Math Concepts
One formula at a time — the plain meaning, the intuition, and where it comes from.
Numbers & Operations
21 conceptsPlace Value
A digit's value depends on which spot (place) it stands in.
What a Fraction Is
A number that tells how many equal parts the whole is cut into (bottom) and how many you have (top).
What a Decimal Is
A number that adds places to the right of a dot (the decimal point) to write parts smaller than one.
a/c ± b/c = (a ± b)/cAdding & Subtracting Fractions
When the slices are the same size (same bottom), just add or subtract how many slices (the tops).
Multiplication & Division
Multiplication is adding the same number many times; division is sharing equally or counting how many groups.
a : b = a ÷ bRatio & Rate
A ratio compares two amounts as 'this to that'; a rate turns that into one number by dividing.
p = (a ÷ b) × 100Percent
A number that says how much the part is when the whole is treated as 100. The sign % means 'out of 100'.
N = p₁^a₁ × p₂^a₂ × … × pₖ^aₖPrime Factorization
A single number broken down into a product of primes only — 2, 3, 5, 7, and so on.
Integers on the Number Line
The positive integers, 0, and the negative integers laid out in order on one line. Bigger to the right, smaller to the left.
|a| = a (a ≥ 0), |a| = −a (a < 0)Absolute Value
How far a number is from 0. Throw away the direction (the sign) and keep only the distance.
(+)×(+)=(+), (−)×(−)=(+), (+)×(−)=(−)Arithmetic with Rational Numbers
The rules for adding, subtracting, multiplying, and dividing integers and fractions. The key is to settle the sign first, then compute the sizes.
aᵐ · aⁿ = aᵐ⁺ⁿLaws of Exponents
A power is the same number multiplied several times. Multiply or divide those, and you just add or subtract the exponents.
√a = b ⇔ b² = a (a ≥ 0, b ≥ 0)Square Roots
When some number squared gives a, that number is a square root of a. √a is the one that is 0 or positive.
Irrational & Real Numbers
Numbers you can write as a fraction (integer/integer) are rational; those you can't are irrational. Together they form the reals.
0.ddd… = d/9, 0.abab… = ab/99Repeating Decimals
A decimal where the same block of digits repeats forever after the point. It can always be turned back into a fraction.
GCD(a, b) × LCM(a, b) = a × bGCD and LCM
The GCD is the largest number dividing both; the LCM is the smallest number that is a multiple of both — the first place their multiples overlap.
i² = −1, z = a + biComplex Numbers
Introduce a new number i whose square is −1, and every number can be written as a+bi, a real part a plus an imaginary part bi. That's a complex number.
(G, ∗): ① a∗b ∈ G ② (a∗b)∗c = a∗(b∗c) ③ a∗e = e∗a = a ④ a∗a⁻¹ = eGroups: The Language of Symmetry
A group is a set plus an operation, where the operation obeys four rules: (1) results never leak out (closure), (2) grouping doesn't matter (associativity), (3) there is a do-nothing identity, and (4) every element has an inverse that undoes it.
a ≡ b (mod n) ⟺ n | (a − b)Modular Arithmetic
Modular arithmetic is arithmetic that wraps back to 0 after some number n. Two numbers are 'congruent' if they leave the same remainder when divided by n, written a ≡ b (mod n). Only the remainder matters.
e^(iθ) = cos θ + i sin θ, e^(iπ) + 1 = 0The Complex Plane & Euler's Formula
See a complex number a+bi as a point (or arrow) in the plane, and you get the complex plane. Euler's formula e^(iθ)=cos θ+i sin θ says that the complex exponential e^(iθ) is the point on the unit circle rotated by angle θ. In short: an imaginary exponent means rotation.
|ℕ| = |ℤ| = |ℚ| = ℵ₀ < |ℝ|Countable Infinity
If the elements of an infinite set can be numbered 1,2,3,… with none left out, it is 'countably infinite'. The naturals, integers, and rationals all qualify. Yet the reals cannot be numbered that way even in principle — they are a larger infinity.
Algebra
18 concepts2x + 3Variables & Expressions
'Double some number and add 3' written compactly with the letter x as 2x+3.
ax + b = 0 → x = −b/aLinear Equations
An equality that's true for exactly one value of x. Finding that value is what 'solving the equation' means.
a₁x + b₁y = c₁, a₂x + b₂y = c₂Systems of Linear Equations
Finding the one pair (x, y) that satisfies both unknowns at once. You need two equations to pin the answer to a single point.
ax + b > 0Linear Inequalities
A statement that x is greater or less than something. The answer isn't one number but a 'range'.
y = ax + bLinear Functions & Their Graphs
A rule where you feed in x and out comes y = 'a times x plus b'. Graph it and you always get a straight line.
