20 Equations That Built Our World

**By: Tao Steven Zheng (鄭濤)**

## **1. Pythagorean Theorem**

a
2

b
2

=

c
2

{\displaystyle a^2+b^2=c^2}

**Discovered:** c. 1800 BCE (Babylonians and Egyptians)

**Influence:** The most fundamental definition of a metric (measurement of length). Although the theorem was named in honour of the Greek philosopher Pythagoras (fl. 6th century BCE), the statement of the theorem was well-known to many ancient civilizations: Babylonia, Egypt, China, and India. The Pythagorean theorem can be extended to calculate the length of vectors in n-dimensional space.

# i
1

n

r

i

M

i

s

i

(
m
o
d
M
)

{\displaystyle N=\sum \_{i=1}^{n}{{r}\_{i}M\_{i}s\_{i}(modM)}}

**Discovered:** 3rd Century CE (Master Sun)

   **Influence:** Originally developed by the Chinese for computing intricate calendars, the utility of the Chinese remainder theorem has appreciated in modern times for number theoretic proofs, computational efficiency, and the encryption found in internet security.

## **3. Binomial Theorem**

(
a
b
)

n

=

∑

# k
0

n

(

n

k

)

a

k

b

n
−
k

{\displaystyle {(a+b)}^{n}=\sum \_{k=0}^{n}{\left({\begin{matrix}n\\k\end{matrix}}\right){a}^{k}{b}^{n-k}}}

**Discovered:** 11th Century CE (Jia Xian and Omar Khayyam)

   **Influence:** Initially discovered for its arithmetic and algebraic properties, the binomial theorem eventually opened the doors to combinatorics, probability, and statistics. These mathematical discoveries are now indispensable in the modern world of finance and insurance.

## **4. Fundamental Theorem of Calculus**

∫

a

b

f
(
x
)
d
x

=
F
(
b
)
−
F
(
a
)

{\displaystyle \int \_{a}^{b}{f(x)dx}=F(b)-F(a)}

**Discovered:** 1667 CE (James Gregory)

    **Influence:** Modern science is built on the foundation of calculus. Without the fundamental theorem of calculus, scientists and engineers would not be able to accurately quantify the changing world.

## **5. Compound Interest**

P
(
t
# )

P

0

(

1

r
n

)

n
t

{\displaystyle P(t)={P}\_{0}{\left(1+{\frac {r}{n}}\right)}^{nt}}

**Discovered:** 1683 CE (Jacob Bernoulli)

    **Influence:** The study of compound interest quantitatively explained the power of finance. It was also responsible for discovering the exponential number

(
# e
2.71828...
)

{\displaystyle (e=2.71828...)}

, which is perhaps the most important constant of mathematics.

# F

d
p

d
t

{\displaystyle F = \frac{dp}{dt}}

**Discovered:** 1687 CE (Isaac Newton)

    **Influence:** Newton's second law of motion states that forces are responsible for motion and any change in motion. This equation ignited a revolution in science, which paved the road to quantitatively describe the motions of the heavens and the Earth with great accuracy.

## **7. Taylor’s Theorem**

f
(
x
# )
f
(
a
)

∑

# n
1

∞

f

(
n
)

(
a
)

n
!

(
x
−
a
)

n

{\displaystyle f(x)=f(a)+\sum \_{n=1}^{\infty }{{\frac {{f}^{(n)}(a)}{n!}}{(x-a)}^{n}}}

**Discovered:** 1712 CE (Brook Taylor)

    **Influence:** Alongside the fundamental theorem of calculus, Taylor's theorem was responsible for the formal study of analyticity of functions, and the approximations used in theoretical physics and engineering.

## **8. Wave Equation**

∂

2

u

∂

t

2

=

c

2

∂

2

u

∂

x

2

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

**Discovered:** 1746 CE (Jean le Rond d’Alembert)

    **Influence:** The wave equation expanded elevated study of partial differential equations to explain the physical phenomena of vibrations and oscillations. This quickly accelerated science in Western Europe, most notably the study of heat, which led to the invention and development of many industrial technologies.

## **9. Euler’s Formula**

e

i
x

=
cos
⁡

x

i
sin
⁡

x

{\displaystyle {e}^{ix}=\cos {x}+i\sin {x}}

**Discovered:** 1748 CE (Leonhard Euler)

    **Influence:** The formula

e

i
π

# 1
0

{\displaystyle {e}^{i\pi} + 1 = 0 }

 is considered by many mathematicians to be the most beautiful equation in the realm of mathematics because it ties together the exponential number

e

{\displaystyle e}

, imaginary unit

i

{\displaystyle i}

, the identity element of multiplication 1, and the identity element of addition 0. Aside from its beauty, the general Euler's formula

e

i
x

=
cos
⁡

x

i
sin
⁡

x

{\displaystyle {e}^{ix}=\cos {x}+i\sin {x}}

 is rather useful for solving differential equations.

## **10. Euler-Lagrange Equation**

d

d
t

∂
L
(
x
,

x
˙

)

∂

x
˙

−

∂
L
(
x
,

x
˙

)

∂
x

=
0

{\displaystyle {\frac {d}{dt}}{\frac {\partial L(x,{\dot {x}})}{\partial {\dot {x}}}}-{\frac {\partial L(x,{\dot {x}})}{\partial x}}=0}

**Discovered:** 1750s CE (Leonhard Euler and Joseph-Louis Lagrange)

    **Influence:** The transformation of Newton's laws of motion into a differential equation with generalized coordinates tremendously expanded the power and capabilities of physics. The Euler-Lagrange equation is an indispensable tool for the development of modern physics.

