Monocurl Essays

The cycle double cover conjecture, and its proof

2026-10-12T00:00:00Z

Every bridgeless graph has cycles that use each edge exactly twice. The whole proof, from bridges and paint to three-bit labels and a parity count.

Why is a negative times a negative positive?

2026-10-11T00:00:00Z

Multiplying by a negative number flips the number line about zero, and two flips cancel. We see it, then prove it from the distributive law.

The distributive law as a picture: FOIL and (a+b)²

2026-10-11T00:00:00Z

Cutting a rectangle into pieces explains FOIL, why (a+b)² has a middle term 2ab, and why (a+b)(a−b) = a² − b². Each identity is then proved.

Completing the square and the quadratic formula

2026-10-11T00:00:00Z

Solve x² + 6x = 16 by rearranging a square and a rectangle into a bigger square with one corner missing, then turn that picture into the quadratic formula.

What slope means, and why y = mx + b is a line

2026-10-11T00:00:00Z

A phone bill that grows by the same amount per gigabyte plots as a straight line. Similar triangles explain why, and give slope and y = mx + b their meaning.

Solving two equations at once: where lines meet

2026-10-11T00:00:00Z

Ten heads and 28 legs: how many chickens and rabbits? Each clue is a line of possibilities, and elimination pivots a line about the point where they cross.

Why x⁰ = 1, and negative and fractional exponents

2026-10-11T00:00:00Z

Powers of 2 halve as the exponent drops, and one rule, add exponents when you multiply, forces 2⁰ = 1, 2⁻¹ = 1/2 and 2^(1/2) = √2.

Absolute value is distance: solving |x − a| < r

2026-10-11T00:00:00Z

Read |x − 3| < 2 as "x is within 2 of 3" and the answer, 1 < x < 5, can be seen on a number line. Also why multiplying by a negative flips an inequality.

Why the angles of a triangle add to 180°

2026-10-11T00:00:00Z

Tear the corners off any triangle and they line up along a straight edge. A single parallel line explains why, and a sphere shows the line is needed.

The Pythagorean theorem: a proof you can see

2026-10-11T00:00:00Z

Why the squares on the two short sides of a right triangle add up to the square on the long side, proved by moving four triangles inside a box.

Why the area of a circle is πr²

2026-10-11T00:00:00Z

The number π is defined by the distance around a circle. Cutting the circle into sectors, and then into rings, shows why the same π gives its area.

Similar triangles and why area grows as the square

2026-10-11T00:00:00Z

Scaling multiplies every length by one factor and keeps every angle. That explains shadows, maps, and why a pizza twice as wide is four times the pizza.

The inscribed angle theorem and Thales' semicircle

2026-10-11T00:00:00Z

From every point on one arc of a circle, a chord is seen under the same angle, half the angle at the center. Two isosceles triangles explain why.

Why a cone is a third of a cylinder, and a sphere's volume

2026-10-11T00:00:00Z

A cone holds exactly one third of the cylinder around it. A cube cut into three pyramids shows why, and the same slicing idea gives the volume of a sphere.

The unit circle: sine and cosine as coordinates

2026-10-11T00:00:00Z

Sine and cosine start as ratios in a right triangle, which stops at 90°. Placing the triangle in a circle of radius 1 extends them to every angle.

Radians: measuring an angle by the arc it cuts

2026-10-11T00:00:00Z

A radian is the angle whose arc is as long as the radius. With it, arc length, sector area and the rule sin θ ≈ θ need no conversion constants.

Pythagorean identity: why sin² + cos² = 1

2026-10-11T00:00:00Z

The identity sin²θ + cos²θ = 1 is the Pythagorean theorem on the unit circle. Tangent and secant are lengths in the same picture and give two more.

Angle addition formulas: sin(a + b) from one picture

2026-10-11T00:00:00Z

The sine of a sum is not the sum of the sines. Two stacked right triangles show what it is, and a rotated chord proves the formulas for every angle.

Law of cosines and law of sines: solving any triangle

2026-10-11T00:00:00Z

The law of cosines is the Pythagorean theorem corrected for angles other than 90°. With the law of sines it solves any triangle, two-answer case included.

Sine waves: amplitude, frequency and phase

2026-10-11T00:00:00Z

A sine wave is the height of a point on a turning wheel. Amplitude, frequency and phase change the wheel, and explain how two waves add, cancel or beat.

Shifting and stretching graphs: why f(x - 2) moves right

2026-10-11T00:00:00Z

Adding 2 outside a function lifts its graph, but subtracting 2 inside moves it right. A graph is a set of points, and tracking one point explains why.

Compound interest and where e comes from

2026-10-11T00:00:00Z

Pay 100% interest in more and more instalments and a dollar grows to 2, 2.25, 2.44, 2.61, and then stops short of 2.72. That ceiling is the number e.

What a logarithm is: turning multiplication into addition

2026-10-11T00:00:00Z

A logarithm answers "what exponent gives this number?". Because exponents add when powers multiply, logarithms turn every product into a sum.

Complex numbers: why multiplying by i is a quarter turn

2026-10-11T00:00:00Z

No number on the line squares to -1. Read multiplication by -1 as a half turn and the missing number appears: a quarter turn, which needs a second axis.

