The sum of the first $latex {n}&fg=000000$ odd numbers is equal to the square of $latex {n}&fg=000000$: $latex \displaystyle 1 + 3 + 5 + \cdots + (2n-1) = n^2 \,. &fg=000000$ We can check this for the first few: $latex {1 = 1^2,\ \ 1+3=2^2,\ \ 1+3+5 = 3^2}&fg=000000$. But how do we prove … Continue reading Proofs without Words
Tag: Algebra
Closeness in the 2-Adic Metric
When is 144 closer to 8 than to 143? The usual definition of the norm of a real number $latex {x}&fg=000000$ is its modulus or absolute value $latex {|x|}&fg=000000$. We measure the ``distance'' between two real numbers by means of the absolute value of their difference. This gives the Euclidean metric $latex {\rho(x,y) = |x-y|}&fg=000000$ … Continue reading Closeness in the 2-Adic Metric
A Grand Unification of Mathematics
There are numerous branches of mathematics, from arithmetic, geometry and algebra at an elementary level to more advanced fields like number theory, topology and complex analysis. Each branch has its own distinct set of axioms, or fundamental assumptions, from which theorems are derived by logical processes. While each branch has its own flavour, character and … Continue reading A Grand Unification of Mathematics
Finding Fixed Points
An isometry on a metric space is a one-to-one distance-preserving transformation on the space. The Euclidean group $latex {E(n)}&fg=000000$ is the group of isometries of $latex {n}&fg=000000$-dimensional Euclidean space. These are all the transformations that preserve the distance between any two points. The group depends on the dimension of the space. For the Euclidean plane … Continue reading Finding Fixed Points
Dimension Reduction by PCA
We live in the age of ``big data''. Voluminous data collections are mined for information using mathematical techniques. Problems in high dimensions are hard to solve --- this is called ``the curse of dimensionality''. Dimension reduction is essential in big data science. Many sophisticated techniques have been developed to reduce dimensions and reveal the information … Continue reading Dimension Reduction by PCA
Bang! Bang! Bang! Explosively Large Numbers
Enormous numbers pop up in both mathematics and physics. The order of the monster group, the largest of the 26 sporadic groups, is $latex \displaystyle 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 &fg=000000$ which is approximately $latex {8\times 10^{53}}&fg=000000$. The number of atoms in the universe is estimated to be about $latex {10^{80}}&fg=000000$. When we consider permutations of large sets, even … Continue reading Bang! Bang! Bang! Explosively Large Numbers
The Rambling Roots of Wilkinson’s Polynomial
Finding the roots of polynomials has occupied mathematicians for many centuries. For equations up to fourth order, there are algebraic expressions for the roots. For higher order equations, many excellent numerical methods are available, but the results are not always reliable. James Wilkinson (1963) examined the behaviour of a high-order polynomial $latex \displaystyle p(x,\epsilon) = … Continue reading The Rambling Roots of Wilkinson’s Polynomial
Adjoints of Vector Operators
We take a fresh look at the vector differential operators grad, div and curl. There are many vector identities relating these. In particular, there are two combinations that always yield zero results: $latex \displaystyle \begin{array}{rcl} \mathbf{curl}\ \mathbf{grad}\ \chi &\equiv& 0\,, \quad \mbox{for all scalar functions\ }\chi \\ \mathrm{div}\ \mathbf{curl}\ \boldsymbol{\psi} &\equiv& 0\,, \quad \mbox{for all … Continue reading Adjoints of Vector Operators
Some Fundamental Theorems of Maths
Every branch of mathematics has key results that are so important that they are dubbed fundamental theorems. The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is … Continue reading Some Fundamental Theorems of Maths
The Brief and Tragic Life of Évariste Galois
On the morning of 30 May 1832 a young man stood twenty-five paces from his friend. Both men fired, but only one pistol was loaded. Évariste Galois, a twenty year old mathematical genius, fell to the ground. The cause of Galois's death is veiled in mystery and speculation. Whether both men loved the same woman … Continue reading The Brief and Tragic Life of Évariste Galois
The Many Modern Uses of Quaternions
The story of William Rowan Hamilton's discovery of new four-dimensional numbers called quaternions is familiar. The solution of a problem that had bothered him for years occurred to him in a flash of insight as he walked along the Royal Canal in Dublin. But this Eureka moment did not arise spontaneously: it was the result … Continue reading The Many Modern Uses of Quaternions
The Flight of the Bumble Bee
Alice and Bob, initially a distance l apart, walk towards each other, each at a speed w. A bumble bee flies from the nose of one to the nose of the other and back again, repeating this zig-zag flight at speed f until Alice and Bob meet. How far does the bumble bee fly? There … Continue reading The Flight of the Bumble Bee
