The Mathematics Of Happy New Year
My 2012 New Year Wish has some beautiful mathematics: beautifully simple, beautifully abstract, and beautifully mysterious. Want to peek under her skirts?
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Mathematics
Late Night Meeting of Gödel and Turing?
Hacking together concepts from mathematical analysis, mathematical logic, and computer science and finding similarities while half asleep can be fun! :-)
Infinity of Infinities
Our long journey through the infinite lands is coming to an end. What end is there to infinity, you ask? I’d have to put on a theologian’s hat to answer that. But as a mathematician, I can answer a question much more daring: what end is there to infinities?
The Set of Subsets of a Set
Can there be any fact more shocking than that there are two infinities, one bigger than the other? Well, take a guess… But to crack the mystery underlying the existence of two different infinities, we first need to learn a little more from set theory… Monkeys, sheep, and leopards return!
The Bigger Infinity
Let’s challenge our notion of infinity by stating a downright crazy idea… and then proving it beyond any doubt.
All The Numbers In The World
We know that the natural numbers 1,2,3,… go off to infinity. But what happens when we consider negative numbers -1,-2,-3,… as well? How many numbers do we get? And what about fractions? Do we have multiple infinities?
Aleph-Null Bottles Of Beer On The Wall
Let’s quickly recap the most important (and weird) properties of the “natural numbers infinity” we call <$\aleph_0$>.
Reintroducing Maths For Normal People
Back in spring I started a series on interesting mathematical topics for the layman. Now I’m back with some fresh stuff. I’m still aiming to explain thrilling areas of pure mathematics to a non-mathematical audience.
Cheat Sheet: Classification of States In Markov Chains
I created a brief summary of definitions and theorems related to basic classification of Markov chain states. There is even a couple of diagrams that helped me learn some of the rules. Download, enjoy, and let me know what I can improve!
How To Solve First Order Linear ODEs Using Integrating Factor
Multiplication by integrating factor is my favorite method of solving first order linear ordinary differential equations…
On The Finiteness Of Life Experience
There’s a great new comic at Spiked Math: It’s a small world (after all). Be sure to check it out even if you are not a mathematician. Using mathematical reasoning, the comic asserts that the number of ways you could lead your life is finite, in other words, there is a limit to what you could do in your life. I strongly disagree, and I can disprove this assertion using the very same tool: mathematics :-)
Mother’s Day 2e3 + 11
Did you remember to give your mother a flower? :-)
Simulating Markov Chains In Mathematica
A Markov chain is a sequence of random variables (states) satisfying the Markov property: the probability of the current state depends only on the state that immediately preceded it. In other words, the past state and the future state are stochastically independent. How can we simulate such chains in Wolfram Mathematica?
Deriving Common Trigonometric Identities
There are quite a few identities concerning trigonometric functions. The most commonly used are the cosine and sine of double angles and the cosine and sine of the sum or difference of two angles. It might be perhaps surprising that these formulae can be easily derived with the help of basic complex analysis and algebra. Personally, I find it much easier to derive most of the formulae whenever I need them instead of remembering them all.
Poincaré on Mathematical Elegance
“Mathematicians attach a great importance to the elegance of their results, and this is not mere dilettantism. What is it that gives us the feeling of elegance in a solution or a demonstration?…”
Solving Separable Ordinary Differential Equations
Simple guide to solving separable differential equations, plus explanation how it works.
Infinity Plus Three (Maths For Normal People III)
Are there more even numbers than odd numbers? How much can we take away from infinity to keep it infinite? And how much is “ten times infinity”? Is it more than “twenty times infinity”? Do these questions have a meaning at all? Let us delve into the depths of the infinite oceans once again!
Of Macbeth And 100,000 Monkeys (Maths For Normal People II)
Do you know what the largest natural number is? Where do the borders of the infinite realm lie? And how many monkeys does it take to write the complete works of Shakespeare? Come hither, ye finite mortal, and behold the glory of eternity that men have created in their minds!
My Cardinal Is Bigger Than Yours (Maths For Normal People I)
Are you a non-mathematician? Do you want to know a little about interesting mathematical phenomena? I am starting a series of articles on selected mathematical topics. I am going to explain them very simply and comprehensibly. No prior knowledge of higher mathematics will be needed. Today, we are going to talk about sets and their sizes. In the next article, we will use this knowledge to understand the mightiest beast itself – infinity.
What Are Natural Numbers?
We use natural numbers 1, 2, 3,… (sometimes including zero) in everyday life so obviously, so effortlessly, and so automatically that it hardly ever occurs to us to ask what they actually are. What is a natural number? How do you define twenty-seven? Let us take a brief look at three approaches, ranging from Plato to the present day, that try to set down a formal definition of the fundamental term number.
Bertrand Russell’s Subtle Humor
The works of the philosopher, logician, and mathematician Bertrand Russell are always a pleasant reading. Be it because of the appeal of the topic, eloquent style, profound treatment,… or his occasional kind jest.
Happy Mother’s Day 2e3 + 10
Dear mother! Let your parametric curves be smooth, continuous, and differentiable ∀ϑ ∊ <0, 2π>. Happy Mother’s Day 2010! (that exclamation mark is not a factorial)
Matrix Multiplication in Galois Fields With Python
A few weeks ago I needed to quickly verify a result of matrix multiplication. Having only a few minutes, I had to resort to a DIY solution instead of finally learning how to use numpy (yeah, still on the TO DO list…)
Inclusion-Exclusion Principle: Proof by Mathematical Induction For Dummies
Inclusion-Exclusion Principle is one of those double-faced theorems. Based on a transparent idea, but nearly incomprehensible when formulated and proved in mathematical terms. After seeing the proof demonstrated in a Discrete Mathematics course, the only comment I could think of was “What the hell?!”