What does it mean: The square of the hypotenuse of a right triangle is equal to the
SUM of the squares of its legs.
History: Attributed to Pythagoras, it isn't certain that he first proved it. The first clear
proof came from Euclid, and it is possible the concept was known 1000 years before
Pythoragas by the Babylonians.
Importance: The equation is at the core of geometry, links it with algebra, and is the
foundation of trigonometry. Without it, accurate surveying, mapmaking, and navigation
would be impossible.
Modern use: Triangulation is used to this day to pinpoint relative location for GPS
navigation.
The logarithm and its identities
What does it mean: You can multiply numbers by adding related numbers.
History: The initial concept was discovered by the Scottish Laird John Napier of Merchiston
in an effort to make the multiplication of large numbers, then incredibly tedious and time
consuming, easier and faster. It was later refined by Henry Briggs to make reference tables
easier to calculate and more useful.
in an effort to make the multiplication of large numbers, then incredibly tedious and time
consuming, easier and faster. It was later refined by Henry Briggs to make reference tables
easier to calculate and more useful.
Importance: Logarithms were revolutionary, making calculation faster and more accurate
for engineers and astronomers. That's less important with the advent of computers, but they're
still an essential to scientists.
for engineers and astronomers. That's less important with the advent of computers, but they're
still an essential to scientists.
Modern use: Logarithms still inform our understanding of radioactive decay.
The fundamental theorem of calculus
What does it mean?: Allows the calculation of an instantaneous rate of change.
History: Calculus as we currently know it was described around the same in the late
17th century by Isaac Newton and Gottfried Leibniz. There was a lengthy debate over
plagiarism and priority which may never be resolved. We use the leaps of logic and
parts of the notation of both men today.
17th century by Isaac Newton and Gottfried Leibniz. There was a lengthy debate over
plagiarism and priority which may never be resolved. We use the leaps of logic and
parts of the notation of both men today.
Importance: According to Stewart, "More than any other mathematical technique,
it has created the modern world." Calculus is essential in our understanding of how
to measure solids, curves, and areas. It is the foundation of many natural laws, and
the source of differential equations.
it has created the modern world." Calculus is essential in our understanding of how
to measure solids, curves, and areas. It is the foundation of many natural laws, and
the source of differential equations.
Modern use: Any mathematical problem where an optimal solution is required.
Essential to medicine, economics, and computer science.
Essential to medicine, economics, and computer science.
Newton's universal law of gravitation
What does it mean?: Calculates the force of gravity between two objects.
History: Isaac Newton derived his laws with help from earlier work by Johannes
Kepler. He also used, and possibly plagiarized the work of Robert Hooke.
Kepler. He also used, and possibly plagiarized the work of Robert Hooke.
Importance: Used techniques of calculus to describe how the world works.
Even though it was later supplanted by Einstein's theory of relativity, it is
still essential for practical description of how objects interact with each other.
We use it to this day to design orbits for satellites and probes.
Even though it was later supplanted by Einstein's theory of relativity, it is
still essential for practical description of how objects interact with each other.
We use it to this day to design orbits for satellites and probes.
Value: When we launch space missions, the equation is used to find optimal
gravitational "tubes" or pathways so they can be as energy efficient as possible.
Also makes satellite TV possible.
gravitational "tubes" or pathways so they can be as energy efficient as possible.
Also makes satellite TV possible.
The origin of complex numbers
What does it mean?: The square of an imaginary number is negative.
History: Imaginary numbers were originally posited by famed gambler/mathematician
Girolamo Cardano, then expanded by Rafael Bombelli and John Wallis. They still existed
as a peculiar, but essential problem in math until William Hamilton described this definition.
Girolamo Cardano, then expanded by Rafael Bombelli and John Wallis. They still existed
as a peculiar, but essential problem in math until William Hamilton described this definition.
Importance: According to Stewart ".... most modern technology, from electric lighting to
digital cameras could not have been invented without them." Imaginary numbers allow for
complex analysis, which allows engineers to solve practical problems working in the plane.
digital cameras could not have been invented without them." Imaginary numbers allow for
complex analysis, which allows engineers to solve practical problems working in the plane.
Modern use: Used broadly in electrical engineering and complex mathematic theory.
Euler's formula for polyhedra
What does it mean?: Describes a space's shape or structure regardless of alignment.
History: The relationship was first described by Descartes, then refined, proved, and published
by Leonhard Euler in 1750.
by Leonhard Euler in 1750.
Importance: Fundamental to the development of topography, which extends geometry to
any continuous surface. An essential tool for engineers and biologists.
any continuous surface. An essential tool for engineers and biologists.
Modern use: Topology is used to understand the behavior and function of DNA.