(a + b)² = a² + 2ab + b²Multiplication Formulas
(a+b)² is not a²+b²; a 2ab squeezes into the middle, making it a²+2ab+b².
x² + (a + b)x + ab = (x + a)(x + b)Factoring
Turning something like x²+5x+6 into a 'product of two expressions' such as (x+2)(x+3). It's the exact reverse of expanding.
ax² + bx + c = 0 → x = (−b ± √(b² − 4ac)) / (2a)Quadratic Equations
An equation with an x² in it, ax²+bx+c=0. It has at most two solutions, and plugging a, b, c into the quadratic formula always produces them.
y = a(x − p)² + qQuadratic Functions
y = a(x−p)² + q. The graph is a parabola, and the vertex (p, q) is its 'pointed tip' (the highest or lowest point).
y = aˣ ⇔ x = logₐ yExponential & Logarithmic Functions
aˣ multiplies a by itself x times; logₐ y asks back 'how many times must I multiply a to reach y?'. Exponent and log undo each other.
aₙ = a₁ + (n−1)d, aₙ = a₁·rⁿ⁻¹Arithmetic & Geometric Sequences
An arithmetic sequence adds a fixed number d to the previous term each time; a geometric sequence multiplies the previous term by a fixed number r each time.
Σₖ₌₁ⁿ k = n(n+1)/2Sigma Notation & Series
Σ is a command: 'plug in from the bottom value up to the top value and add them all'. Σₖ₌₁ⁿ k is the sum from 1 to n.
(AB)ᵢⱼ = Σₖ aᵢₖ bₖⱼMatrices
A matrix arranges numbers in rows and columns. To multiply two matrices, you pair a 'row of the left' with a 'column of the right', multiply, and add.
(a + b)ⁿ = Σₖ₌₀ⁿ ₙCₖ aⁿ⁻ᵏ bᵏThe Binomial Theorem
When you expand (a+b)ⁿ, each term's coefficient is ₙCₖ — the number of ways to choose b from k of the n brackets.
T(v) = Av, T(u+v) = T(u)+T(v), T(cu) = c·T(u)Linear Transformations
A linear transformation moves every point of the plane (or space) while keeping grid lines straight, evenly spaced, and the origin fixed. This orderly motion is captured completely by a single matrix.
Av = λv (v ≠ 0); det(A − λI) = 0Eigenvalues & Eigenvectors
Apply the transformation A and most vectors get knocked off their direction. But certain special vectors v keep their direction and only get scaled by a factor λ. Such a v is an eigenvector, and the factor λ is its eigenvalue.
u, v ∈ V, a, b ∈ ℝ ⟹ a·u + b·v ∈ VVector Spaces
A vector space is a set whose elements never escape it when you add them together or multiply by a real number. Those two operations just need to obey a few natural rules.
det([[a, b], [c, d]]) = ad − bcThe Determinant
The determinant of a 2×2 matrix [[a,b],[c,d]] is ad − bc. This number tells you the factor by which the matrix scales any shape's area.
Geometry
16 conceptsAngles
How far apart two lines (sides) spread from one point. We measure that spread in degrees (°).
▭ S = a × b △ S = c × h ÷ 2Area of Rectangles & Triangles
The amount of flat space a shape covers, counted as how many unit squares fit inside.
▭ P = 2 × (a + b) ⬛ P = 4 × aPerimeter
The total length of the outline once around — just add up all the sides.
V = a × b × cVolume of a Box
The amount of space a solid takes up, counted as how many 1 cm cubes fit inside.
Line Symmetry
The property that folding along a line (the axis) makes the two halves match exactly.
a² + b² = c²The Pythagorean Theorem
In a right triangle, square each of the two shorter sides and add them, and you land exactly on the square of the hypotenuse.
△ABC ≅ △DEFCongruent Triangles
Two triangles being congruent means you can rotate or flip one onto the other perfectly. Every corresponding side and angle is equal.
a : a' = b : b' = c : c' = kSimilarity
Two figures being similar means all corresponding angles are equal and all corresponding sides are the same multiple (k times) of each other.
(n − 2) × 180°Interior & Exterior Angles of Polygons
Add up the interior angles of an n-gon and you get (n−2) × 180°. And the exterior angles of a convex polygon always add up to 360°.
π r² × (θ / 360)Arcs and Sectors of a Circle
A sector is part of a circle. Its central angle θ takes up some fraction of a full turn (360°), and it grabs that same fraction of the circle's total area (or circumference).
d = √((x₂ − x₁)² + (y₂ − y₁)²)Distance in the Coordinate Plane
Find the horizontal gap and the vertical gap between the two points and they form a right triangle; the segment joining the points is its hypotenuse. So the distance is √(horizontal² + vertical²).
