OpenStem

@openstem · Joined Jul 2026

7420 public items8 groups

Note~282 words · 1 min

Slicing a circle two ways Central versus inscribed angles So a 90° central angle means every inscribed angle standing on that arc is 45°, wherever on the circle you put its vertex. Move the vertex all you like; the answer doesn't budge. The

Mathematics

Mathematics · L3 · Circles: Arcs, Sectors & Equations

@openstem

Note~293 words · 1 min

Identify the family first One question sorts nearly every solid: what does its cross-section do? The formula table For cones and pyramids, l is the SLANT height along the surface, not the vertical height h. Where the one-third comes from Sa

Mathematics

Mathematics · L3 · Surface Area & Volume of Solids

@openstem

Note~314 words · 2 min

When SOH-CAH-TOA runs out Those ratios need a right angle. Most triangles don't have one. Oblique triangles need two new tools. Law of Sines Every side divided by the sine of its opposite angle gives the same value. Use it when you know two

Mathematics

Mathematics · L3 · Law of Sines & Cosines

@openstem

Note~305 words · 2 min

Two kinds of series, two formulas Arithmetic series The trick behind it: pair the first term with the last, the second with the second-last, and so on. Every pair adds to the same total, and there are n/2 pairs. Worked, a concert hall. Fron

Mathematics

Mathematics · L3 · Arithmetic & Geometric Series

@openstem

Note~216 words · 1 min

Counting without listing How many passwords, seating plans, card hands? Break it into steps and multiply. The fundamental counting principle m ways for the first choice, n for the second, so m × n altogether. Extend it to as many steps as y

Mathematics

Mathematics · L3 · Permutations, Combinations & Probability

@openstem

Note~306 words · 2 min

What it is, and what breaks it A rational expression is a fraction with a polynomial on top and a polynomial underneath: (x + 2)/(x − 5). Because a zero denominator is undefined, it comes with restrictions attached. Simplify by factoring, t

Mathematics

Mathematics · L3 · Rational Expressions & Equations

@openstem

Note~252 words · 1 min

|x| is a distance |x| is how far x sits from 0 on the number line. Distances are never negative, so |x| ≥ 0 always. Equations: two points, two cases Two different points sit distance b from a: one above, one below. So the equation splits. W

Mathematics

Mathematics · L3 · Absolute Value Equations & Inequalities

@openstem

Note~270 words · 1 min

Distance is Pythagoras in disguise Draw a right triangle between two points: a horizontal leg |x₂ − x₁| and a vertical leg |y₂ − y₁|. The distance you want is the hypotenuse. From A(1, 2) to B(4, 6): d = √(3² + 4²) = √25 = 5. Midpoint is ju

Mathematics

Mathematics · L3 · Coordinate Geometry

@openstem

Note~321 words · 2 min

The General Transformed Equation Every transformation of a sine or cosine graph can be written using four constants, A, B, C, and D: Each one acts alone. Change B and the amplitude doesn't care. Amplitude and period Slide a and b below on t

Mathematics

Mathematics · L3 · Graphing Trig Functions

@openstem

Note~190 words · 1 min

A matrix is a labelled grid Rows and columns of numbers, each number an entry. Dimensions are always rows first, then columns. Two rows, three columns, so that's a 2 × 3 matrix. Adding and subtracting Both matrices must have identical dimen

Mathematics

Mathematics · L3 · Introduction to Matrices

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Note~289 words · 1 min

What a rational function is A polynomial over a polynomial. The domain excludes every x that makes the denominator zero, so (x − 1)/(x² − 4) is defined for all reals except x = 2 and x = −2. Vertical asymptote or hole? Factor the top and bo

Mathematics

Mathematics · L3 · Rational Functions & Asymptotes

@openstem

Note~240 words · 1 min

Sets and membership A set is a collection of distinct objects, its elements. List them in braces; use ∈ to say something belongs. Set-builder notation describes a set by a rule rather than a list, which is essential once the set is large or

Mathematics

Mathematics · L3 · Sets & Set Operations

@openstem

Note~292 words · 1 min

Propositions A proposition is a statement with a definite truth value: true or false, never both, never neither. The three connectives NOT (¬) just flips a single proposition's value. Building a truth table The conditional: if p, then q Con

Mathematics

Mathematics · L3 · Logic & Truth Tables

@openstem

Note~320 words · 2 min

Vertices, edges, degree A graph has vertices (dots) and edges (lines joining pairs of them). It's the right model whenever the relationships matter more than the things. The degree of a vertex is how many edges touch it. Degree 0 means isol

Mathematics

Mathematics · L3 · Graph Theory Basics

@openstem

Note~311 words · 2 min

Two ways to define a sequence Fibonacci Base cases F(1) = 1 and F(2) = 1. Then for n > 2, each term is the sum of the two before it. This one needs TWO base cases, because the rule reaches back two steps. Give it only one and the sequence c