The normal distribution
What does it mean?: Defines the standard normal distribution, a bell shaped curve
in which the probability of observing a point is greatest near the average, and declines
rapidly as one moves away.
in which the probability of observing a point is greatest near the average, and declines
rapidly as one moves away.
History: The initial work was by Blaise Pascal, but the distribution came into its own
with Bernoulli. The bell curve as we currently comes from Belgian mathematician Adolphe Quetelet.
with Bernoulli. The bell curve as we currently comes from Belgian mathematician Adolphe Quetelet.
Importance: The equation is the foundation of modern statistics. Science and social
science would not exist in their current form without it.
science would not exist in their current form without it.
Modern use: Used to determine whether drugs are sufficiently effective relative to negative
side effects in clinical trials.
side effects in clinical trials.
The wave equation
What does it mean?: A differential equation that describes the behavior of waves,
originally the behavior of a vibrating violin string.
originally the behavior of a vibrating violin string.
History: The mathematicians Daniel Bournoulli and Jean D'Alembert were the first to
describe this relationship in the 18th century, albeit in slightly different ways.
describe this relationship in the 18th century, albeit in slightly different ways.
Importance: The behavior of waves generalizes to the way sound works, how earthquakes
happen, and the behavior of the ocean.
happen, and the behavior of the ocean.
Modern use: Oil companies set off explosives, then read data from the ensuing sound
waves to predict geological formations.
waves to predict geological formations.
The Fourier transform
What does it mean?: Describes patterns in time as a function of frequency.
History: Joseph Fourier discovered the equation, which extended from his famous
heat flow equation, and the previously described wave equation.
heat flow equation, and the previously described wave equation.
Importance: The equation allows for complex patterns to be broken up, cleaned up,
and analyzed. This is essential in many types of signal analysis.
and analyzed. This is essential in many types of signal analysis.
Modern use: Used to compress information for the JPEG image format and
discover the structure of molecules.
discover the structure of molecules.
The Navier-Stokes equations
What does it mean?: The left side is the acceleration of a small amount of fluid
, the right indicates the forces that act upon it.
, the right indicates the forces that act upon it.
History: Leonhard Euler made the first attempt at modeling fluid movement,
French engineer Claude-Louis Navier and Irish mathematician George Stokes made
the leap to the model still used today
French engineer Claude-Louis Navier and Irish mathematician George Stokes made
the leap to the model still used today
Importance: Once computers became powerful enough to solve this equation,
it opened up a complex and very useful field of physics. It is particularly useful in
making vehicles more aerodynamic.
it opened up a complex and very useful field of physics. It is particularly useful in
making vehicles more aerodynamic.
Modern use: Among other things, allowed for the development of modern passenger jets.
Maxwell's equations
What does it mean?: Maps out the relationship between electric and magnetic fields.
History: Michael Faraday did pioneering work on the connection between electricity and
magnetism, James Clerk Maxwell translated it into equations, fundamentally altering physics.
magnetism, James Clerk Maxwell translated it into equations, fundamentally altering physics.
Importance: Helped predict and aid the understanding of electromagnetic waves,
helping to create many technologies we use today.
helping to create many technologies we use today.
Modern use: Radar, television, and modern communications.
Second law of thermodynamics
What does it mean?: Energy and heat dissipate over time.
History: Sadi Carnot first posited that nature does not have reversible processes.
Mathematician Ludwig Boltzmann extended the law, and William Thomson formally stated it.
Mathematician Ludwig Boltzmann extended the law, and William Thomson formally stated it.
Importance: Essential to our understanding of energy and the universe via the concept
of entropy. It helps us realize the limits on extracting work from heat, and helped lead to a better steam engine.
of entropy. It helps us realize the limits on extracting work from heat, and helped lead to a better steam engine.
Modern use: Helped prove that matter is made of atoms, which has been somewhat useful.
Einstein's theory of relativity
What does it mean?: Energy equals mass times the speed of light squared.
History: The less known (among non-physicists) genesis of Einstein's equation was
an experiment by Albert Michelson and Edward Morley that proved light did not move
in a Newtonian manner in comparison to changing frames of reference. Einstein followed
up on this insight with his famous papers on special relativity (1905) and general relativity (1915).
an experiment by Albert Michelson and Edward Morley that proved light did not move
in a Newtonian manner in comparison to changing frames of reference. Einstein followed
up on this insight with his famous papers on special relativity (1905) and general relativity (1915).
Importance: Probably the most famous equation in history. Completely changed our view
of matter and reality.
of matter and reality.
Modern use: Helped lead to nuclear weapons, and if GPS didn't account for it, your directions
would be off thousands of yards.
would be off thousands of yards.
