Mathematical Equations That Changed The World.
Sources: Five Equations That Changed the World by Michale Guillen In pursuit of the Unknown by Ian Stewart Business Insider, BBC, The Guardian
The Pythagorean Theorem
Function:
a² + b² = c²
Working out the lengths of the various sides of a triangle.
The Pythagorean theorem is named after the Greek mathematician Pythagoras (ca. 570 BC—ca. 495 BC),
Importance: Foundation of modern trigonometry - used in surveying, mapping and navigation.
Pythagoras (Πυθαγόρας) (c. 570 BC - c. 495 BC)
The Black- Scholes Model
Function:
∂²V ∂V ∂V ½σ²S² ― + rS ― + ― - rV = 0 ∂S² ∂S ∂t
Calculates the price of financial derivatives.
The model was first articulated by Fischer Black and Myron Scholes in their 1973 paper-The Pricing of Options and Corporate Liabilities"
Importance: Helped create the trilion dollar derivatives market but incorrect use contributed to the current financial crisis.
Fischer Black (1938 – 1995)
Myron Scholes
Fundamental theorem of Calculus
Function:
df f (t+h) - f(t) ― = Lim ― dt h–›0 h
Working out instanteneous rates of change. Importance: Measurments of areas of solids and curves - used in medicin, economics and computer science.
The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory . Isaac Barrow proved a more generalized version of the theorem while Barrow's student Isaac Newton completed the development of the surrounding mathematical theory. Gottfried Leibniz systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today.
James Gregory (1638–1675)
Isaac Barrow (1630–1677)
Isaac Newton (1643–1727)
Gottfried Leibniz (1646–1716)
The Fourier Transform
Function:
f (ξ) = ∫+∞∞ f (x) eˉ² πixξ dx
Describes patterns in time as a function of frequency.
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The Fourier transform and Fourier's Law are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect.
Importance: Used in JPEG compression, signal analysis and discovering the structure of molecules.
Joseph Fourier (1768 - 1830 )
Universal law of Gravitation
Function:
m1m² F = G― d²
Calculates the force of gravity between two objects.
This is a general physical law derived from empirical observations by what Newton called induction, was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687.
Importance: Critical to astronomy- helped discover Pluto, used in space exploration & putting satellites in orbit.
Isaac Newton (1643–1727)
Euler’s Formula for Polyhedra
Function:
F-E+V=2
Calculating the spaces of shapes.
The formula bears the name of the famous Swiss mathematician Leonhard Euler , who would have celebrated his 300th birthday this year.
Importance: Led to the development of topography - used in 3D mapping & by biologists to understand DNA.
Leonhard Euler (1707 - 1783)
Second Law of Thermodynamics dS ≼ 0
The second law may be expressed in many specific ways, but the first formulation is credited to the French scientist Sadi Carnot in 1824.
Function: Explains entropy - the dissipation of energy and heat over time. Importance: Proved perpetual motion is impossible & helped prove matter is made from atoms.
Nicolas LĂŠonard Sadi Carnot ( 1796 - 1832)
Maxwell’s Equations ∆.E=0 ∆.H=0
∂H ∆ x E = -― c ∂t ∂E ∆xH=-― c ∂t
Function: Explain relationships between elctric and magnetic fields.
They are named after the Scottish physicist and mathematician James Clerk Maxwell who published an early form of those equations between 1861 and 1862.
Importance: Prediction and underesting of electromagnetic waves used in radar, TV and modern communications.
James Clerk Maxwell (1831–1879)
The Schrödinger Equation
Function:
∂ - Ψ = ĤΨ iħ — ∂t
Models matter as a wave rather than a particle.
It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger.
Importance: Revolutionized small-scale physics- crucial in developing modern computers via semiconductor and transistor.
Erwin Schrödinger (1887-1961)
Theory of Special Relativity
Function:
E = mc²
Mass and energy are interchangeable and explains behaviour of objects at very high speeds. Importance: Introduce the idea of time dilation and helped lead to nuclear weapons.
This physical theory proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"
(1879-1955)