In the vast mathematical universe, formulas occupy a central place. These famous equations are found in every mathematical theorem from ancient times to the recent discoveries of the 20th century. With every new mathematical equation comes new questions and new answers. The following is an anthology of some of the math equations that have changed the way we see the world.
| 🙋♂️ Equation | 🔢 Equation or theory name | 🔎 What it does |
|---|---|---|
 | Pythagorean Theorem | Helps solve problems involving right angles. |
| Natural logarithms Decimal logarithms Binary logarithms | Logarithms | Used to measure the complexity of algorithms and fractals, and appear in formulas for counting prime numbers. |
 | The Universal Law of Gravitation | It states that two bodies in the universe attract each other with a force that depends on their masses and the distance between them. |
 | The Theory of Relativity | It explains the fundamental relationship between space, time, and gravity. |
 | Area of a Circle | Calculates the area of a circle. Also helps measure wave forms, lens curvatures, and light distribution. |
 | Chaos Theory | Used to understand and predict possible behaviors of complex systems that are sensitive to initial conditions. |
 | Euler's Identity | Useful in the analysis of waves and oscillations. Provides a connection between different areas of mathematics. |
ΔS=∫  | The Second Law of Thermodynamics | It dictates the direction of potential energy transfers. |
| f(x)=12a0+∑∞n=1ancosnx+∑∞n=1bnsinnx | The Fourier Transform | It divides time into several frequencies and simple waves. |
 | The Schrödinger Equation | Governs wave functions of non-relativistic quantum-mechanical systems. |
 | Maxwell's Equations | These equations are the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. |
Pythagoras' Theorem
This theorem, dating back to 530 BC, is one of the most famous math equations to exist.
The Pythagorean Theorem is one of the cornerstones of modern mathematics; it has significantly contributed to the development of mathematics as a discipline. Even after many years, the formula of this theorem remains ingrained in our memory.
Thanks to Pythagoras and his famous equation, we can calculate lengths, angles, and determine whether a triangle is right-angled or not. This equation is just one of the many examples of how math is a part of our daily life, whether we are aware of it or not.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The Pythagorean Theorem continues to be used in practical fields such as construction, architecture, and carpentry, among others.
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Logarithms
Logarithms, popularized by John Napier in 1610, combine inverse and exponential functions, and opposites. They are common in formulas used in science, to measure the complexity of algorithms and fractals, and appear in formulas for counting prime numbers.
Until the development of the modern computer, calculating with logarithms was the most usual way of multiplying large numbers together and made leaps and bounds in the fields of math, physics, engineering, and astronomy.
There are 3 types of logarithm:
Natural Logarithms
The fundamental basis in mathematical analysis
Decimal Logarithms
Used in mathematical calculations
Binary Logarithms
Used in computational theory and for applied calculations
Such calculations are useful in poker, and in solving puzzles.
Universal Law of Gravitation
Isaac Newton formulated the Universal Law of Gravitation in the 17th century. It states that two bodies in the universe attract each other with a force that depends on their masses and the distance between them.
This force is directly proportional to the product of the masses of the bodies and inversely proportional to the square of the distance between them. For example, the Earth attracts the moon, and vice versa.
The same law applies to the attraction between an apple and the Earth, and the attraction between the planets and the sun. All these relationships fall under the Universal Law of Gravitation.
, where
F = Gravitational force (N: Newtons)
M = Mass of the first body (usually a planet, kg)
m = Mass of the second body (kg)
R = Distance between the bodies (meters)
G = Gravitational constant
Remember: gravitational interaction is universal; it applies to all types of matter. Gravity only manifests as attraction; it does not account for the repulsion of bodies.
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The Theory of Relativity
Pure mathematics is, in its way, the poetry of logical ideas.
Albert Einstein
Albert Einstein's Theory of Relativity is the most famous among mathematical theories. It proves how important math is in real life and the main impact math can have on the course of history. And yet, no one fully understands it.
This formula, which illustrates the theory of relativity (restricted relativity and general relativity) revolutionized our understanding of physics up to that point.
It remains crucial to this day, as it shows that matter can be converted into energy and vice versa. 🤓
Note that the mass of any object is a form of energy. In the formula, the mass (𝑚) of an object can be converted into energy (𝐸), and vice versa. The “c” represents the speed of light, and it is squared to indicate that even a small amount of mass can be transformed into a large amount of energy.
, where
E = energy
m = mass
c = velocity (the speed of light)
Restricted relativity introduced the idea that the speed of light was a universal constant that did not change. It further explains that the passage of time was not the same for bodies moving at different speeds.
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The Area of a Circle
A circle is a closed, flat curve where all points are equidistant from the center. In simple terms, a circle is a closed line, like a ring or a tire.
, where
A = the area of the circle
π = 3.1416
r = the radius of the circle
The radius is the distance between the circle's center point and any point on its boundary.
Whether for practical purposes or in physics calculations, knowing the formula for calculating the area of a circle is essential. Thanks to this famous maths equation, we can study wave phenomena, electromagnetic fields, lenses, and the distribution of light in greater depth.
