[x.com](https://twitter.com/burny_tech/status/1771877026334404900) Mathematical iceberg Explaining various mathematical concepts using examples with growing difficulty 1. Arithmetic: Arithmetic is the foundation of mathematics, dealing with basic operations like addition, subtraction, multiplication, and division. These operations are used in everyday life, such as calculating change at a store or determining the total cost of items. Example: If you buy 3 apples at $0.50 each and 2 oranges at $0.75 each, the total cost can be calculated using the equation: Total cost = (3 × $0.50) + (2 × $0.75) = $1.50 + $1.50 = $3.00 2. Algebra: Algebra introduces the concept of variables and equations, allowing us to solve problems with unknown quantities. It is used in various fields, including finance, engineering, and physics. Example: A rectangular garden has a length that is 3 meters longer than its width. If the perimeter of the garden is 26 meters, we can find the dimensions using the equation: 2(w + l) = 26, where w is the width and l is the length. Substituting l = w + 3, we get: 2(w + (w + 3)) = 26 Simplifying: 2(2w + 3) = 26 4w + 6 = 26 4w = 20 w = 5 Therefore, the width is 5 meters, and the length is 5 + 3 = 8 meters. 3. Geometry: Geometry deals with the properties and relationships of points, lines, angles, shapes, and solids. It has applications in architecture, design, and navigation. Example: The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (a² + b² = c²). If a ladder 5 meters long leans against a wall, and the base of the ladder is 3 meters from the wall, we can find the height it reaches using the Pythagorean theorem: 3² + h² = 5² 9 + h² = 25 h² = 16 h = 4 Therefore, the ladder reaches a height of 4 meters on the wall. 4. Trigonometry: Trigonometry is the study of the relationships between the angles and sides of triangles. It has applications in physics, engineering, and navigation. Example: In navigation, the distance to a lighthouse can be calculated using the angle of elevation and the height of the lighthouse. If the angle of elevation to the top of a 100-meter tall lighthouse is 35°, the distance to the lighthouse can be found using the tangent function: tan(35°) = 100 / d, where d is the distance to the lighthouse. Solving for d: d = 100 / tan(35°) ≈ 141.1 meters 5. Calculus: Calculus is the study of continuous change and has two main branches: differential calculus, which deals with rates of change and slopes of curves, and integral calculus, which deals with accumulation and areas under curves. Calculus has applications in physics, engineering, economics, and more. Example: In physics, the velocity of an object can be found by taking the derivative of its position function. If an object's position is given by the function s(t) = 3t² + 2t, where s is in meters and t is in seconds, the velocity at any time t can be found by taking the derivative: v(t) = ds/dt = 6t + 2 Therefore, the velocity of the object at t = 2 seconds is: v(2) = 6(2) + 2 = 14 meters per second 6. Probability: Probability is the study of the likelihood of events occurring. It has applications in various fields, such as insurance, finance, and weather forecasting. Example: In a fair coin toss, the probability of getting heads (H) or tails (T) is equal and can be expressed as: P(H) = P(T) = 1/2 or 0.5 If we toss a fair coin 3 times, the probability of getting exactly 2 heads can be calculated using the binomial probability formula: P(X = 2) = (3C2) × (1/2)^2 × (1/2)^(3-2) = 3 × (1/2)^2 × (1/2)^1 = 3/8 ≈ 0.375 where (3C2) is the number of ways to choose 2 heads out of 3 tosses, and (1/2)^2 and (1/2)^(3-2) represent the probabilities of getting a head and a tail, respectively. 7. Statistics: Statistics involves collecting, analyzing, and interpreting data. It has applications in various fields, such as social sciences, healthcare, and business. Example: The mean (average) of a dataset can be calculated using the formula: mean = (sum of all values) / (number of values) If the monthly sales of a store for the past 5 months were $10,000, $12,000, $15,000, $11,000, and $13,000, the mean monthly sales can be calculated as: mean = ($10,000 + $12,000 + $15,000 + $11,000 + $13,000) / 5 = $61,000 / 5 = $12,200 8. Linear Algebra: Linear algebra deals with linear equations, matrices, and vector spaces. It has applications in computer graphics, machine learning, and quantum mechanics. Example: In computer graphics, linear transformations like rotation can be represented using matrices. To rotate a 2D point (x, y) by an angle θ counterclockwise around the origin, we can use the rotation matrix: [cosθ -sinθ] [sinθ cosθ] If we want to rotate the point (3, 4) by 90° (π/2 radians) counterclockwise, we can multiply the point by the rotation matrix: [0 -1] [3] = [-4] [1 0] [4] = [3] Therefore, the rotated point is (-4, 3). 