Marden’s Marvel
Although polynomial equations have been studied for centuries, even millennia, surprising new results continue to emerge. Marden's Theorem, published in 1945, is one such -- delightful -- result. For centuries, mathematicians have struggled to find roots of polynomials like p(x) ≡ xn + an-1 xn-1 + an-2 xn-2 + an-3 xn-3 + … a1 x … Continue reading Marden’s Marvel
Sophus Lie
“It is difficult to imagine modern mathematics without the concept of a Lie group.” (Ioan James, 2002). Sophus Lie grew up in the town of Moss, south of Oslo. He was a powerful man, tall and strong with a booming voice and imposing presence. He was an accomplished sportsman, most notably in gymnastics. It was no … Continue reading Sophus Lie
Cubic Skulduggery & Intrigue
Babylonian mathematicians knew how to solve simple polynomial equations, in which the unknown quantity that we like to call x enters in the form of powers, that is, x multiplied repeatedly by itself. When only x appears, we have a linear equation. If x-squared enters, we have a quadratic. The third power of x yields … Continue reading Cubic Skulduggery & Intrigue
Metallic Means
Everyone knows about the golden mean. It must be one of the most written-about numbers, certainly in recreational mathematics. It is usually denoted by $latex {\phi}&fg=000000$ and is the positive root of the quadratic equation $latex \displaystyle x^2 - x - 1 = 0 \ \ \ \ \ (1)&fg=000000$ with the value $latex {\phi … Continue reading Metallic Means
The Beginning of Modern Mathematics
The late fifteenth century was an exciting time in Europe. Western civilization woke with a start after the slumbers of the medieval age. Johannes Gutenberg's printing press arrived in 1450 and changed everything. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. Leonardo da Vinci was in his prime and Christopher Columbus was … Continue reading The Beginning of Modern Mathematics
Raphael Bombelli’s Psychedelic Leap
The story of how Italian Renaissance mathematicians solved cubic equations has elements of skullduggery and intrigue. The method originally found by Scipione del Ferro and independently by Tartaglia, was published by Girolamo Cardano in 1545 in his book Ars Magna. The method, often called Cardano's method, gives the solution of a depressed cubic equation t3 … Continue reading Raphael Bombelli’s Psychedelic Leap
Andrew Wiles wins 2016 Abel Prize
A recent post described the Abel Prize, effectively the Nobel Prize for Mathematics, and promised a further post when the 2016 winner was announced. This is the follow-up post [also at TM091, or search for “thatsmaths” at irishtimes.com]. Next Tuesday, HRH Crown Prince Haakon will present the Abel Medal to Sir Andrew Wiles at a ceremony … Continue reading Andrew Wiles wins 2016 Abel Prize
James Joseph Sylvester
James Joseph Sylvester was born in London to Jewish parents in 1814, just 201 years ago today. The family name was Joseph but, for reasons unclear, Sylvester – the name of an anti-Semitic Pope from the Roman period – was adopted later. [TM075; or search for “thatsmaths” at irishtimes.com ] Sylvester's mathematical talents became evident at … Continue reading James Joseph Sylvester
Increasingly Abstract Algebra
In the seventeenth century, the algebraic approach to geometry proved to be enormously fruitful. When René Descartes (1596-1650) developed coordinate geometry, the study of equations (algebra) and shapes (geometry) became inextricably interlinked. The move towards greater abstraction can make mathematics appear more abstruse and impenetrable, but it brings greater clarity and power, and can lead … Continue reading Increasingly Abstract Algebra
Emmy Noether’s beautiful theorem
The number of women who have excelled in mathematics is lamentably small. Many reasons may be given, foremost being that the rules of society well into the twentieth century debarred women from any leading role in mathematics and indeed in science. But a handful of women broke through the gender barrier and made major contributions. … Continue reading Emmy Noether’s beautiful theorem
Mode-S: Aircraft Data improves Weather Forecasts
A simple application of vectors yields valuable new wind observations for weather forecasting [see this week’s That’s Maths column (TM065) or search for “thatsmaths” at irishtimes.com]. It has often happened that an instrument designed for one purpose has proved invaluable for another. Galileo observed the regular swinging of a pendulum. Christiaan Huygens derived a mathematical … Continue reading Mode-S: Aircraft Data improves Weather Forecasts
The Klein 4-Group
What is the common factor linking book-flips, solitaire, twelve-tone music and the solution of quartic equations? Answer: $latex {K_4}&fg=000000$. Symmetries of a Book --- or a Brick Take a book, place it on the table and draw a rectangle around it. How many ways can the book fit into the rectangle? Clearly, once any … Continue reading The Klein 4-Group
The Steiner Minimal Tree
