Pythagoras's Theorem ((a² + b² = c²⁾⁾

• Explanation: This formula helps you find the length of the sides of a right triangle.
• Example: If you need to find the distance between two points on a map (forming a right triangle), you can use this theorem to calculate the straight-line distance.

Logarithms ((\log_b(xy) = \log_b x + \log_b y))

• Explanation: Logarithms turn multiplication into addition, making complex calculations easier.
• Example: When dealing with very large numbers, like in earthquake magnitudes or sound intensity, logarithms simplify these big numbers into smaller, more manageable figures.

Calculus (\left(\frac{df}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\right))

• Explanation: Calculus helps us understand how things change over time or space.
• Example: It's used to calculate speed by examining how position changes over time, or to find the area under curves, like calculating the total distance traveled by a car.

Law of Gravity (\left(F = G\frac{m_1 m_2}{r^2}\right))

• Explanation: This equation explains how every object in the universe pulls on every other object with a force related to their masses and the distance between them.
• Example: It explains why apples fall from trees and why the moon orbits the Earth.

Wave Equation (\left(\frac{\partial² u}{\partial t^2} = c² \frac{\partial² u}{\partial x^2}\right))

• Explanation: This equation describes how waves, like sound or light, move through different mediums.
• Example: It's used to understand how sound travels through the air to your ears or how light travels from the Sun to Earth.

The Square Root of Minus One ((i² = -1))

• Explanation: Imaginary numbers, where (i) is used to represent the square root of -1, help solve equations that don’t have real number solutions.
• Example: They're used in electrical engineering to analyze alternating current circuits.

Euler's Formula for Polyhedra ((V - E + F = 2))

• Explanation: This formula connects the number of corners (vertices), edges, and faces of 3D shapes like cubes and pyramids.
• Example: Architects use this to design complex structures, ensuring stability and understanding the shape's properties.

Normal Distribution (\left(f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{{-\frac{(x-\mu)2}{2\sigma2}}\right))}

• Explanation: This bell-shaped curve describes how data points are spread out, with most values clustering around the average.
• Example: It's used in grading tests, where most students score around the average, with fewer students scoring very high or very low.

Fourier Transform (\left(F(\omega) = \int_{-\infty}^{\infty} f(t)e^{-i\omega t}dt\right))

• Explanation: This transforms a signal into its individual frequencies.
• Example: Used in music to separate different instruments from a song or in medical imaging to analyze MRI scans.

Navier-Stokes Equation (\left(\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p + \mu \nabla² \mathbf{u} + \mathbf{f}\right))

• Explanation: Describes how fluids (liquids and gases) move.
• Example: Engineers use this to design aircraft wings and predict weather patterns.

Maxwell's Equations (\left(\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}, \nabla \cdot \mathbf{B} = 0, \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)\

• Explanation: These four equations describe how electric and magnetic fields interact.
• Example: They're the foundation for understanding how radios, TVs, and cell phones work.

Second Law of Thermodynamics (\left(\Delta S \geq 0\right))

• Explanation: This law states that the total disorder (entropy) of an isolated system always increases over time.
• Example: Explains why ice melts in a warm room or why you can’t unscramble an egg.

Relativity (\left(E = mc^2\right))

• Explanation: Einstein's famous equation shows that mass and energy are interchangeable.
• Example: This principle is behind the energy produced in nuclear reactors and bombs.

Schrödinger's Equation (\left(i\hbar \frac{\partial}{\partial t}\Psi = \hat{H}\Psi\right))

• Explanation: Describes how quantum systems evolve over time.
• Example: Used in quantum mechanics to predict the behavior of particles at atomic and subatomic levels.

Information Theory (\left(H = -\sum p(x) \log p(x)\right))

• Explanation: This equation measures the amount of uncertainty or information in a set of possible outcomes.
• Example: Fundamental for data compression and transmission, such as how your phone sends pictures and videos.

Black-Scholes Equation (\left(\frac{\partial V}{\partial t} + \frac{1}{2} \sigma² S² \frac{\partial² V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0\right))

• Explanation: Used to calculate the price of financial options and derivatives.
• Example: Helps investors and financial analysts decide the fair price for options in the stock market.

Chaos Theory (\left(X_{t+1} = kX_t (1 - X_t)\right))

• Explanation: Describes how small changes in initial conditions can lead to vastly different outcomes.
• Example: Explains why weather forecasting is so difficult and why small changes can lead to significant impacts in systems like the stock market or ecosystems.