9. Number Theory: Number theory is the study of the properties of integers. It has applications in cryptography and computer science. Example: The Euclidean algorithm is used to find the greatest common divisor (GCD) of two integers. It states that the GCD of two integers a and b is the same as the GCD of b and the remainder of a divided by b. If we want to find the GCD of 1071 and 462, we can use the Euclidean algorithm: 1071 = 2 × 462 + 147 462 = 3 × 147 + 21 147 = 7 × 21 + 0 Therefore, the GCD of 1071 and 462 is 21. 10. Graph Theory: Graph theory deals with the study of graphs, which are mathematical structures used to model pairwise relations between objects. It has applications in network analysis, social networks, and route optimization. Example: Dijkstra's algorithm is used to find the shortest path between nodes in a graph. It assigns a tentative distance value to every node and updates these values if a shorter path is found. If we have a weighted graph with nodes A, B, C, and D, and we want to find the shortest path from A to D, we can use Dijkstra's algorithm: Initialization: A = 0, B = ∞, C = ∞, D = ∞ Step 1: A to B = 4, A to C = 2; A = 0, B = 4, C = 2, D = ∞ Step 2: C to B = 1, C to D = 4; A = 0, B = 3, C = 2, D = 6 Step 3: B to D = 2; A = 0, B = 3, C = 2, D = 5 Therefore, the shortest path from A to D is A → C → B → D with a total distance of 5. 11. Differential Equations: Differential equations are equations that involve derivatives of functions. They are used to model various phenomena in physics, engineering, and economics. Example: The exponential growth of a population can be modeled using a first-order differential equation: dP/dt = kP where P is the population size, t is time, and k is the growth rate constant. If a bacterial population starts with 1000 cells and grows at a rate of 20% per hour, the equation becomes: dP/dt = 0.2P, with P(0) = 1000 Solving this equation yields: P(t) = 1000e^(0.2t) This means that after 5 hours, the population will be: P(5) = 1000e^(0.2×5) ≈ 2718 cells. 12. Topology: Topology is the study of the properties of spaces that are preserved under continuous deformations, such as stretching or twisting, but not tearing or gluing. It has applications in physics, chemistry, and biology. Example: The Euler characteristic (χ) is a topological invariant that relates the number of vertices (V), edges (E), and faces (F) of a polyhedron: χ = V - E + F For a cube, we have 8 vertices, 12 edges, and 6 faces, so: χ = 8 - 12 + 6 = 2 This invariant helps classify surfaces and has implications in the study of manifolds and higher-dimensional spaces. 13. Combinatorics: Combinatorics is the study of counting and arranging objects. It has applications in probability theory, computer science, and operations research. Example: The number of ways to choose k objects from a set of n objects (without replacement and where the order doesn't matter) is given by the binomial coefficient: (nCk) = n! / (k!(n-k)!) If we have a group of 10 people and want to form a committee of 4, the number of possible committees is: (10C4) = 10! / (4!(10-4)!) = 10! / (4!6!) = 210 14. Numerical Analysis: Numerical analysis deals with the development and analysis of algorithms for solving mathematical problems using numerical approximation. It has applications in scientific computing, engineering, and finance. Example: The Newton-Raphson method is used to find the roots of a function f(x) by iteratively improving an initial guess x₀. The iteration formula is: x_{n+1} = x_n - f(x_n) / f'(x_n) If we want to find the square root of 2 (i.e., solve f(x) = x^2 - 2 = 0), we can start with an initial guess x₀ = 1.4 and apply the Newton-Raphson method: f(x) = x^2 - 2, f'(x) = 2x x₁ = 1.4 - (1.4^2 - 2) / (2×1.4) ≈ 1.4143 x₂ ≈ 1.4142 x₃ ≈ 1.4142 The method quickly converges to the square root of 2 (≈ 1.4142) after just a few iterations. 15. Chaos Theory: Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions. It has applications in weather forecasting, population dynamics, and economics. Example: The logistic map is a simple model of population growth that exhibits chaotic behavior. It is defined by the equation: x_{n+1} = rx_n(1 - x_n) where x_n is the population at time n, and r is a growth rate parameter. For r = 3.7 and x₀ = 0.1, the first few iterations of the logistic map are: x₁ = 3.7 × 0.1 × (1 - 0.1) ≈ 0.333 x₂ ≈ 0.8214 x₃ ≈ 0.5291 As the iterations continue, the population values exhibit chaotic behavior, meaning that small changes in the initial condition (x₀) can lead to drastically different outcomes. 