number can be factored into primes in a unique way. If 1 was a prime, this would fail (e.g. 2x3, 1x2x2, 1x1x2x3, …). “One is the loneliest number.”
  • Number 2 is the only even prime. “Two, is the loneliest number since the number one.”
  • Twin primes are pairs separated by a non-prime number (e.g. 11 and 13, 17 and 19). They get progressively rarer, but never stop occurring. The prime number theorem was discovered by Carl Friedrich in 1792 (aged 15) and states the average gap between primes is approximately the natural logarithm (base e) of N.
  • Number theory provides the basis for encryption, which relies on the difficulty in decomposing a very large number into its prime factors.

Group Theory

  • One of the most versatile parts of mathematics. Necessarily abstract. Distills symmetry to its essence.
  • Practical applications include choreography, laws of particle physics, fractal mosaics, and when to flip your mattress for even wear (spin in the spring, flip in the fall”). Bridges art and science.
  • Instead of thinking of symmetry as a property of a shape, group theorists focus more on what you can do to a shape. Four transformations define the symmetries of the shape.
  • Do-nothing transformation provides the same role that 0 does for addition, or 1 for multiplication. Called the identity element, I.
  • Commutative law applies — the indifference to the order of transformations.


  • Offshoot of geometry, where two shapes are regarded as the same if they can be contorted — without ripping or puncturing — from one to the other.
The National Library of Kazakhstan is a Möbius strip
  • Möbius strips only have one side and one edge. If you cut it neatly down the middle, the strip will remain intact and grow twice as long! A conveyor belt can use a Möbius strip lasts twice as long by wearing out evenly on “both” sides.

Differential Geometry

  • Spherical geometry, pioneered by Carl Friedrich Gauss and Bernhard Reimann about 200 years ago. Studies the effects of small local differences on various shapes.
  • Shortest paths are critical for routing traffic on the internet.
  • The usual map of the world — the Mercator projection — is misleading. The most direct route takes the Earth’s curvature into account.
  • Great (i.e. largest) circles on a sphere contain the straightest (in the sense that there is no additional curving beyond following the surface) and shortest paths between any two points. Examples include the equator and longitudinal (North to South Pole) circles.
  • There are infinitely many locally shortest helical paths on a cylinder! They are the shortest of the candidate paths nearby, but none are the globally shortest path — which is a straight line.
  • Geodesics are locally shortest paths. Light beams follow them as curve through the space-time of the universe.
Two-holed torus


  • Archimedes realised the power of the infinite and came close to inventing calculus nearly 2,000 years earlier than Newton and Leibniz. He taught us the power of approximation and iteration — what is now the modern field of numerical analysis.
  • pi is the ratio of the distance across the circle through the centre (diameter) and the circumference around the circle. The area within the circle is A = πr².
  • Thinking mathematically about curved shapes by pretending they are made up of lots of little straight pieces. As you slice a circle the new shape approaches a rectangle. If there an infinite slices, the shape is a rectangle.
  • Exhaustion is the method of trapping an unknown number between two tightening bounds. The current record of pi is over 2.7 trillion decimal places; it will never be known completely.
  • Infinite sums reveal some unpleasant surprises — adding an infinite string of one and negative ones, depending on where you place the parenthesis, appears to be both 0 and 1! The debate went for 150 years until the two key notions of partial sums and convergence were introduced. The answer is S = 1 — S, or 1/2.
  • An alternating harmonic series (e.g. 1–1/2 + 1/3–1/4…) converges to the limiting value of the natural logarithm of two, denoted as ln2. But this series can be rearranged to converge to any (0, pi, 297.126) real number! This astonishing fact can be proven by the Riemann rearrangement theorem.
  • Georg Cantor discovered, shockingly, in the 1800s that some infinites are bigger than others!
The Hilbert Hotel
  • The Hilbert Hotel thought experiment. A hotel with infinite rooms that is always booked solid, yet there’s always a vacancy! Whenever a new guest arrives the manager shifts each occupant to the next room (room 1 goes to room 2, 2 to 3, etc.). What if an infinite number of new guests arrive? Shift all guests by doubling their room number (e.g. 1 to 2, 2 to 4, 3 to 6), which opens up all (infinitely many of them) the odd-numbered rooms. Now what if an infinity number of buses arrive, each with an infinite number of guests?! Infinity squared, whatever that means. To make sure all guests get a room eventually, a zig-zag pattern must be used — otherwise you never reach the second bus.
  • Cantor proved that there are exactly as many positive fractions (rations of p/q of positive whole numbers p and q) as there are natural numbers (1, 2, 3…) — a buddy system between the two pairs corresponding with passenger p on bus q. Implies we could make an exhaustive list of all positive fractions — even though there’s no smallest one.
  • Cantor proved (by contradiction) that some infinite sets are bigger than others! A set of real numbers between 0 and 1 is uncountable — it can’t be put in correspondence with the natural numbers. There won’t be enough room for all of them at the Hilbert Hotel.


  • Ezra Cornell invented the telegraph and worked for Samuel Morse. Created Western Union which connecting the North American continent. The first telegram from Baltimore the Washington DC in 1844 was “What hath God wrought.”
  • Complex dynamics is a vibrant blend of chaos theory, complex analysis, and fractal geometry.
  • Mathematical modelling is the valuable skill of making simplifying assumptions.
  • It’s hard to fold a piece of paper in half more than eight times as the thickness doubles exponentially while the length decreases exponentially fast. In 2002, a junior high schooler derived a formula then used a special roll of toilet paper 3/4 of a mile long — which she folded twelve times over seven hours smashing the world record!
  • Mathematical proofs don’t just convince; they illuminate.
  • The equals sign was created by Robert Recorde in 1557, which he described with “no two things can be more equal.”
  • How to slice a bagel in half such that the two pieces are locked together!
  • Vi Hart YouTube
  • Other books: Calculated Risks: How to Know When Numbers Deceive You, Visual Group Theory.