The Schrödinger equation
What does it mean?: Models matter as a wave, rather than a particle.
History: Louis-Victor de Broglie pinpointed the dual nature of matter in 1924.
The equation you see was derived by Erwin Schrodinger in 1927, building off of
the work of physicists like Werner Heisenberg.
The equation you see was derived by Erwin Schrodinger in 1927, building off of
the work of physicists like Werner Heisenberg.
Importance: Revolutionized the view of physics at small scales. The insight that
particles at that level exist at a range of probable states was revolutionary.
particles at that level exist at a range of probable states was revolutionary.
Modern use: Essential to the use of the semiconductor and transistor, and thus,
most modern computer technology.
most modern computer technology.
Shannon's information theory
What does it mean?: Estimates the amount of data in a piece of code by
the probabilities of its component symbols.
the probabilities of its component symbols.
History: Developed by Bell Labs engineer Claude Shannon in the years after
World War 2.
World War 2.
Importance: According to Stewart, "It is the equation that ushered in the information
age." By stopping engineers from seeking codes that were too efficient, it established
the boundaries that made everything from CDs to digital communication possible.
age." By stopping engineers from seeking codes that were too efficient, it established
the boundaries that made everything from CDs to digital communication possible.
Modern use: Pretty much anything that involves error detection in coding.
Anybody use the internet lately?
Anybody use the internet lately?
The logistic model for population growth
What does it mean?: Estimates the change in a population of creatures across
generations with limited resources.
generations with limited resources.
History: Robert May was the first to point out that this model of population growth
could produce chaos in 1975. Important work by mathematicians Vladimir Arnold
and Stephen Smale helped with the realization that chaos is a consequence of differential
equations.
could produce chaos in 1975. Important work by mathematicians Vladimir Arnold
and Stephen Smale helped with the realization that chaos is a consequence of differential
equations.
Importance: Helped in the development of chaos theory, which has completely changed
our understanding of the way that natural systems work.
our understanding of the way that natural systems work.
Modern use: Used to model earthquakes and forecast the weather.
The Black–Scholes model
What does it mean?: Prices a derivative based on the assumption that it is riskless
and that there is no arbitrage opportunity when it is priced correctly.
and that there is no arbitrage opportunity when it is priced correctly.
History: Developed by Fischer Black and Myron Scholes, then expanded by Robert Merton.
The latter two won the 1997 Nobel Prize in Economics for the discovery.
The latter two won the 1997 Nobel Prize in Economics for the discovery.
Importance: Helped create the now multi trillion dollar derivatives market. It is argued that
improper use of the formula (and its descendants) contributed to the financial crisis.
In particular, the equation maintains several assumptions that do not hold true in real financial markets.
improper use of the formula (and its descendants) contributed to the financial crisis.
In particular, the equation maintains several assumptions that do not hold true in real financial markets.
Modern use: Variants are still used to price most derivatives, even after the financial crisis,
Bonus: Hodgkin-Huxley equations
From an email interview with Dr. Stewart:
"At one stage I planned to include the Hodgkin-Huxley equations, which gave
mathematical biology a huge boost by using equations to model the way nerve cells
send signals to each other. It formed the basis of theoretical neuroscience, and is still important.
But it made the book too long, and in the end I felt that its impact on human history has not yet
been quite great enough. However, that is likely to change by the middle of this century, as
mathematical methods become a major part of mainstream biology — which I think they will."
mathematical biology a huge boost by using equations to model the way nerve cells
send signals to each other. It formed the basis of theoretical neuroscience, and is still important.
But it made the book too long, and in the end I felt that its impact on human history has not yet
been quite great enough. However, that is likely to change by the middle of this century, as
mathematical methods become a major part of mainstream biology — which I think they will."
Bonus: "The Google equation"
From an email interview with Dr. Stewart:
"My current candidate for and 18th equation (number 1 in ‘Seventeen MORE Equations
That Changed the World’ -- I joke... I think...) is the basic equation behind Google.
This describes how to rate the importance of a website in terms of the links to it,
and it’s a clever application of basic undergraduate linear algebra. It deserved to be
in the book, but I was running out of space — and worried that my readers’ enthusiasm
for yet another equation might be drying up."
That Changed the World’ -- I joke... I think...) is the basic equation behind Google.
This describes how to rate the importance of a website in terms of the links to it,
and it’s a clever application of basic undergraduate linear algebra. It deserved to be
in the book, but I was running out of space — and worried that my readers’ enthusiasm
for yet another equation might be drying up."
Read more: http://www.businessinsider.com/17-equations-that-changed-the-world-2013-1?op=1#ixzz2IcBuMDMQ