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adjTrigonometric Ratios
Pick one acute angle θ in a right triangle, and the ratios between its sides are fixed by the angle alone. Those ratios are sin, cos, and tan.
sin²θ + cos²θ = 1Trigonometric Functions
On a circle of radius 1, after turning by angle θ, the point's vertical position is sinθ and its horizontal position is cosθ. tanθ is their ratio.
(x − a)² + (y − b)² = r²The Equation of a Circle
The points (x,y) whose distance from the center (a,b) is always r form the circle. Writing that condition as an equation gives (x−a)²+(y−b)²=r².
|v| = √(x² + y²)Vectors
A vector is an arrow carrying both size and direction. Moving x sideways and y up is written (x,y).
a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cosθ; |a×b| = |a||b|sinθDot & Cross Products
The dot product a·b is a single number measuring how aligned two vectors are. The cross product a×b is a new vector standing perpendicular to the plane the two vectors span.
Analysis & Calculus
21 conceptslim(n→∞) aₙ = LThe Limit of a Sequence
As you push the term number n to infinity, if the terms keep getting closer to some single number L, then L is the limit of the sequence.
lim(x→a) f(x) = LLimits
Push x closer and closer to some value a, and watch which number the function crowds toward.
lim(x→a) f(x) = f(a)Continuity
If the value you expected on approach (the limit) exactly equals the value actually marked at that point, the curve runs through it unbroken.
f'(x) = lim(h→0) [f(x+h) − f(x)] / hThe Derivative — Slope at an Instant
Take the slope between two points (rise ÷ run), then slide the two points infinitely close to get 'the slope at a single point'.
d/dx xⁿ = n·xⁿ⁻¹The Power Rule
The slope function of xⁿ is: multiply out front by the exponent n, and lower the exponent to n−1.
(f·g)' = f'·g + f·g'The Product Rule
To differentiate a product of two functions, differentiate one at a time and add the two results.
(f(g(x)))' = f'(g(x))·g'(x)The Chain Rule
To differentiate a composite (a function inside a function), multiply the outer function's derivative by the inner function's derivative.
f'(x) = 0Critical Points — Where the Slope Is Zero
Set the derivative equal to 0 and solve; the x's you get are where the tangent goes flat — candidates for a max or min.
f''(x) > 0 ⇒ ∪ , f''(x) < 0 ⇒ ∩Concavity and the Second Derivative
The second derivative f'' is the rate of change of the slope. Its sign tells you whether the curve bends like a cup (∪) or a cap (∩).
∫ₐᵇ f(x) dx = lim Σ f(xᵢ)·ΔxThe Integral — Accumulating Area
The area under a curve, filled in with many very narrow rectangles whose areas you add together. Send their width to 0 and you get the exact area.
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)The Definite Integral
The area over [a,b], computed not by a Riemann sum but by finding an antiderivative F and taking F(b)−F(a).
d/dx ∫ₐˣ f(t) dt = f(x)The Fundamental Theorem of Calculus
Differentiate the function that has been accumulating area from a to x, and the original function f(x) pops right back out. Differentiation and integration are inverses.
e = lim(n→∞) (1 + 1/n)ⁿ ≈ 2.71828The Number e
Take 100% interest, chop it into ever-finer compounding steps, and as the number of steps goes to infinity the result approaches e ≈ 2.718.
ln x = y ⇔ eʸ = xThe Natural Logarithm
ln x answers 'to what power must e be raised to get x'. In other words, it's the inverse of the exponential eˣ.
d/dx eˣ = eˣDeriving the Exponential — e^x Is Its Own Slope
The slope function of eˣ is eˣ again. At any point, the height there is exactly the slope there.
f(x) ≈ f(a) + f'(a)(x−a) + ½f''(a)(x−a)²Taylor Approximation — Curves as Polynomials
A complicated function approximated, near one point, by a simple polynomial tuned to share the same value, slope, and bend there.
lim(x→a) f(x) = L ⟺ ∀ε>0, ∃δ>0 : 0 < |x−a| < δ ⟹ |f(x)−L| < εThe ε–δ Definition of a Limit
If, however small an error ε your opponent names, you can shrink the input window δ so that f(x) stays within ε of L, then the limit is L.
S = Σ(n=1..∞) aₙ = lim(N→∞) (a₁ + a₂ + ⋯ + a_N); Σ(n=0..∞) arⁿ = a/(1−r), |r|<1Convergence of Series
The 'sum' of an infinite series is not literally added to the end. You form the partial sum of the first N terms; if that value approaches some number S as N grows, the series 'converges to S'.
z = f(x, y) → ∂f/∂x, ∂f/∂yPartial Derivatives
For a function of two or more variables, you freeze every variable except the one you care about, then take an ordinary derivative.