16. Fourier Analysis: Fourier analysis is the study of representing functions as a sum of simpler trigonometric functions. It has applications in signal processing, image compression, and quantum mechanics. Example: A square wave can be represented as an infinite sum of sine waves using the Fourier series: f(x) = (4/π) × [sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + ...] This equation shows that a square wave can be decomposed into a fundamental sine wave and its odd harmonics, each with a specific amplitude. This concept is used in audio synthesis and digital signal processing. 17. Game Theory: Game theory is the study of strategic decision-making in situations where multiple players interact. It has applications in economics, political science, and psychology. Example: The Prisoner's Dilemma is a classic example in game theory. Two prisoners, A and B, are being interrogated separately. If both confess, each serves 2 years in prison. If A confesses and B does not, A goes free and B serves 3 years (and vice versa). If neither confesses, both serve 1 year. The payoff matrix can be represented as: B stays silent B confesses A stays (-1, -1) (-3, 0) silent A (0, -3) (-2, -2) confesses The Nash equilibrium (a stable state where no player can benefit by changing their strategy unilaterally) is for both prisoners to confess, even though it leads to a suboptimal outcome. 18. Knot Theory: Knot theory is the study of mathematical knots, which are closed curves in three-dimensional space. It has applications in physics, chemistry, and molecular biology. Example: The Jones polynomial is a knot invariant that distinguishes between different knots. It is defined by the recursive relations: V(∅) = 1 t^(-1)V(L₊) - tV(L₋) = (t^(1/2) - t^(-1/2))V(L₀) where L₊, L₋, and L₀ represent the link diagrams obtained by changing a single crossing in a specific way. For the trefoil knot, the Jones polynomial is: V(trefoil) = t + t^3 - t^4 This invariant helps classify knots and has implications in the study of DNA topology and quantum field theory. 19. Cryptography: Cryptography is the study of secure communication techniques in the presence of adversaries. It has applications in computer security, e-commerce, and digital currencies. Example: The RSA encryption algorithm is based on the difficulty of factoring large numbers. It uses modular exponentiation to encrypt and decrypt messages. To generate an RSA key pair: 1. Choose two large prime numbers, p and q. 2. Compute n = pq and φ(n) = (p-1)(q-1). 3. Choose an integer e such that 1 < e < φ(n) and gcd(e, φ(n)) = 1. 4. Compute d such that de ≡ 1 (mod φ(n)). The public key is (n, e), and the private key is (n, d). To encrypt a message m, compute c ≡ m^e (mod n). To decrypt a ciphertext c, compute m ≡ c^d (mod n). 20. Fractals: Fractals are complex geometric shapes that exhibit self-similarity at different scales. They have applications in computer graphics, antenna design, and modeling natural phenomena. Example: The Mandelbrot set is a famous fractal defined by the complex quadratic equation: z_{n+1} = z_n^2 + c where z₀ = 0, and c is a complex number. The Mandelbrot set consists of all values of c for which the sequence z_n remains bounded. It can be visualized by plotting the complex plane and coloring each point based on how quickly the sequence diverges. The equation generates intricate patterns with infinite detail, demonstrating the beauty and complexity of fractals. 21. Tensor Analysis: Tensor analysis is a generalization of vector calculus to higher dimensions. It has applications in general relativity, continuum mechanics, and machine learning. Example: In general relativity, the curvature of spacetime is described by the Riemann curvature tensor: R^a_{bcd} = ∂_c Γ^a_{bd} - ∂_d Γ^a_{bc} + Γ^a_{ce} Γ^e_{bd} - Γ^a_{de} Γ^e_{bc} where Γ^a_{bc} are the Christoffel symbols, which describe how the metric changes in different coordinate systems. The Einstein field equations relate the curvature of spacetime to the distribution of matter and energy: G_{μν} + Λg_{μν} = (8πG/c^4) T_{μν} where G_{μν} is the Einstein tensor, Λ is the cosmological constant, g_{μν} is the metric tensor, G is Newton's gravitational constant, c is the speed of light, and T_{μν} is the stress-energy tensor. These equations form the basis of our understanding of gravity and the large-scale structure of the universe. 