Nearly every conceivable modern technology taken for granted today would not exist if it were not for specific revelations in the fields of mathematics, physics, and engineering. From the internal combustion engine used in cars to the nuclear arsenals that shape international relations, much of our modern life can be traced to several mathematical formulas and theorems that paved the way for progress.

24/7 Wall St. compiled a list of the most important mathematical formulas and theorems. While many of mathematical discoveries helped shape our world, we gave preference to those that have had the farthest-reaching practical implications and those that are the most widely revered.

Nearly every formula on this list was revolutionary, but none came out of thin air. Nearly all of these theories evolved from decades and centuries of accumulated scientific knowledge. In other words, rather than being standalone revelations, almost all of them were derived from, or expanded on, theories and equations that had been previously developed.

To have real world practicality, the equations on this list are often used with other discoveries — on this list or not. For example, modern flight would not have been possible without the contribution of Isaac Newton to our understanding of physics. Newton made countess scientific breakthroughs possible, including the lift formula the Wright brothers conceived. Not to mention, none of the electrical components on modern aircraft would be possible without the equations of James Clerk Maxwell, which describe how magnetic and electric fields behave.

In some cases, the equations on this list represent a conceptual breakthrough, even if they have since been discovered to be partially flawed. For example, Wright brothers’ lift formula of the early 20th century has since been modified into what is now known as the modern lift equation, which swaps the original measure of drag for the dynamic pressure of the air flow.

Most concepts on this list, like lift formula and Maxwell’s equations, led to feats in technology in engineering. The Dirac equation did not have immediate practical applications, but did much to support and advance modern particle physics. Meanwhile others, like Euler’s identity, are considered mathematical achievements in and of themselves.

Many concepts on this list are still shaping cutting edge scientific thought, as they contain implications beyond what was initially realized by their creator. For example, the famous formula E=mc^{2} was a critical part of Einstein’s theory of special relativity and though Einstein will forever be known as the formula creator, scientists today have a better understanding of the theory of relativity than Einstein himself.

To identify the mathematical discoveries that changed the world forever, 24/7 Wall St. reviewed mathematical formulas and theories from a range of sources. Those that had the farthest-reaching practical implications were favored, in addition to those that are highly revered in the field of mathematics. The list is not ranked in order of importance and is not comprehensive.