∬_R f(x, y) dA = ∫∫ f(x, y) dx dyMultiple Integrals
You chop the base region R into tiny rectangles, multiply each patch's area by the function's height above it to make a little column, and add up every column.
f(x) = a₀/2 + Σₙ (aₙ cos nx + bₙ sin nx)Fourier Series
A periodic function can be written exactly as a sum of sines and cosines whose frequencies are integer multiples of a base. You only need to decide how much of each pure tone goes in (the coefficients aₙ, bₙ).
Probability & Statistics
14 conceptsN = m × nCounting Cases
Things that can't happen together you ADD (the sum rule); things that happen one after another you MULTIPLY (the product rule).
P(A) = a / nWhat Probability Means
Count the equally likely outcomes, then take the share of them that are the ones you're hoping for.
P(Aᶜ) = 1 − P(A)Complementary Events
The probability that something does NOT happen is the whole probability 1 minus the probability that it does.
x̄ = (x₁ + x₂ + ⋯ + xₙ) / nMeasures of Center
The mean is the sum divided by the count, the median is the middle value in order, and the mode is the value that shows up most.
σ² = { (x₁−x̄)² + ⋯ + (xₙ−x̄)² } / nSpread: Variance & Standard Deviation
The average of the squared deviations of each value from the mean is the variance; its square root is the standard deviation.
ₙPᵣ = n!/(n−r)!, ₙCᵣ = n!/(r!(n−r)!)Permutations & Combinations
Picking r from n items: if the order you picked them in matters, it's a permutation (P); if only 'who got picked' matters, it's a combination (C).
P(B|A) = P(A∩B) / P(A)Conditional Probability
Given that event A has already happened, it's the probability B happens under that condition. You shrink the world to A, then weigh B.
P(X=k) = ₙCₖ pᵏ (1−p)ⁿ⁻ᵏThe Binomial Distribution
It's the distribution of the probability of getting k successes when you run n independent trials, each with the same success probability p.
P(μ−σ ≤ X ≤ μ+σ) ≈ 0.68The Normal Distribution
A bell-shaped distribution, symmetric about its mean. Data clusters most densely near the mean and thins out sharply as you move away.
E(X) = Σ xᵢ pᵢExpected Value & Variance
When you know the probabilities of the outcomes, it's each value times its probability, all summed. Repeat enormously many times and it's the 'long-run average' things settle to.
P(A|B) = P(B|A)·P(A) / P(B)Bayes' Theorem
Bayes' theorem is the rule for updating a belief with new evidence. Take the probability of A before seeing B (the prior), multiply by how likely B is if A holds, divide by the total chance of B, and you get the probability of A after seeing B (the posterior).
x̄ₙ = (X₁ + X₂ + ⋯ + Xₙ) / n → μ (n → ∞)The Law of Large Numbers
The law of large numbers says that if you repeat the same random experiment many times, the average of the results gets closer and closer to the theoretical expected value μ. Individual outcomes stay unpredictable, but the average becomes predictable.
(x̄ₙ − μ) / (σ/√n) → N(0, 1) (n → ∞)The Central Limit Theorem
The central limit theorem says that whatever the shape of the original distribution—flat or skewed—if you gather many independent samples and take their average (or sum), the distribution of those averages approaches the normal (bell) shape.
P(Xₙ₊₁ | X₀…Xₙ) = P(Xₙ₊₁ | Xₙ); πP = πMarkov Chains
It is a process that hops randomly among states, where the probability of the next state depends only on the current state and not at all on the path taken to get there. This 'memorylessness' is the Markov property.
Discrete & Logic
4 conceptsn(A∪B) = n(A) + n(B) − n(A∩B)Sets and Venn Diagrams
The size of the union is the two sets' sizes added, then the overlap (the intersection) subtracted exactly once.
(p → q) ≡ (~q → ~p)Propositions: True or False
A proposition is a sentence you can clearly judge true or false. A conditional like 'if p then q' always shares the same truth value as its contrapositive.
G = (V, E), Σ(v∈V) deg(v) = 2|E|Graph Theory
A graph is a collection of dots (vertices) and lines (edges) joining them. It records only what is connected to what, throwing away position, shape, and distance. It's a map of relationships stripped to the skeleton.
[ P(1) ∧ ( P(k) ⟹ P(k+1) ) ] ⟹ ∀n ≥ 1, P(n)Mathematical Induction
Mathematical induction proves statements true 'for every natural number n'. (1) Show it holds at n=1 (the base), and (2) show that if it holds at n=k it must hold at n=k+1 (the step); like dominoes, they all fall.
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