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!

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F.T.A. via Complex Analysis

Although this requires a bit of knowledge on Complex Anlaysis, I recently discovered this new way to prove the Fundamental Theorem of Algebra and I couldn’t help but share it.

First of all, what is the Fundamental Theorem of Algebra (FTA)? This very important (hence the name!) result states that:

Every non-constant polynomial with complex coefficients has a complex root. 

In order to prove this, we must first be aware of Liouville’s Theorem:

Every bounded, entire function is constant. 


Definitions

Bounded: a function on a set X is said to be bounded if there exists a real number M such that

|f(x)|\le M

for all x in X.

Entire: An entire function is a holomorphic function on the entire complex plane.


Liouville’s theorem is proved using the Cauchy integral formula for a disc, one of the most important results in Complex Analysis. Although I will not describe how to prove it or what it states in this blog post, I encourage you to read about here it as it is truly a remarkable result.

Now armed with Liouville’s Theorem we can prove the FTA.

Proof

Let P(z) = zn + cn-1zn-1 + … + c1z + c0 be a polynomial of degree n > 0. Then |P(z)| –> ∞ as |z| –> ∞, so there exists R such that |P(z)| > 1 for all z with |z| > R.

Consider f(z) = 1/P(z). If P has no complex zeros then f is entire. So, as f is continuous, f is bounded on {|z| ≤ R}.

As |f(z)| < 1 when |z| > R, f is a bounded entire function, so by Liouville’s Theorem f is constant, which is a contradiction.

The only thing we assumed was that P had no complex zeros, and so we contradicted this fact. Hence, P must have at least one complex zero. Amazing right!

 

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Julia Set

The Julia Set is named after the French mathematician Gatson Julia who investigated their properties in 1915, culminating in a paper published in 1918.

Julia was interested in the iterative properties of the more general expression:

z4 + z3/(z-1) + z2/(z3 + 4z2 + 5) + c.

However, now the Julia Set is associated with those points z0 = x + iy on the complex plane for which the series of form zn+1 = zn2 + c, does not tend to infinity.

How are the images of the Julia Set generated?

You may have seen some really beautiful images of the Julia Set:

Computing the Julia Set is quite straightforward using the brute force method approach. One must simply assign each pixel a number in the complex plane, which is then the starting point of the series. The series is iterated for each starting point and two colours are assigned for the two cases which can arise: the series diverges to infinity (usually white) or it does not (usually black).

Below is the Julia Set for f(z) = z2 – 0.75. Note that the other colours in the image indicate how soon the iterates left towards infinity (going from red, yellow, green, blue and magenta in decreasing order of speed).

For almost every c, the Julia Set generates a fractal.

Sources: 1 | 2 | 3

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Vectors #1: A History

This week I am going to be doing a series on Vectors, starting off with the history of vectors.

Complex Numbers

Vectors were born during the start of the 19th century, due to the need to represent complex numbers geometrically. The mathematicians Caspar Wessel (1745-1818), Jean Robert Argand (1768-1822) and Carl Friedrich Gauss (1777-1855) were the first to show complex numbers as being points in a two-dimensional plane, and hence as two-dimensional vectors.

Source: Wikipedia

This idea was part of the effort to extend two-dimensional numbers to three dimensions, however at the time no one was able to accomplish this whilst still preserving the basic algebraic properties of real and complex numbers.

In 1827, August Ferdinand Möbius published a book entitled ‘The Barycentric Calculus’, where he introduced line segments which had a direction and where denoted by letters. Basically, these were vectors in all but the name! In the book, he showed how to perform calculations with these line segments – how to add them and multiply them with a real number. However, these accomplishments and their importance were not noticed by the mathematical community.

Quaternions

In 1843, William Hamilton introduced quaternions – a four dimensional system. Hamilton expressed:

“I was walking in to attend and preside along the Royal Canal, an under-current of thought was going on in my mind, which at last gave a result, whereof it is not too much to say that I felt at once the importance. An electric circuit seemed to close; and a spark flashed forth, I could not resist the impulse to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formulae.”

Quaternions were of the form: q = w + ix + jy + kz, where w, x, y and z are real numbers. Hamilton realised that quaternions consisted of two parts – the scalar and the vector part. Hamilton used his ‘fundamental formula’: i2 =j2 = k2 = –ijk = -1, to multiply them together and thus discovered that the product of quaternions was not commutative.

In the 1850s, Peter Guthrie Tait applied quaternions to problems involving electricity and magnetism and other problems in physics.

Grassmann Algebra

Around the same time, Hermann Grassman published the book ‘The Calculus of Extension’, which developed a new geometric calculus. He used this to simply large parts of two classical works – Analytical Mechanics by Lagrange and Celestial Mechanics by Laplace. Grassmann expanded the concept of a vector from two or three dimensions to n dimensions, which greatly extended the ideas of space. Furthermore, he anticipated a large amount of modern matrix, linear algebra and vector and tensor analysis. However, his work was largely ignored as it was lacking in explanatory examples and written in a strange style with overcomplicated notation.

Gibbs

The development of the algebra of vectors and vector analysis is largely credited to J. Willard Gibbs. Although British scientists, including Maxwell, relied on Hamilton’s quaternions in order to express the dynamics of physical quantities, Gibbs was the first to note that the product of quaternions always had to be separated into two parts. Hence, calculations with quaternions introduced unnecessary complications and redundancies that could be removed. Therefore, he proposed defining distinct dot and cross products for pairs of vectors and introduced the vector notation we use today.

