Watching TV, making a phone call, and even turning on the light are routine acts that only exist in our lives because of some mathematical equations. Although we do not see them or even remember their existence, these equations are fundamental to a better understanding of the universe.

It would be impossible to list all the ways in which mathematics influences our lives, but NewScientist recently published a list of the main equations that make much of our routine possible. More interesting than knowing them is also knowing how they were used in practice, something that often ends up not being taught in school.

Wave equation

We live in a world full of waves. Our ears detect wave-like sounds, while our eyes see light in the same shape. The waves are all over the place, from the radio station we tune to the quakes caused by earthquakes. Even so, it takes some time to understand them.

And what helped to better understand the workings of waves was art. In 1727, Swiss mathematician Johann Bernoulli studied a violin string and found that the simplest vibration of that string was a sine curve.

Twenty years later, the study was revised by Frenchman Jean Le Rond d'Alembert, who focused his calculations on the possibility of simplifying wave equations. With this, he achieved a very elegant equation that demonstrates how the waveform varies over time by defining its propagation.

One of the major applications of this equation concerns the study of earthquakes, allowing seismologists to detect what is happening to Earth hundreds of miles below the ground.

Maxwell's Four Equations

But the great achievement of the wave equation was to serve as the basis for studies of magnetism conducted by British physicist James Clerk Maxwell. The equations created during this period define the modern foundations of electromagnetism, combining it with electricity, magnetism and optics.

By 1830 most physicists were looking for some analogy with gravity to explain the phenomena of electricity and magnetism. Michael Faraday, one of the world's most influential physicists, postulated that electrical and magnetic phenomena were caused by fields that penetrated space, changed over time, and could be detected by the force they produce.

In 1864 Maxwell reformulated Faraday's ideas and wrote four equations of the basic interactions between electric and magnetic fields. Two of these equations roughly say that these fields cannot "escape, " while the other two stipulate that when a region of an electric field rotates in the form of a small circle, it creates a magnetic field. When these rotations happen in a small portion of the magnetic field, they create an electric field.

Portrait of James Clerk Maxwell. Image source: Reproduction / Wikimedia

However, Maxwell's big balcony came shortly after, when the scientist decided to derive his equations and thereby deduced that light could be an electromagnetic wave. This was a surprising discovery, as no one imagined a relationship between light, electricity and magnetism.

According to the scientist, the color of the light would vary with the wavelength, and Maxwell came to the conclusion that there were waves long enough to be invisible to humans. These waves would transform the world and become known as radio waves.

In 1887, radio waves were demonstrated in practice in a presentation by Heinrich Hertz. This trajectory has given rise to a multitude of technologies, such as radio, TV, radar, mobile, etc.

Schrödinger's equations

Later scholars discovered that light, although behaving like a wave, also behaved like particles. From this came the revolutionary concept that matter is made of quantum waves and that a tightly knit group of these waves behave like a particle.

In 1927, physicist Erwin Schrödinger developed equations for these quantum waves, and from them a strange new world emerged: a world in which electrons, for example, were not well-defined particles, but a cloud of probabilities. It was not long before these quantum oddities led scientists to worry about the theories of multiverse and Schrödinger's famous cat.

Modern gadget transistors use concepts of semiconductor quantum mechanics. Image Source: Playback / Keyboard

For us who don't study physics, these discoveries materialized in the form of modern gadgets such as computers, cell phones and video games. All these devices have transistor-based memory chips that operate through quantum semiconductor mechanics.

And the applications of this knowledge do not stop there, since we have constant examples of innovations made with the help of Schrödinger's equations. There are, for example, quantum dot applications - very tiny semiconductors that can emit light of various colors - being used in biological material imaging processes, for example eliminating the use of toxic dyes. And if that wasn't enough, the future promises us the wonders of the quantum computer.

Fourier Transform

Finally, the seventh equation of the article was created by French mathematician and physicist Jean-Baptiste Joseph Fourier, who used wave concepts to better explain how heat flow occurred in a heated metal bar, ie how the temperature varied. over time. After a few years and many critical discussions about these studies, the world has gained the definitive version of the Fourier ideas, its Transform.

This equation has changed the lives of human beings in many ways. To begin with, it is possible to analyze, for example, the signal produced by an earthquake and calculate the frequencies at which the energy released by the earthquake is greatest.

The above JPG image of Fourier's bust was compressed using an equation he created. Image source: Reproduction / Wikipedia

In addition, the Fourier Transform can be used to remove noise from audio recordings, find DNA structure in x-ray images, improve radio wave reception, and even prevent a car from vibrating more than expected. Not only that, the equation is also present in one of the compression steps used in the JPG image format.

Isn't it amazing that mathematical equations can help shape our world? Some say they are far more influential than kings, queens, and global leaders, making much more impact on the world than rulers. Analyzing the above data, it is really hard to not believe it.

* Originally posted on 07/03/2013 .

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