Influential Mathematicians: Gauss (3)

Probability and Statistics

Gauss introduced what is now known as the Gaussian distribution: he showed how probability can be represented by a bell-shaped curve, with peaks around the mean when falls off quickly towards plus or minus infinity.

File:Normal Distribution PDF.svg
Source: Wikipedia

He also created the Gaussian function: a function of the form

{\displaystyle f(x)=ae^{-{\frac {(x-b)^{2}}{2c^{2}}}}}

for arbitrary real constants a, b and c.

Modular Arithmetic

The modern approach to modular arithmetic was developed by Gauss in his book Disquisitiones Arithmeticae, published in 1801.  This now has application in number theory, abstract algebra, computer science, cryptography, and even in visual and musical art.

Surveying

Whilst doing a surveying job for the Royal House of Hanover in the years after 1818, Gauss was also looking into the shape of the Earth and started to question what the shape of space itself was. This led him to question Euclidean geometry – one of the central tenets of the whole mathematics, which premised a flat universe, rather than a curved one. He later claimed that as early as 1800 he had already started to consider types of non-Euclidean geometry (where the parallel axiom does not hold), which were consistent and free of contradiction. However, to avoid controversy, he did not publish anything in this area and left the field open to Bolyai and Lobachevsky, although he is still considered by some to be the pioneer of non-Euclidean geometry.

This survey work also fuelled Gauss’ interest in differential geometry, which uses differential calculus to study problems in geometry involving curves and surfaces. He developed what has become known as Gaussian curvature. This is an intrinsic measure of curvature that depends only on how distances are measured on the surface, not on the way it is embedded in space.

Positive, negative and zero Gaussian curvature of a shell
Source: shellbuckling.com

His achievements during these years, however, was not only limited to pure mathematics. He invented the heliotrope, which is an instrument that uses a mirror to reflect sunlight over great distances to mark positions in a land survey.

Image result for heliotrope gauss
Heliotrope | Source: Wikipedia

All in all, this period of time was one of the most fruitful periods of his academic life; he published over 70 papers between 1820 and 1830.

In later years, he worked with Wilhelm Weber to make measurements of the Earth’s magnetic field, and invented the first electric telegraph.

Read part 1 here and part 2 here.

Let me know what you think of this new series! M x

 

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Influential Mathematicians: Gauss (2)

Read the first part of this series here.

Although Gauss made contributions in many fields of mathematics, number theory was his favourite. He said that

“mathematics is the queen of the sciences, and the theory of numbers is the queen of mathematics.”

A way in which Gauss revolutionised number theory was his work with complex numbers.

Gauss gave the first clear exposition of complex numbers and of the investigation of functions of complex variables. Although imaginary numbers had been used since the 16th century to solve equations that couldn’t be solved any other way, and although Euler made huge progress in this field in the 18th century, there was still no clear idea as to how imaginary numbers were connected with real numbers until early 19th century. Gauss was not the first to picture complex numbers graphically (Robert Argand produced the Argand diagram in 1806). However, Gauss was the one who popularised this idea and introduced the standard notation a + bi. Hence, the study of complex numbers received a great expansion allowing its full potential to be unleashed.

Furthermore, at the age of 22 he proved the Fundamental Theorem of Algebra which states:

Every non-constant single-variable polynomial over the complex numbers has at least one root.

This shows that the field of complex numbers is algebraically closed, unlike the real numbers.

Gauss also had a strong interest in astronomy, and was the Director of the astronomical observatory in Göttingen. When Ceres was in the process of being identifies in the late 17th century, Gauss made a prediction of its position. This prediction was very different from those of other astronomers, but when Ceres was discovered in 1801, it was almost exactly where Gauss had predicted. This was one of the first applications of the least squares approximation method, and Gauss claimed to have done the logarithmic calculations in his head.

gauss_least_squares.gif
Source: The Story of Mathematics

Part 3 coming next week!

M x

Influential Mathematicians: Gauss (1)

I decided to start a new series on influential mathematicians, starting with Gauss, one of my personal favourites. Carl Friedrich Gauss (1777-1855) is considered to be one of the greatest mathematicians in the 19th century, and is sometimes referred to as the “Prince of Mathematics”.

His discoveries influenced and left a lasting mark in a variety of different areas, including number theory, astronomy, geodesy, and physics, particularly the study of electromagnetism.

Born in Brunswick, Germany to poor, working class parents, he was discouraged from attending school from his father, a gardner and brick-layer, who expected Gauss would follow one of the family trades. However, Gauss’ mother and uncle recognised Gauss’ early genius and knew he must develop this gift with a proper education.

In arithmetic class, at the age of 10, Gauss showed his skills as a maths prodigy. A well known anecdote about Gauss and his early school education is about when the strict schoolmaster gave the following assignment:

“Write down all the whole numbers from 1 to 100 and add up their sum.”