Diagram representing the cross product of two vectors
Source: Wikipedia

While working on vector analysis, Gibbs realised that his approach was similar to that of Grassmann and thus sought to publicise Grassmann’s work, stressing that is was more general than Hamilton’s quaternions.

Oliver Heaviside also developed his own vector analysis of the same style.

In the 1880s and 1890s, quaternions was eventually abandoned by physicists who preferred the vectorial approach proposed by Gibbs and Heaviside.

Hope you enjoyed today’s short introduction to vectors. Make sure you return for the rest of the series! M x

 

 

Roots of Unity

The nth Roots of Unity appear when we consider the complex roots of an equation of the form:

z^n = 1.

Solving the Equation

As we have an nth degree polynomial, we will have n complex roots. By converting this to the polar form (by letting  Screen Shot 2016-04-21 at 10.05.09 AM.png and noting that $1 = e^{2\pi ik}$ for $k\in \mathbb{Z}$), we get the expression:

$r^ne^{ni\theta} = e^{2\pi ik}$

As the magnitude of the right hand side is 1, we can deduce that r = 1, leaving us with $e^{ni\theta} = e^{2\pi ik}$. Quick algebraic manipulation gives us:

$\theta=\frac{2\pi k}n$

Hence, we can conclude that the solutions the polynomial are given by Screen Shot 2016-04-21 at 10.12.27 AM which can be converted to the trigonometric equation using Euler’s formula:

Screen Shot 2016-04-21 at 10.14.41 AM.png

Geometry

All roots of unity lie on the unit circle in the complex plane, as all roots have a magnitude of 1.

The roots of unity
Source: mathworld.wolfram.com

Additionally, the nth roots of unity are connected in order, they form a regular n sided polygon. This can easily be seen by analysing the arguments of the roots.

Properties

  • The sum of the nth roots of unity is 0.
  • If $\zeta$ is a primitive nth root of unity, then the roots of unity can be expressed as $1, \zeta, \zeta^2,\ldots,\zeta^{n-1}$.
    • A primitive nth root of unity is such that $\zeta^m\neq 1$ for $1\le m\le n-1$.
    • This sequence of powers is n periodic because z j + n = z jz n = z j⋅1 = z j for all values of j.
  • For each nth root of unity, $\zeta$, we have that $\zeta^n=1$. Although obvious, this property should not be forgotten as, for example, it can aid with algebraic manipulation.

Representations of Roots of Unity

Colorful "string" art on the 17 th roots of unity
17th roots of unity | Source: nyates314.wordpress.com
Complex Polynomial Functions of degree 3: cubic roots of unity | matematicasVisuales
z^3-1 | Source: matematicasVisuales
8th roots of unity | isaacdooley.com

Pretty huh? M x

Fractals

A fractal is a never-ending pattern that repeats itself at different scales and is created by a simple process that repeats continuously.

Fractals in Nature

Romanesco Broccoli is an example of the ‘ultimate’ fractal vegetable. Its pattern represents the golden spiral. fractal_10

The golden spiral is a logarithmic spiral where every turn is farther from the origin by a factor of the golden ratio (phi).

golden spiral

Salt flats also contain remarkably consistent but random patterns created by the encrusted salt.

fractal_11a

Ammonites have been extinct for 65 million years and were predatory, squid-like creatures that lived inside coil-shaped shells. The walls between the chambers inside their spiral shells were complex fractal curves. Similarly to the romanesco broccoli, the shells of ammonites also grow as a logarithmic spiral.

fractal_1a

Mountains are formed from tectonic forces that push the crust upwards, joined with erosion, which breaks some of the crust down, resulting in a fractal pattern.

IDL TIFF file

Fractal patterns are also present in many plants, as they generate their branching shapes and leaf patterns through simple recursive formulas.

fractal_6a

When water crystallises it forms repeating patterns in snowflakes or on frosty surfaces. fractal_12a

These patterns inspired the first described fractal curves – the Koch snowflake – in a 1904 paper by Swedish mathematician Helge von Koch.

The first four iterations of the Koch snowflake
The first four iterations of the Koch snowflake

Finally, the path that lightning takes is formed step by step as it moves towards the ground and closely resembles a fractal pattern.

fractal_8a

Fractals in Mathematics

Mathematical fractals are formed by calculating a simple equation thousands of times, feeding the answer back to the start. The mathematical beauty of fractals is that infinite complexity – meaning we can zoom into them forever – is formed from relatively simple equations.

A very famous example of a mathematical fractal is the Mandelbrot Set, which was discovered by Benoit Mandelbrot in 1980.

The Mandelbrot set is a collection of numbers that are generated from the recurrence equation:

NumberedEquation1

Firstly we specify an initial value of z and c (a constant). We are looking for starting values of z, for which the sequence of numbers generated by the equation remains bounded. For example, if we start with z = -1 and c = 0, then the values will always be 1 or -1, so the sequence is bounded. Hence, -1 is included in our set of solutions.

When extending this to use complex numbers, the results become very interesting; we get the Mandelbrot Set:

Mandel_zoom_00_mandelbrot_set

In this picture, the black indicates numbers in the set, blue are numbers not in the set, and white is the boundary.

For more information, I suggest clicking here or here.

Additionally, Fractals are closely related to Chaos Theory (as they are complex systems that have definite properties) which is a subset of and area in mathematics called Dynamical Systems. These allow us to determine the general behaviour of solutions to systems of equations without actually solving the equations.

Let me know what you think of fractals! 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