## **11. Normal Distribution**

ϕ
(
x
# )

1

2
π

σ

2

e

−

(
x
−
μ
)

2

2

σ

2

{\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi {\sigma }^{2}}}}{e}^{-{\frac {{(x-\mu )}^{2}}{2{\sigma }^{2}}}}}

**Discovered:** 1809 CE (Carl Friedrich Gauss)

    **Influence:** Essential to the study of large data sets, the normal distribution greatly contributed to the understanding of central tendency in statistics. This tremendously enhanced the validity research not only in the physical sciences, but also in business, finance, demography, and psychology.

## **12. Fourier Transform**

F
(
k
# )

∫

−
∞

∞

f
(
x
)

e

−
2
π
i
k
x

d
x

{\displaystyle F(k)=\int \_{-\infty }^{\infty }f(x){e}^{-2\pi ikx}dx}

**Discovered:** 1822 CE (Jean-Baptiste Fourier)

    **Influence:** Initially developed for the study of heat, the Fourier transform is now vital to the study of waves and signals. The numerous applications of this important mathematical tool ranges from radio astronomy to music editing.

## **13. Maxwell's Equations**

∇
×
# E
−

∂
B

∂
t

{\displaystyle \nabla \times E=-{\frac {\partial B}{\partial t}}}

∇
×
# B

μ

0

(

J

ϵ

0

∂
E

∂
t

)

{\displaystyle \nabla \times B={\mu }\_{0}\left(J+{\epsilon }\_{0}{\frac {\partial E}{\partial t}}\right)}

∇
∙
# E

ρ

ϵ

0

{\displaystyle \nabla \bullet E={\frac {\rho }{{\epsilon }\_{0}}}}

∇
∙
# B
0

{\displaystyle \nabla \bullet B=0}

**Discovered:** 1861 CE (James Clerk Maxwell)

    **Influence:** Perhaps the most elegant set of equations known to man, Maxwell's equations unites two fundamental forces of the universe: electricity and magnetism. Maxwell's equations takes advantage of the field theory in mathematical physics to great heights, producing a succinct set of physical and mathematical truth and beauty.

# S

k

B

ln
⁡

Ω

{\displaystyle S=k\_{B}\ln {\Omega }}

**Discovered:** 1875 CE (Ludwig Boltzmann)

    **Influence:** Ludwig Boltzmann tied together the disorder of a thermal system to the number of configurations of matter and energy called microstates. There are many physical and philosophical ramifications from this result that extend into the realm of information theory.

# F
q
(
E
v
×
B
)

{\displaystyle F=q(E+v\times B)}

**Discovered:** 1889 CE (Oliver Heaviside), 1892 CE (Hendrik Lorentz)

    **Influence:** As the Maxwell equations united electric and magnetic fields, the Lorentz force united the electric and magnetic forces. There are countless applications of the Lorentz force in modern technology, most notably the massive particle accelerators used in search of elementary particles.

## **16. Lorentz Transformations**

x

′

=

γ

(
x
−
v
t
)

{\displaystyle {x}^{\prime }={\gamma }(x-vt)}

t

′

=
γ
(
t
−
v
x

/

c

2

)

{\displaystyle {t}^{\prime }=\gamma (t-vx/{c}^{2})}

**Discovered:** 1890s CE (Hendrik Lorentz and Joseph Larmor)

    **Influence:** The Lorentz transformations was the stepping stone for Einstein's discovery of special relativity. It linked the relation between the measurement of space and time, as well as the meaning of simultaneity and causality.

## **17. Einstein Mass-Energy Relation**

E

2

=

m

2

c

4

p

2

c

2

{\displaystyle {E}^{2}={m}^{2}{c}^{4}+{p}^{2}{c}^{2}}

**Discovered:** 1905 CE (Albert Einstein)

    **Influence:** There is a famous photograph of Einstein writing the equation

# E
m

c
2

{\displaystyle E=mc^2}

 on the blackboard. This equation only captured half of his mass-energy relation (when objects with mass are at rest). For moving objects (with or without mass), the mass-energy relation should be renamed Einstein's mass-momentum-energy relation.

## **18. Schrödinger Equation**

i
ℏ

∂
Ψ

∂
t

=
H
Ψ

{\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}=H\Psi }

**Discovered:** 1925 CE (Erwin Schrödinger)

**Influence:** The Schrödinger equation is a monumental discovery in modern physics. It encapsulates many fundamental concepts of quantum mechanics, most notably the wave-particle duality. The Schrödinger equation is essentially a complex heat-equation that is characteristic to describing the time evolution of quantum mechanical systems.

# Y
A

K

α

L

β

{\displaystyle Y=A{K}^{\alpha }{L}^{\beta }}

**Discovered:** 1927 CE (Charles Cobb and Paul Douglas)

    **Influence:** Economic models are often difficult to model, often times leaving out many details that contribute to production. The Cobb-Douglas production function makes a good attempt to describe the production of goods based on two dominant factors: capital and labour.

## **20. Black-Scholes Equation**

∂
V

∂
t

1
2

σ

2

S

2

=
r
V
−
r
S

∂
V

∂
S

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

**Discovered:** 1973 CE (Fischer Black and Myron Scholes)

    **Influence:** Finance markets are a nightmare to model. Few equations are as successful at simulating the derivative trading as the Black-Scholes equation. The equation provides valuable insight to the trading of stocks and bonds, promoting its practice and proliferation.