❓Are math formulas invented or discovered?

A noticeable pattern emerges: humans invent mathematical concepts by abstracting elements from their surroundings and then explore the connections between these ideas. Discover the most famous equations in our article.

1️⃣ What was the first formula in math?

The Thales Theorem from around 550 B.C. was the first formula in math. Learn more about the most famous math equations in our article.

🟰 What makes Euler's Identity famous in math?

Euler's Identity is a famous equation, e^(iπ) + 1 = 0, bringing together key mathematical constants (0, 1, π, e, i). It's celebrated for its elegance, showing unexpected connections between these constants and revealing profound mathematical relationships.

💡 Why is E=mc² iconic in physics and math?

Einstein's Mass-Energy Equivalence, E=mc², is a revolutionary equation stating that energy equals mass multiplied by the speed of light squared. It transformed physics, showcasing the equivalence of mass and energy. This simple formula has far-reaching implications, impacting fields beyond mathematics and leading to advancements in nuclear physics.

Famous Math Equations That Shape Our World

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
$\text{[math]}$	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.
$\text{[math]}$	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.
$\text{[math]}$	The Theory of Relativity	It explains the fundamental relationship between space, time, and gravity.
$\text{[math]}$	Area of a Circle	Calculates the area of a circle. Also helps measure wave forms, lens curvatures, and light distribution.
$\text{[math]}$	Chaos Theory	Used to understand and predict possible behaviors of complex systems that are sensitive to initial conditions.
$\text{[math]}$	Euler's Identity	Useful in the analysis of waves and oscillations. Provides a connection between different areas of mathematics.
F - E + V = 2	Euler's Formula for Polyhedra	Allows you to find the simple relationship with a solid shape with flat surfaces.
ΔS=∫ $\text{[math]}$	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.
$\text{[math]}$	The Schrödinger Equation	Governs wave functions of non-relativistic quantum-mechanical systems.
$\text{[math]}$	Maxwell's Equations	These equations are the foundation of classical electromagnetism, classical optics, electric and magnetic circuits.
$\text{[math]}$	Derivative Calculus Equation	Measures the instantaneous rate of change of a function.
$\text{[math]}$	The Square Root of Minus One	Combines the use of real and imaginary numbers to form complex numbers that are essential to solving equations.
$\text{[math]}$	Normal Distribution	Find the curve's shape where most values cluster around the average and rarer values taper off symmetrically on either side.
$\text{[math]}$	The Wave Equation	Describes how a disturbance travels through a medium.
$\text{[math]}$	The Navier-Stokes Equations	Describes the motion of fluids
$\text{[math]}$	The Black-Scholes Equation	Allows traders to calculate the fair price of a financial option.

The best Mathematics tutors available

Pythagoras' Theorem

This theorem, dating back to 530 BC, is one of the most famous math equations to exist.

Four different colors featuring triangles and right angles.

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.

Pythagoras' Theorem

$\text{[math]}$
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.

Here is where to find out about the best math lessons for kids.

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 17^th 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.

The Universal Law of Gravitation

$\text{[math]}$ , 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.

Find a reputable math tutor in Boston here on Superprof.

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.

E=mc2 on a chalkboard — There are some key differences between online tutorials and face-to-face tutorials. (Source: freephotocc)

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.

The Theory of Relativity

$\text{[math]}$ , 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.

Listen to Einstein's Theory of General Relativity by Astronomy Cast on Spotify!

Learn more with a Superprof math tutor, wherever you are in the US!

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.

The Area of a Circle

$\text{[math]}$ , 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.

Blue and pink waves in an abstract formation.

The butterfly effect means that small changes in initial conditions can lead to drastic changes in outcomes.

Chaos Theory

$\text{[math]}$
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.

Chaos came into focus in the 1960s, when meteorologist Ed Lorenz tried to make a basic computer simulation of Earth's atmosphere.

Don't forget, Superprof can help you find the perfect online math tutor if you need additional help understanding these math theories!

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

Euler's Identity

$\text{[math]}$ , 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.

Euler's Formula for Polyhedra

F - E + V = 2

V = number of vertices
E = number of edges
F = number of faces

Euler discovered the groundwork of the math branch called topology through his work regarding the Polyhedra Formula. When you take a solid shape with flat surfaces and count the vertices, edges, and faces, you find that a simple relationship holds true every single time.

Practice Problem

Try Euler's Formula for Polyhedra on a cube:
8 vertices - 12 edges + 6 faces = 2

Today, the formula underpins fields as varied as computer graphics, molecular chemistry, etc., reminding us that hidden patterns shape even the simplest objects.

The best Mathematics tutors available

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.

Second Law of Thermodynamics

ΔS=∫ $\text{[math]}$ , 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.

Search for an experienced math tutor on Superprof US.

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.

Black Board Famous Math — Source: Monstera Production on Unsplash

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

$\text{[math]}$

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:

How many math papers are published each year?

30,000

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

A diagram of four mathematical formulas and their names. — Source: www.powerelectronictips.com/intuitive-view-of-maxwells-equations-faq/

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:

They are simple, elegant, and concise.

They help connect fundamental mathematics with simpler concepts.