Mathematics

Mathematics · L3 · Recursive Sequences & Fibonacci

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Note~350 words · 2 min

Mod is the remainder For a whole number a and a positive n, a mod n is what's left over after dividing. It always lands between 0 and n − 1. The clock is the picture A 12-hour clock counts to 12 then wraps back to 1. Adding hours is additio

Mathematics

Mathematics · L3 · Modular Arithmetic

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Note~295 words · 1 min

Two things, not infinitely many Induction proves P(n) for every natural number without checking each one. It does this by proving two much smaller statements. The base case starts the chain; the inductive step passes truth along it, one lin

Mathematics

Mathematics · L3 · Mathematical Induction

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Note~338 words · 2 min

Big-O measures shape, not seconds It answers one question: as the input n grows, how does the work grow? Five growth rates worth memorising Logarithmic growth is the good kind O(log n) is the flattest useful curve there is. Watch it below:

Mathematics

Mathematics · L3 · Algorithms & Big-O Basics

@openstem

Note~239 words · 1 min

Magnitude AND direction A vector carries both a size and a direction. A scalar carries only size. Speed is a scalar: 60 km/h. Velocity is a vector: 60 km/h north. Draw them as arrows. The length is the magnitude, the way it points is the di

Mathematics

Mathematics · L3 · Introduction to Vectors

@openstem

Note~276 words · 1 min

Different rules on different stretches A piecewise function uses one rule here and a different rule there. To evaluate it, first find which interval your x lives in, then apply only that piece. The floor function is a staircase The greatest

Mathematics

Mathematics · L3 · Piecewise Functions & Step Functions

@openstem

Note~358 words · 2 min

Three routes to the same destination Most theorems say "if P then Q". There's more than one valid way to get there, and the right choice can turn a painful proof into a three-line one. Direct proof Prove: if n is even, then n² is even. Assu

Mathematics

Mathematics · L3 · Proof Techniques: Direct, Contrapositive & Contradiction

@openstem

Note~267 words · 1 min

The greatest common divisor GCD(a, b) is the largest whole number dividing evenly into both. You could list every factor of each and pick the biggest shared one, but for large numbers that's slow. The Euclidean algorithm never factors anyth

Mathematics

Mathematics · L3 · Number Theory: GCD, LCM & the Euclidean Algorithm

@openstem

Note~387 words · 2 min

Distance and angle instead of across and up Cartesian coordinates locate a point by how far across and how far up. Polar coordinates use how far out and which way round. Read (r, θ) as: turn through θ, then go out a distance r. It's the nat

Mathematics

Mathematics · L3 · Polar Coordinates & Graphing

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Note~298 words · 1 min

Why you need new formulas The Pythagorean identity relates sin and cos of the SAME angle. It says nothing about sin(45° + 30°). The sum and difference formulas The cosine sign flip is the detail most people misremember. Sine keeps your sign

Mathematics

Mathematics · L3 · Sum, Difference & Double-Angle Identities

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Note~347 words · 2 min

Population versus sample All of statistical inference is one move: use what the sample tells you to say something responsible about the population you couldn't measure. Why not just ask everyone? Random versus biased In a random sample ever

Mathematics

Mathematics · L3 · Sampling & Statistical Inference Basics

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Note~342 words · 2 min

Adding and subtracting A complex number is a + bi: a real part and an imaginary part. Combine real with real, imaginary with imaginary, and never mix the two. Multiplying Expand like two binomials, then apply the one rule that makes complex

Mathematics

Mathematics · L3 · Complex Number Operations

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Note~350 words · 2 min

Bring in a clock Usually a curve is y in terms of x. Parametric form writes BOTH coordinates in terms of a third variable, the parameter t. As t changes, the point (x(t), y(t)) moves and traces a path. Tracing a curve Take x = t and y = t².

Mathematics

Mathematics · L3 · Parametric Equations & Motion

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Note~315 words · 2 min

Belief updating, written as arithmetic Bayes' theorem tells you how to revise a probability once evidence arrives. It's one equation. The denominator is just the law of total probability, summing both branches of the tree: The medical test,

Mathematics

Mathematics · L4 · Bayesian reasoning and Bayes' theorem

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Note~448 words · 2 min

Making "approaches" mean something "f(x) gets close to L as x gets close to a" is a sentence, not a definition. Cauchy and Weierstrass turned it into one. Read it as a game Note the strict 0 < |x − a|. The value f(a) is irrelevant, and may

Mathematics

Mathematics · L4 · ε–δ definition of a limit

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Note~380 words · 2 min

The vectors a matrix doesn't turn Most vectors get rotated and stretched by a matrix. A few only get stretched. Those are the eigenvectors, and the stretch factor is the eigenvalue. Finding them For a 2×2 matrix the characteristic polynomia

Mathematics

Mathematics · L4 · Eigenvalues and eigenvectors

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