Engineering is the practice of using natural science, mathematics, and the engineering design process to solve technical problems, increase efficiency and productivity, and improve systems. Modern engineering comprises many subfields which include designing and improving infrastructure, machinery, vehicles, electronics, materials, and energy systems.
The steam engine, the major driver in the Industrial Revolution, underscores the importance of engineering in modern history. This beam engine is on display in the Technical University of Madrid.
The Ancient Romans built aqueducts to bring a steady supply of clean and fresh water to cities and towns in the empire.
A water-powered mine hoist used for raising ore, c. 1556
The application of the steam engine allowed coke to be substituted for charcoal in iron making, lowering the cost of iron, which provided engineers with a new material for building bridges. This bridge was made of cast iron, which was soon displaced by less brittle wrought iron as a structural material.
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
This is the Ulam spiral, which illustrates the distribution of prime numbers. The dark diagonal lines in the spiral hint at the hypothesized approximate independence between being prime and being a value of a quadratic polynomial, a conjecture now known as Hardy and Littlewood's Conjecture F.
On the surface of a sphere, Euclidean geometry only applies as a local approximation. For larger scales the sum of the angles of a triangle is not equal to 180°.
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC
A page from al-Khwārizmī's Algebra
