{"id":1336,"date":"2025-05-16T14:46:57","date_gmt":"2025-05-16T11:46:57","guid":{"rendered":"https:\/\/planeta21vek.ru\/artemiada\/eng\/?p=1336"},"modified":"2025-11-20T19:36:59","modified_gmt":"2025-11-20T16:36:59","slug":"equations-that-changed-the-world","status":"publish","type":"post","link":"https:\/\/planeta21vek.ru\/artemiada\/eng\/2025\/05\/16\/equations-that-changed-the-world\/","title":{"rendered":"Equations that changed the world"},"content":{"rendered":"

Mathematical equations give meaning to reality and help us see things we previously couldn’t see. So it’s no surprise that new developments in mathematics have often gone hand in hand with advances in our understanding of the universe. Here are a few equations that have revolutionized history, from tiny particles to the vast cosmos.<\/p>\n

<\/p>\n

Pythagoras’ theorem<\/strong><\/span><\/p>\n

One of the first basic rules of trigonometry that people learn in school is the relationship between the sides of a right triangle: the length of each of the two shorter sides, squared and added together, equals the length of the longest side squared. This is usually written as a^2 + b^2 = c^2, and it has been known for at least 3,700 years, since the time of the ancient Babylonians.<\/p>\n

\"\"
\nPythagorean Theorem. dzen.ru<\/em><\/p>\n

According to the University of St Andrews in Scotland, the Greek mathematician Pythagoras is credited with writing the version of the equation used today. Along with its uses in construction, navigation, cartography, and other important processes, the Pythagorean theorem helped expand the very concept of numbers. In the fifth century B.C., mathematician Hippasus of Metapontum noticed that an isosceles right triangle with two base sides each 1 unit long would have a hypotenuse that was the square root of 2, an irrational number. No one in history had encountered such numbers up to that point. For his discovery, Hippasus was said to have been thrown into the sea because Pythagoras\u2019 followers (including Hippasus) were so concerned about the possibility of numbers that continued forever past the decimal point without repeating, according to a paper from the University of Cambridge.<\/p>\n

F=ma and the law of universal gravitation<\/strong><\/span><\/p>\n

British luminary Sir Isaac Newton is credited with a number of world-shaking discoveries. Among them is his second law of motion, which states that force is equal to an object\u2019s mass multiplied by its acceleration, commonly written as F = ma. An expansion of this law, combined with Newton\u2019s other observations, led him to describe in 1687 what is now called his law of universal gravitation.<\/p>\n

\"\"
\nLaw of universal gravitation. multiring.ru<\/em><\/p>\n

It is usually written as F = G (m1*m2)\/ r^2, where m1 and m2 are the masses of the two objects, and r is the distance between them. G is a fundamental constant whose value must be discovered experimentally. Since then, these concepts have been used to understand many physical systems, including the motion of the planets in the solar system and how to travel between them using rockets.<\/p>\n\n

Wave equation<\/strong><\/span><\/p>\n

Using Newton’s relatively new laws, 18th-century scientists began to analyze everything around them. In 1743, French polymath Jean-Baptiste Le Rond d’Alembert derived an equation that describes the oscillations of a vibrating string, or the motion of a wave, according to a 2020 paper published in the journal Advances in Historical Studies. The equation can be written as follows:<\/p>\n

1\/v^2 * \u2202^2y\/\u2202t^2 = \u2202^2y\/\u2202x^2<\/p>\n

\"\"
\nSound waves. awesomeworld.ru<\/em><\/p>\n

In this equation, v is the speed of the wave, and the other parts describe the displacement of the wave in one direction. Extended to two or more dimensions, the wave equation allows researchers to predict the movement of water, seismic, and sound waves, and is the basis for things like the Schr\u00f6dinger equation in quantum physics, which underlies many modern computer gadgets.<\/p>\n

Fourier equation<\/strong><\/span><\/p>\n

Even if you haven\u2019t heard of French Baron Jean-Baptiste Joseph Fourier, his work has impacted your life. That\u2019s because the mathematical equations he wrote down in 1822 allowed researchers to break down complex and confusing data into combinations of simple waves that are much easier to analyze.<\/p>\n

\"\"
\nFourier equation. Neslihan Gorudzhu\/istock<\/em><\/p>\n

The Fourier transform, as it is called, was a radical idea in its day, and many scientists refused to believe that complex systems could be reduced to such elegant simplicity, according to an article in Yale Scientific. But Fourier transforms are workhorses in many modern scientific fields, including data processing, image analysis, optics, communications, astronomy, and engineering.<\/p>\n\n

