What's in an equation? Letters and numbers and strange symbols – but also a new understanding of the relationships between fundamental components of our physical reality. All equations crystallize something important about the universe, be it the relationship between mass and energy, the attraction between two bodies, or the sides of a triangle.
The most important part of any equation is the equal sign in the heart. Those two horizontal lines tell us that when we change one thing, we will see a corresponding change in another, seemingly separate thing. In this way, equations reveal the relationships between superficially different quantities or properties. Once connected, that newly discovered relationship can serve as the foundation for future insights.
Choosing the most important equations is an almost hopeless task. The importance of a particular relationship will differ greatly depending on the context. For example, if you fall out of an airplane, the law of gravity will feel much more important than the Schrödinger equation. Likewise, equations are of different importance to scientists depending on the field they are in.
But it is possible to choose a few equations that have had too much of an impact on the way we see the world. While not an exhaustive list, these five comparisons all sum up something completely new – whether it's a new relationship between things or just a new way of looking at the world. And once put on paper, these equations all made for future breakthroughs as generations of thinkers used their powers to make new discoveries.
E = mc ^ 2
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First, let's take perhaps the most famous comparison of all. Albert Einstein & # 39; s The 1905 equation linking mass and energy is both elegant and superficially counterintuitive. It says energy is equal to the mass of an object in its resting frame multiplied by the speed of light squared. In doing this, Einstein revealed that mass and energy can be considered equivalent to each other and unites what were until then two separate domains.
From Einstein's equationwe can see that changing the mass of an object will also change the energy it contains, and vice versa. This is made terribly clear in a nuclear explosion, when small changes in the mass of radioactive elements correspond to enormous amounts of energy.
There is a common misconception that the equation shows that mass can be converted into energy and back again. Einstein didn't really mean that. Instead, he simply showed that changing mass must result in a change in energy – albeit a very large one.
The Pythagorean theorem
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The essential relationship between the two legs of a right triangle and the hypotenuse is named after the Greek philosopher Pythagoras, although he was not necessarily the first to come up with it.
The theorem shows that for each right triangle we can add the two shorter legs squared and get the length of the longest leg squared. The insight brought together the disciplines of geometry and algebra, and it is a good early example of using the relationships between shapes to derive a basic observation about numbers. Later discoveries in this vein live on today in the field of topology, and, more prosaically, we rely on the theorem every time we have a GPS triangulation in front of us.
Second Law of Thermodynamics
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The laws of thermodynamics arose from observations of how energy moves. The first law states that energy must always be conserved – an important finding in itself. But the second law, which initially described how heat was transferred in a system, would turn out to have extremely far-reaching implications.
The law can be formulated in many ways, depending on the situation, but the most basic observation is that heat – and thus energy – naturally flows in one direction only, from hot to cold. While we can heat something up by using energy, it is only a temporary fix.
It's something we see every day, but the significance is huge. This irreversibility underlies intoxicating concepts such as the arrow of time and entropy. Ultimately, it leads to the inevitability of the heat death of the universe – when mass and energy are spread so thinly and evenly across the cosmos that nothing more can happen.
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Calculus involves many different equations, but it started with a single breakthrough. Two 17th century thinkers, Isaac Newton and Gottfried Leibniz, independently found a way to formalize the convergence of infinite series as they approach a defined limit. This was partly due to the problem of calculating the slope of a curved line at some point. It's a question that mathematicians had previously answered partially, but never quite as elegantly and completely as Leibniz and Newton.
Their work led to the derivative and the integral, the two cornerstones of calculus. Derivatives give us the rate of immediate change of a function, and integrals give the area under a curve on a graph. Today, calculus is part of engineering, physics, economics and many other scientific disciplines.
The two mathematicians bitterly disagreed as to who should be considered the true father of calculus. Today, both men are credited with inventing it independently. However, we can thank Leibniz for the term calculus itself. If Newton had his way, today we would refer to the "Method of Fluxions
Universal Law of Gravity
While Newton must share credit for calculus, he can unilaterally claim credit for his Universal Law of Gravity. The equation is based on the work of scientists such as Galileo and Johannes Kepler to argue that every particle of matter in the universe has an attraction for every other particle of matter. This force increases with mass and decreases exponentially with distance.
Newton's work united Galileo & # 39; s observations of the motion of objects on Earth with Kepler's study of the motion of astronomical bodies. The result was an equation that showed the same rules for the movements of both planets and cannonballs – not necessarily a given in his day.
Today, Newton's laws have been replaced by Einstein's theory of relativity, which explains, among other things, things that are very close to each other or very burdensome. But Newton's observations still hold true for most of the interactions we see around us. Not bad for someone in the 17th century.