They expected this assignment to take a while to complete but after a few seconds, to the teacher’s surprise, Carl placed his slate on the desk in front of the teacher, showing he was done with the question. His other classmates took a much longer time to complete the assignment. At the end of class time, although most other students answers were wrong, Gauss’ was correct: 5050. Carl then explained to the teacher that he found the result as he could see that 1+100 = 101, 2+99=101, etc. So he could find 50 pairs of numbers that each add up to 101, and so 50*101 = 5050. I don’t know about you but I definitely could not come up with this sort of argument at the age of 10…

Although his family was poor, Gauss’ intellectual abilities drew the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum at the age of 15, and then to the University of Göttingen – a very prestigious university – where he stayed from 1795 to 1798. During this period, Gauss discovered many important theorems.

Prime Numbers

No pattern had previously been found in the occurrence of prime numbers until Gauss. Although the occurrence of the primes seems to be completely random, by approaching the problem from a different angle and graphing the incidence of primes as the numbers increased, he noticed a rough trend: as numbers increased by 10, the probability of the numbers reduced by a factor of around 2. However, as his method only gave him an approximation, and as he could not definitively prove his findings, he kept them a secret until much later in his life.

Graphs of the density of prime numbers

1796

1796 is known as Gauss’ “annus mirabilis” (means “wonderful year” and is used to refer to several years during which events of major importance are remembered).  In 1796:

  • Gauss constructed a regular 17-sided heptadecagon, which had previously been unknown, using only a ruler and a compass. This was a major advance in geometry since the time of the Greeks.
  • Gauss formulated this prime number theorem on the distribution of prime numbers among the integers, which states that \displaystyle \lim_{n\rightarrow\infty}\left[ \frac{\pi(n)}{n/\log n} \right] = 1 \,.  Here {\pi(n)} is the number of primes less than or equal to n. We can also write {\pi(n) \sim n/\log n}.
  •  Gauss proved that every positive integer can be represented as the sum of at most 3 triangular number

More about Gauss in the next post!

M x

MATHS BITE: Heptadecagon

A heptadecagon (or a 17-gon) is a seventeen sided polygon.

File:Regular polygon 17 annotated.svg
Regular Heptadecagon | Wikipedia

Constructing the Heptadecagon

In 1796, Gauss proved, at the age of 19 (let that sink in…) that the heptadecagon is constructible with a compass and a straightedge, such as a ruler. His proof of the constructibility of an n-gon relies on two things:

  • the fact that “constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions“;
  • the odd prime factors of n are distinct Fermat primes.

Constructing the regular heptadecagon involves finding the expression for the cosine of  2\pi /17 in terms of square roots, which Gauss gave in his book Disquistiones Arithmeticae:

{\displaystyle {\begin{aligned}16\,\cos {\frac {2\pi }{17}}=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\\=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}.\end{aligned}}}
Source: Wikipdia

An explicit construction was given by Herbert Willian Richmond in 1893.

Regular Heptadecagon Using Carlyle Circle.gif
Source: Wikipedia

M x

MATHS BITE: Shoelace Theorem

The Shoelace theorem is a useful formula for finding the area of a polygon when we know the coordinates of its vertices. The formula was described by Meister in 1769, and then by Gauss in 1795.

Formula

Let’s suppose that a polygon P has vertices (a1, b1), (a2, b2), …, (an, bn), in clockwise order. Then the area of P is given by

\[\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|\]

The name of this theorem comes from the fact that if you were to list the coordinates in a column and mark the pairs to be multiplied, then the image looks like laced-up shoes.

Screen Shot 2017-08-04 at 11.59.29 AM.png

Proof

(Note: this proof is taken from artofproblemsolving.)

Let $\Omega$ be the set of points that belong to the polygon. Then

\[A=\int_{\Omega}\alpha,\]

where $\alpha=dx\wedge dy$.

Note that the volume form $\alpha$ is an exact form since $d\omega=\alpha$, where

\[\omega=\frac{x\,dy}{2}-\frac{y\,dx}{2}.\label{omega}\]

Substitute this in to give us

\[\int_{\Omega}\alpha=\int_{\Omega}d\omega.\]

and then use Stokes’ theorem (a key theorem in vector calculus) to obtain

\[\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.\]

where

$\partial \Omega=\bigcup A(i)$

and $A(i)$ is the line segment from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$, i.e. Screen Shot 2017-08-04 at 12.05.20 PM.png is the boundary of the polygon.

Next we substitute for $\omega$:

\[\sum_{i=1}^n\int_{A(i)}\omega=\frac{1}{2}\sum_{i=1}^n\int_{A(i)}{x\,dy}-{y\,dx}.\]

Parameterising this expression gives us

\[\frac{1}{2}\sum_{i=1}^n\int_0^1{(x_i+(x_{i+1}-x_i)t)(y_{i+1}-y_i)}-{(y_i+(y_{i+1}-y_i)t)(x_{i+1}-x_i)\,dt}.\]

Then, by integrating this we obtain

\[\frac{1}{2}\sum_{i=1}^n\frac{1}{2}[(x_i+x_{i+1})(y_{i+1}-y_i)- (y_{i}+y_{i+1})(x_{i+1}-x_i)].\]

This then yields, after further manipulation, the shoelace formula:

\[\frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i).\]

M x

Prime Number Theorem

Today I thought I’d quickly discuss a extremely important theorem in one of my favourite areas in mathematics: Number Theory (as you can probably tell by the number of posts that I’ve published about primes!).