Maxwell’s equations<\/strong><\/span><\/p>\n

Electricity and magnetism were still new concepts in the 1800s, when scientists were exploring how to capture and harness these strange forces. Scottish scientist James Clerk Maxwell greatly expanded our understanding of both phenomena in 1864 when he published a list of 20 equations describing how electricity and magnetism function and how they are related.<\/p>\n

\"\"
\nMaxwell’s equations. profazu.ru<\/em><\/p>\n

Later refined to four, Maxwell’s equations are now taught to first-year college physics students and provide the basis for everything electronic in our modern technological world.<\/p>\n

E = ms^2. Equivalence of mass and energy<\/strong><\/span><\/p>\n

No list of transformation equations would be complete without the most famous equation of them all. First formulated by Albert Einstein in 1905 as part of his groundbreaking theory of special relativity, the equation E = mc^2 showed that matter and energy are two aspects of the same thing. In the equation, E stands for energy, m represents mass, and c is the constant speed of light.<\/p>\n

\"\"
\nE = mc2. habr.com<\/em><\/p>\n

The concepts contained in such a simple statement are still difficult for many people to understand, but without E = mc^2 we would not understand how the stars or the universe work, and we would not be able to build giant particle accelerators like the Large Hadron Collider to probe the nature of the subatomic world.<\/p>\n

Friedman equations<\/strong><\/span><\/p>\n

It seems arrogant to think that you can create a set of equations that define the entire cosmos, but that’s exactly what Russian physicist Alexander Friedmann did in the 1920s. Using Einstein’s theory of relativity, Friedmann showed that the characteristics of the expanding universe could be expressed since the Big Bang using two equations.<\/p>\n

\"\"
\nFriedman equations. myshared.ru<\/em><\/p>\n

They combine all the important aspects of the cosmos, including its curvature, the amount of matter and energy it contains, its rate of expansion, and a number of important constants such as the speed of light, the gravitational constant, and the Hubble constant, which reflects the accelerating expansion of the universe. Einstein famously disliked the idea of \u200b\u200ban expanding or contracting universe, which his general theory of relativity suggested could be due to the effects of gravity. He tried adding a variable to the result, denoted by the Greek letter lambda, which acted in opposition to gravity to make the cosmos static. Although he later called this his greatest mistake, decades later the idea was dusted off and shown to exist in the form of a mysterious substance called dark energy, which is the driving force behind the accelerating expansion of the universe.<\/p>\n

Shannon’s information equation<\/strong><\/span><\/p>\n

Most people are familiar with the ones and zeros that make up computer bits. But this crucial concept would not have become popular without the pioneering work of American mathematician and engineer Claude Shannon. In a landmark 1948 paper, Shannon laid out an equation showing the maximum efficiency with which information can be transmitted, often given as C = B * 2log(1+S\/N).<\/p>\n

\"\"
\n70 years of the information age by K. Shannon. shilov-sss.ru<\/em><\/p>\n

In the formula, C is the achievable capacity of a particular information channel, B is the bandwidth of the line, S is the average signal power, and N is the average noise power. (S over N gives the famous signal-to-noise ratio of the system.) The output of the equation is expressed in units of bits per second. In a 1948 paper, Shannon attributes the idea of \u200b\u200bthe bit to mathematician John W. Tukey as shorthand for the phrase “binary digit.”<\/p>\n\n","protected":false},"excerpt":{"rendered":"

Mathematical equations give meaning to reality and help us see things we previously couldn’t see. So it’s no surprise that new developments in mathematics have often gone hand in hand with advances in our understanding of the universe. Here are a few equations that have revolutionized history, from tiny particles to the vast cosmos.<\/p>\n","protected":false},"author":1,"featured_media":1337,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[238,40],"tags":[463,464,465],"class_list":{"0":"post-1336","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-exact-sciences","8":"category-philosophy","9":"tag-equations","10":"tag-speed-of-light","11":"tag-wave"},"_links":{"self":[{"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/posts\/1336","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/comments?post=1336"}],"version-history":[{"count":2,"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/posts\/1336\/revisions"}],"predecessor-version":[{"id":1384,"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/posts\/1336\/revisions\/1384"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/media\/1337"}],"wp:attachment":[{"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/media?parent=1336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/categories?post=1336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/planeta21vek.ru\/artemiada\/eng\/wp-json\/wp\/v2\/tags?post=1336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}