Euler's formula: why e^(iπ) = -1

2026-10-11T00:00:00Z

You cannot multiply e by itself iπ times, so the famous equation needs a definition first. Compound interest supplies one, and it leads around a circle.

Geometric series, and why 0.999... equals 1

2026-10-11T00:00:00Z

What it means to add infinitely many numbers, when a geometric series has a sum, the formula a/(1 - r) with proof, and why 0.999... is exactly 1.

Roots, factors and the shape of a polynomial

2026-10-11T00:00:00Z

A cubic crosses the x axis once or three times, never exactly twice. The factor theorem, multiplicity and the leading term explain why, with proofs.

What is a derivative? Speed at a single instant

2026-10-11T00:00:00Z

A falling ball has a speed at every instant, but an instant is too short to measure a speed over. The derivative is the precise idea that resolves this.

Why the derivative of x² is 2x: the power rule from pictures

2026-10-11T00:00:00Z

A growing square gains two thin strips and a tiny corner. That picture explains the 2 in 2x, and counting the same pieces in general proves the power rule.

The product rule and the chain rule, seen

2026-10-11T00:00:00Z

A rectangle whose sides both change explains the product rule, and a nudge passed along a chain of number lines explains the chain rule. Both are then proved.

Why the derivative of sine is cosine

2026-10-11T00:00:00Z

Nudge a point around the unit circle and a tiny triangle appears, similar to the big one. Its legs are the derivatives of sine and cosine.

What is a limit? The epsilon-delta definition, explained

2026-10-11T00:00:00Z

A limit is a game. You name a tolerance, I answer with a window. From that one definition we prove the limit laws and the squeeze theorem.

What is an integral? Adding up to get area

2026-10-11T00:00:00Z

Distance from a changing speed, found by slicing time into pieces and adding. The definite integral is the limit of those sums, defined precisely and computed.

The fundamental theorem of calculus, explained and proved

2026-10-11T00:00:00Z

Why area and slope undo each other. The rate at which an area grows is the height of the curve, and that turns every integral into a subtraction.

Taylor series: building functions out of polynomials

2026-10-11T00:00:00Z

How a calculator finds cos(0.3) with only addition and multiplication, why factorials appear, how large the error can be, and where the method breaks.

Maxima and minima: where the slope is zero

2026-10-11T00:00:00Z

To fence the largest field or make the cheapest can, look where the derivative is zero, and also at the ends. Why that works, and when it does not.

What is a vector? Arrows, lists of numbers, and adding them

2026-10-11T00:00:00Z

A vector is a list of numbers read as a movement. Adding entry by entry matches placing arrows tip to tail, and two non-parallel arrows reach the whole plane.

A matrix is a transformation of space

2026-10-11T00:00:00Z

A 2x2 matrix moves every point of the plane, and its two columns say where the basis arrows land. Linearity is why those four numbers settle everything else.

Why matrix multiplication is defined the way it is

2026-10-11T00:00:00Z

Multiplying matrices means doing one transformation after another. The rows-times-columns rule, the failure of AB = BA and associativity all follow from that.

The determinant is how much area gets scaled

2026-10-11T00:00:00Z

A matrix multiplies every area in the plane by one number, ad - bc. That explains its sign, why det(AB) = det(A)det(B), and which matrices can be undone.

The dot product: projection, angle and why the formula works

2026-10-11T00:00:00Z

Multiply matching coordinates and add. The result measures how far one arrow reaches along another, which gives angles, projections and Cauchy-Schwarz.

Eigenvectors: the directions a transformation leaves alone

2026-10-11T00:00:00Z

Most arrows are knocked off their line by a matrix. The few that are only stretched are its eigenvectors, and they turn matrix powers into powers of numbers.

Solving Ax = b by undoing a transformation

2026-10-11T00:00:00Z

A linear system asks which vector a matrix sends to b. That view explains when there is one solution, none or a whole line, and what the inverse matrix does.

Surfaces and contour maps: functions of two variables

2026-10-11T00:00:00Z

A hiking map shows a mountain on flat paper. Its contour lines are the level sets of a function of two variables, and their spacing measures steepness.

The gradient: which way is uphill?

2026-10-11T00:00:00Z

On a hillside the slope depends on the direction you face. Partial derivatives give two of those slopes, and the gradient packs all of them into one arrow.

Lagrange multipliers: the best point on a path

2026-10-11T00:00:00Z

The highest point of a trail is where the trail just touches a contour line. There the gradients of the height and of the constraint are parallel.

Double integrals: volume under a surface

2026-10-11T00:00:00Z

Volume under a surface is a limit of sums of thin columns. Slicing turns it into two ordinary integrals, and polar coordinates handle round regions.

Divergence and curl: reading a vector field

2026-10-11T00:00:00Z

Two numbers at each point of a flow. Divergence is the net outflow from a tiny square per unit area, and curl is the circulation around it per unit area.

Green's theorem: the inside read from the boundary

2026-10-11T00:00:00Z

A walk around the edge of a region, adding up a vector field along the way, equals the integral of the field's curl over the inside. Here is why.

What probability means: counting equally likely outcomes

2026-10-11T00:00:00Z

Is a total of 7 with two dice three times as likely as a 12, or six times? Settling it leads to sample spaces, events and the rules every probability obeys.