Perhaps the first property of π(x) – the number of primes less than or equal to x – is that π(x) tends to infinity as x tends to infinity. In other words, the prime numbers are infinite, which was proved by Euclid in “Elements”. A more precise result, established by Euler in 1737, was that the series of reciprocals of the prime numbers:

Screen Shot 2016-10-19 at 6.09.35 PM.png

is a divergent series. In doing so, Euler found an alternative way to prove that there was an infinite number of primes, as if there wasn’t then the series would have a finite value.

The Prime Number Theorem states that if π(x) is the number of primes less than or equal to x, then

img39.gif

Although the notation ~ may be unfamiliar, it simply means that π(x) is asymptotically equal to x/lnx, i.e.

 img41.gif

Note that the prime number theorem is equivalent to saying that the nth prime number pn satisfies the following relationship:

img44.gif

The PNT was proposed by Gauss in 1792 when he was only 15 years old! (Makes you wonder what you’ve been doing with your life so far…) He later refined this estimate to

\begin{displaymath}\pi(x) \sim \int_2^x \frac{d u}{\ln{u}}.\end{displaymath}

HAPPY HALLOWEEN! M x

Beautiful Equations II

‘Beautiful Equations I’ was one of my favourite posts to write, so I decided to continue with this series and write a part 2.

Euler’s Identity

Leonhard Euler was one of the most influential and prolific mathematicians in history, laying the foundations of an array of areas in mathematics for his successors to build upon. His output was immense; he published more than 500 books and papers during his lifetime and a further 400 appeared posthumously.

065028gett62angangag6g

Euler’s Identity is often considered the most beautiful equation in mathematics as it combines five of the most fundamental mathematical constants:

  • e: the base of natural logarithms
  • i: the imaginary unit of complex numbers, equivalent to the square root of -1
  • π: the ratio of a circle’s circumference to its diameter
  • 1: the multiplicative identity
  • 0: the additive identity

Feynman described it as “the most remarkable formula in mathematics”, and I must say that I completely agree.

So where does this identity come from?

If you’ve studied complex numbers you will know Euler’s relation:

2000px-Euler's_formula.svg

Simply substitute the angle as π!

euler21

For more information on where Euler’s relation comes from, click here.

Boltzmann’s Entropy Formula

As a chemistry student, one of my favourite topics (apart from my beloved organic chemistry) is entropy. Put simply, entropy is the degree of disorder in the system. For a reaction to occur, entropy must always increase.

boltzmann-equation

Boltzmann’s formula relates entropy (S) of an ideal gas and the number of ways that the atoms or molecules can be arranged (k log W). The more ways the particles can be arranged, the greater the disorder and therefore entropy of the system. K is Boltzmann’s constant and W is the number of microscopic elements of a system in a macroscopic system in a state of balance.

Schrödinger Equation

12759838d4297bdb0bd88c613cbb63c7

Edwin Schrödinger’s famous partial differential equation illustrates how subatomic particles change with time when under the influence of a force. Any particular atom or molecule is described by its wave function (represented by the Greek letter psi), which predicts the probability of where and when the particles appear.

However, physicists are still unsure on how to interpret this equation. Some believe that it’s just a useful calculation tool, but does not actually correspond to anything real, whilst others argue that it demonstrates the limit to the amount that we can learn about the universe, as we can only learn about a particle once it’s measured.

Schrödinger believed that the wavefunction represented a real, physical object and rejected the interpretation that a particle only collapses when it’s measured. In fact, his famous cat experiment actually intended to demonstrate the weakness of this interpretation.

The Gaussian Integral

gaussian integral

The function in the Gaussian integral is a very hard function to integrate. However, when analyzed over the whole real line – from minus infinity to infinity – the answer is surprisingly neat. This formula is of extreme use and has a range of applications. For example, it is used to calculate the normalising constant of the normal distribution.

The Analytic Continuation of the Factorial

The factorial function is commonly defined as

factorial-formula

However, this only works for positive integers. Therefore, by using this integral:

analytic continuation of the factorial

mathematicians are able to compute factorials for fractions, decimals, negative numbers and even complex numbers. The gamma function is an extension of this, using n – 1 instead of n.

gamma function

It’s used in various probability-distribution functions, and so highly applicable to probability, statistics and combinatorics.

The Explicit Formula for the Fibonacci Sequence

The Explicit Formula for the Fibonacci Sequence

This formula, derived by Binet in 1843 (although the result was known to Euler, Daniel Bernoulli and de Moivre more than a century earlier) can be used to calculate the nth Fibonacci number in the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, etc., where each number is the sum of the previous two numbers). Although this sequence a classic and known by the vast majority of people, this formula is known to few. Remarkably, despite the formula having square roots and divisions, the answer is always an exact positive integer.

The phi in the formula represents the golden ratio, where

gr value

Two quantities are in the golden ratio when their ratio is the same as the ratio of their sum to the larger of the two quantities. When a > b > 0, this can be expressed as

gr

I personally find the golden ratio a fascinating part of mathematics. Would you like me to do a blog post on this? x