22. Algebraic Topology: Algebraic topology uses algebraic structures, such as groups and rings, to study topological spaces. It has applications in data analysis, sensor networks, and robotics. Example: Persistent homology is a method in algebraic topology used to analyze the "shape" of data. It involves constructing a filtration of simplicial complexes from the data and computing the homology groups at each stage. The persistence diagram is a visualization of the birth and death times of homology classes across the filtration. It can be used to identify significant topological features and their scales. For a point cloud data set X, the Vietoris-Rips complex VR(X, ε) at scale ε is defined as: VR(X, ε) = {σ ⊆ X | ∀x, y ∈ σ, d(x, y) ≤ ε} where d(x, y) is the distance between points x and y. The persistent homology of the filtration {VR(X, ε) | ε ≥ 0} captures the topological features of the data at different scales, providing insights into its structure and shape. 23. Representation Theory: Representation theory studies the ways in which abstract algebraic structures, such as groups and algebras, can be represented as linear transformations of vector spaces. It has applications in quantum mechanics, coding theory, and harmonic analysis. Example: In quantum mechanics, the symmetries of a physical system are described by a group G, and the states of the system are represented by a vector space V. A representation of G on V is a homomorphism ρ: G → GL(V), where GL(V) is the group of invertible linear transformations on V. The Schrödinger equation for a quantum system with Hamiltonian H is: iℏ ∂/∂t |ψ(t)⟩ = H |ψ(t)⟩ where |ψ(t)⟩ is the state vector, ℏ is the reduced Planck's constant, and H is a linear operator representing the energy of the system. The symmetries of the Hamiltonian correspond to the representations of the symmetry group, and the irreducible representations provide a basis for the energy eigenstates of the system. 24. Stochastic Calculus: Stochastic calculus is the study of differential equations driven by random processes, such as Brownian motion. It has applications in financial mathematics, physics, and biology. Example: In financial mathematics, the Black-Scholes equation describes the price of a European call option under certain assumptions. It is a partial differential equation given by: ∂V/∂t + (1/2)σ^2 S^2 ∂^2V/∂S^2 + rS ∂V/∂S - rV = 0 where V is the option price, S is the stock price, t is time, σ is the volatility, and r is the risk-free interest rate. The equation can be derived using Itô's lemma, a key result in stochastic calculus that describes how a function of a stochastic process evolves: dV = (∂V/∂t + μS ∂V/∂S + (1/2)σ^2 S^2 ∂^2V/∂S^2) dt + σS ∂V/∂S dW where μ is the drift rate and dW is the increment of a Wiener process (Brownian motion). The Black-Scholes equation and Itô's lemma are fundamental tools in the pricing and hedging of financial derivatives. 25. Lie Theory: Lie theory studies continuous symmetry groups and their representations. It has applications in particle physics, control theory, and computer vision. Example: In particle physics, the Standard Model describes the fundamental particles and their interactions. The symmetries of the Standard Model are described by the Lie group SU(3) × SU(2) × U(1). The generators of the Lie algebra su(3) are the Gell-Mann matrices λ_i, which satisfy the commutation relations: [λ_i, λ_j] = i f_{ijk} λ_k where f_{ijk} are the structure constants of su(3). The Lie group SU(3) describes the strong interaction and the color symmetry of quarks, while SU(2) × U(1) describes the electroweak interaction and the symmetries of the Higgs field. The representations of these Lie groups determine the particle content and the allowed interactions in the Standard Model. 26. Conformal Field Theory: Conformal field theory (CFT) studies quantum field theories that are invariant under conformal transformations. It has applications in string theory, statistical mechanics, and condensed matter physics. Example: In two-dimensional CFT, the conformal symmetry is described by the Virasoro algebra, which is an infinite-dimensional Lie algebra with generators L_n satisfying the commutation relations: [L_m, L_n] = (m - n) L_{m+n} + (c/12) (m^3 - m) δ_{m+n,0} where c is the central charge, a parameter that characterizes the CFT. The primary fields φ(z) of the CFT are eigenfunctions of the Virasoro generators: L_n φ(z) = 0, for n > 0 L_0 φ(z) = hφ(z) where h is the conformal weight of the field. The correlation functions of primary fields satisfy the conformal Ward identities: ⟨T(z) φ_1(z_1) ... φ_n(z_n)⟩ = Σ_i (h_i / (z - z_i)^2 + ∂_i / (z - z_i)) ⟨φ_1(z_1) ... φ_n(z_n)⟩ where T(z) is the stress-energy tensor, which generates the conformal transformations. CFTs describe the critical behavior of statistical systems and the worldsheet dynamics of string theory.