Each has extensive applications in everyday life and across various disciplines.

Although mathematics can be somewhat complex, these equations remain compelling.

Calculus - The Definition of a Derivative

In plain terms, the derivative of a function at a point measures the instantaneous rate of change of that function, in other words, the slope of the tangent line to its curve at any point. Developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, the derivative sits at the heart of differential calculus and gave humanity a way to answer a deceptively simple question: how fast is something changing at this exact moment?

Derivate Equation

\frac{df}{dt}=\displaystyle \lim_{h \to 0}\frac{f(t+h)-f(t)}{h}

f(t) = the value of the function at time (or input) t
f(t+h) = the value of the function at a slightly later input t+h
h = a very small change in the input, approaching zero
lim = the limit, which tells us what the expression approaches as h shrinks

The formula works by taking two points on a curve, one at t and one at the distance h, drawing a line between them, and then letting that distance h get infinitely small.

The derivate is not just an abstract concept confined to textbooks. It powers a number of real-world applications such as physics and engineering, economics, and machine learning.

The Square Root of Minus One

What number, when multiplied by itself, equals -1? For centuries, most believed there was "none", but Italian mathematicians Cardano and Bombelli proved this wrong when they discovered that pretending such a number existed actually helped solve real problems. Leonhard Euler later gave it a symbol: i.

i=\sqrt{-1}, i^{2}= -1

Combined with real numbers, "imaginary" numbers form complex numbers that are essential to many equations and problems.

Take a deeper look into imaginary and complex numbers.

Normal Distribution

Heights, test scores, measurement errors, follow the same bell-shaped curve. First described by Abraham de Moivre in the 18th century and refined by Carl Friedrich Gauss, the normal distribution has become the backbone of modern statistics, which is why it's often called the Gaussian distribution.

\Phi (x) = \frac{1}{\sqrt{2\\Pi \sigma }}e\tfrac{(x-\mu )^{2}}{2\sigma^{2} }

u = the mean (center of the curve)
o = the standard deviation (spread of the curve)

The curve's shape means most values cluster around the average, with rarer values tapering off symmetrically on either side. Understanding the normal distribution is one of the most useful skills a math student can develop.

The Wave Equation

Developed in 1746 by Jean le Rond d'Alembert, the wave equation captures the physics behind all of types of waves in a single formula.

\frac{\partial ^{2}u}{\partial t^{2}}= c^{2}\frac{\partial^2 u}{\partial x^2}

An ocean wave with orange clouds above. — Source: Silas Baisch

This equation describes how a disturbance travels through a medium, whether that's sounds through air, light through a vacuum, or seismic waves through the Earth. Today, the wave equation powers everything from concert hall acoustics and medical ultrasound to wireless communication.

The Navier-Stokes Equations

The Navier-Stokes equations describe the motion of fluids, from air flowing over a plane's wing to blood pumping through your arteries.

\rho (\frac{\partial v}{\partial t}+ v \times \triangledown v)= -\triangledown p + \triangledown \times T + f

Despite powering weather forecasting and climate modeling, no one has proven that smooth solutions always exist in three dimensions, leading to a million-dollar reward to anyone who can solve it by the Clay Mathematics Institute.

Solve The Navier-Stokes Equations to Win

$1M

The Black-Scholes Equation

In 1973, Fischeer Black, Myron Scholes, and Robert Merton published a formula that revolutionized finance and later earned Scholes and Merton the Nobel Prize in Economics in 1997 (unfortunately Black had passed away two years earlier).

The equation lets traders calculate the fair price of a financial option. It triggered the explosive growth of modern derivates markets, which are now worth hundreds of trillions of dollars.

\frac{1}{2}\sigma ^{2}S^{2}\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S}+\frac{\partial V}{\partial t}-rV = 0

V = price of the option
S = price of the underlying asset
σ = volatility
r = risk-free interest rate
t + time

Summarize with AI:

Did you like this article? Leave a rating!

4.46 (13 rating(s))

Chloe Galouchko

Passionate about people, languages and different cultures. I live to travel, share, discover, explore and continuously learn from others.

Leave us a comment

Cancel reply

Ahmed

April 2024

I need online lessons and want to know more math

Hailey - Editorial Manager Superprof United States

Hello Ahmed! Thank you for your interest in Superprof. You can visit our website at Superprof.com and enter ‘Math’ along with your desired location to find a list of qualified tutors who match your expectations and specialize in your preferred area!

The Most Famous Math Theories and Formulas Explained

Pythagoras' Theorem

Logarithms

Universal Law of Gravitation

The Theory of Relativity

The Area of a Circle

Chaos Theory

Euler's Identity

Euler's Formula for Polyhedra

The Second Law of Thermodynamics

The Fourier Transform

The Schrödinger Equation

Maxwell's Equations

Calculus - The Definition of a Derivative

Derivate Equation

The Square Root of Minus One

Normal Distribution

The Wave Equation

The Navier-Stokes Equations

The Black-Scholes Equation

Cancel reply

Frequently asked questions

❓Are math formulas invented or discovered?

1️⃣ What was the first formula in math?

🟰 What makes Euler's Identity famous in math?

💡 Why is E=mc² iconic in physics and math?