Chaos Theory
Chaos theory has shown us that it's impossible to predict with certainty what will happen in the future. It is the study of the behavior of dynamic systems. This theory proves that no existing processes may be predicted with certainty. Robert May's theory is more recent, dating from 1975. It describes a process that is constantly evolving over time.
This theory deals with nonlinear phenomena that are practically impossible to predict or control. Such factors include turbulence, weather, stock markets, brain states, and more. We describe them using fractal mathematics, which reflects the infinite complexity of nature.
Chaos Theory encompasses, for example, the butterfly effect. This effect is often described as a butterfly flapping its wings in Australia causing a hurricane in Chile.
The butterfly effect means that small changes in initial conditions can lead to drastic changes in outcomes.
There exists no definitive, unifying Chaos Theory equation.
Chaos Theory mathematics involves repetitions of simple mathematical formulas like this one.
Chaos Theory in mathematics contradicts the idea of using math for poker or gambling. Even small errors in measuring the state of the system will be greatly amplified, making any prediction useless. Chaos is one of the most interesting math theories, as this clip explains.
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Euler's Identity
Swiss polymath Leonhard Euler gave humanity what many mathematicians and experts consider one of the most remarkable equations ever devised.
But why does this equation matter? Because it makes use of three fundamental operations in arithmetic:
✚ Addition
➖ Subtraction
⏨ Exponentiation
, where
e = the base of natural logarithms
i = the imaginary unit
π = 3.1416
1 and 0 = integers
The elegance of Euler's Identity lies in its simplicity, but this formula isn't just a stunning example of mathematical beauty. It connects five important mathematical constants in a single formula.
The five constants represented are "0", the additive identity; "1", the multiplicative identity; the fabulous pi; "e" which is the base of natural logarithms, and "i", the imaginary unit of the complex numbers found in equations with 3 unknowns.
💡This simplicity gives Euler's Identity a major impact on areas such as engineering, number theory, and complex functions, among others.
The Second Law of Thermodynamics
The second law of thermodynamics (also known as the Carnot principle after its discoverer, in 1824) proves irrefutably that physical phenomena are irreversible, especially when thermal changes occur. The principles of thermodynamics are the principal laws governing thermodynamics.
This principle has been modified and reformulated on several occasions, and gained widespread popularity in 1873 thanks to Ludwig Boltzmann and Max Planck.
While the first law of thermal dynamics specifies that energy can be exchanged between physical systems as heat and work. The second law introduces another quantity, known as entropy. It is a principle of change and evolution since it determines in which direction potential energy transformations are possible.
ΔS=∫ , where
ΔS = change of entropy
dQ = amount of heat transferred
T = the absolute temperature at which the transfer occurs.
To put this Law into a real-world context, think of an iced drink. You'll note that the ice melts, it will NOT freeze the drink.
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The Fourier Transform
The Fourier transform divides time into several frequencies and simple waves, just as a prism splits light in its constituent colors. The Fourier transform allows us to deal with non-periodic functions. Another example could be a magnetic or an acoustic field that is defined as a signal. The Fourier transform is its spectrum, in that it deconstructs such a field.
This theory was so earth-shaking because, suddenly, it was possible to understand the structure of more complex waves, such as human speech. Today this theory, goes to the heart of modern signal processing and analysis, as well as data processing. Search for a math class here on Superprof.
The Schrödinger Equation
Austrian physicist Erwin Schrödinger had many brilliant ideas, including the one about his cat being dead and alive at the same time. His love of felines notwithstanding, he delivered one of the most famous formulas in quantum mechanics.
Schrödinger's Equation is essential for understanding how particles change over time. It helps describe the different states in which a particle can be found.
This equation poses a profound question: Does matter exist only in the possible physical states (solids, liquids, gases)?
At the philosophical level, Schrödinger's Equation invites us to reflect on the nature of matter and how we perceive it. On the practical side, it was very useful for the advancement of nuclear energy, as well as for computing and laser applications. Engaging with an online math tutor can illuminate these concepts further.
The Schrödinger equation explains the changes over time of a particle. It describes the states of a particle, from which it is possible to describe any state, while posing a deep philosophical problem. These types of math enigmas are very common, and this equation is a door to answers that will eventually be found.
Maxwell's Equations
The four Maxwell's Equations form one of the most important foundations of modern physics. These equations are significant for both electrodynamics and the theory of relativity. This statistic delivers proof of the following:
A fair number of them use at least one of those equations. The establishment of Maxwell's Equations is considered the pinnacle of classical physics. All equations for calculating electromagnetism can be derived from them.
These are the 4 forms of Maxwell's equations:
- The Maxwell-Gauss equation
- The Maxwell-Thomson equation
- The Maxwell-Faraday equation
- The Maxwell-Ampere Equation
These equations describe the interaction between electric and magnetic fields and their relationship with electric fields and charges. Both the electric and magnetic fields can be represented using field line diagrams. These equations have defined the importance of math in the modern world, making it evident that math is an integral part of our society.
All these famous math equations have something in common:
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