Steiner's minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner's problem allows for additional points – now called Steiner points – to be added … Continue reading The Steiner Minimal Tree
Old Octonions may rule the World
This week’s That’s Maths column in The Irish Times (TM055, or search for “thatsmaths” at irishtimes.com) is about octonions, new numbers discovered by John T Graves, a friend of William Rowan Hamilton. On this day in 1843, the great Irish mathematician William Rowan Hamilton discovered a new kind of numbers called quaternions. Each quaternion has … Continue reading Old Octonions may rule the World
Algebra in the Golden Age
This week’s That’s Maths column in The Irish Times (TM054, or search for “thatsmaths” at irishtimes.com) is about the emergence of algebra in the Golden Age of Islam. The Chester Beatty Library in Dublin has several thousand Arabic manuscripts, many on mathematics and science. "The ink of a scholar is holier than the blood of … Continue reading Algebra in the Golden Age
Cartoon Curves
The powerful and versatile computational software program called Mathematica is widely used in science, engineering and mathematics. There is a related system called Wolfram Alpha, a computational knowledge engine, that can do Mathematica calculations and that runs on an iPad. Mathematica can do numerical and symbolic calculations. Algebraic manipulations, differential equations and integrals are simple, … Continue reading Cartoon Curves
The High-Power Hypar
Maths frequently shows us surprising and illuminating connections between physical systems that are not obviously related: the analysis of one system often turns out to be ideally suited for describing another. To illustrate this, we will show how a surface in three dimensional space --- the hyperbolic paraboloid, or hypar --- pops up in unexpected … Continue reading The High-Power Hypar
The Unity of Mathematics
This week, That’s Maths in The Irish Times ( TM041 ) is about an ambitious program to unify mathematics. Mathematics expands! Results once proven to be true remain forever true. They are not displaced by subsequent results, but absorbed in an ever-growing theoretical web. Thus, it is increasingly difficult for any individual mathematician to have … Continue reading The Unity of Mathematics
The Langlands Program
An ambitious programme to unify disparate areas of mathematics was set out some fifty years ago by Robert Langlands of the Institute for Advanced Study in Princeton. The “Langlands Program” (LP) is a set of deep conjectures that attempt to build bridges between certain algebraic and analytical objects. Substantial strides have been made in the … Continue reading The Langlands Program
Speed Cubing & Group Theory
The article in this week’s That’s Maths column in the Irish Times ( TM038 ) is about Rubik's Cube and the Group Theory that underlies its solution. The Rubik's cube craze ran through the world like wildfire in the 1980s. This simple mechanical puzzle is made from small pieces, called “cubies”, in a 3x3x3 structure … Continue reading Speed Cubing & Group Theory
Bézout’s Theorem
Two lines in a plane intersect at one point, a line cuts a circle at two points, a cubic (an S-shaped curve) crosses the x-axis three times and two ellipses, one tall and one squat, intersect in four places. In fact, these four statements may or may not be true. For example, two parallel lines … Continue reading Bézout’s Theorem
The Watermelon Puzzle
An amusing puzzle appears in a recent book by John A. Adam (2013). The answer is very surprising. The book argues in terms of simultaneous equations. A simpler argument, using the diagram below, should make all clear. The Watermelon Puzzle. A farmer brings a load of watermelons to the market. Before he sets out, he … Continue reading The Watermelon Puzzle
Monster Symmetry
The That's Maths column in the Irish Times this week is about symmetry and group theory, and the possible link, through string theory, with the fundamental structure of the universe ( TM020 ). In the arts, symmetry is intimately associated with aesthetic appeal. In science, it provides insight into the properties of physical systems. In … Continue reading Monster Symmetry
Singularly Valuable SVD
In many fields of mathematics there is a result of central importance, called the "Fundamental Theorem" of that field. Thus, the fundamental theorem of arithmetic is the unique prime factorization theorem, stating that any integer greater than 1 is either prime itself or is the product of prime numbers, unique apart from their order. The … Continue reading Singularly Valuable SVD
Google PageRank
This week's That's Maths article, at TM002, describes how Google's PageRank software finds all those links when you enter a search word, by solving an enormous problem in linear algebra. A comprehensive description of PageRank is given in the book Google's PageRank and Beyond: The Science of Search Engine Rankings, by Amy N. Langville & … Continue reading Google PageRank
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