Rule #1 Two knots are identical if and only if their complements are identical Rule # 2 Two knots can have the same group yet not be the same knots Therefore: . A knot and . . another [more …]

## Math

x = 0.999…

10x = 9.99…

10x-x = 9.99… – 0.999…

9x = 9

x = 1

0.999… = 1

## Two types of people

There are two types of people. Those who say that the decimal notation 0.999… is not equal to 1, and then there are mathematicians who have proven that 0.999… is equal to 1. The Catholic theologian Bernard Lonergan developed a theory of what it means to understand – a theory of knowledge – much like [more …]

## The Mathematician and the Calculator

Not being a mathematician I can say this with great arrogance that mathematics is something quite above and beyond mere calculation. Mathematics is much more than computation. Yes, of course – but then again perhaps not “of course” but rather, “why?” Why did Carl Friedrich Gauss say: “Mathematics is the queen of sciences and number [more …]

## Collection of Large Numbers

The number of galaxies in the observable universe = 10^12 10000000000000 The number of grains of sand to fill an average office = 5×10^13 50000000000000 Meters from the earth to the edge of the observable universe = 4.3×10^26 (46 billion light years ) 4300000000000000000000000000 Meters in the observable universe = 8.8×10^26 (92 billion light years) [more …]

The Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100 BC. Technological artifacts of similar complexity and workmanship did not reappear until the 14th century, when mechanical astronomical clocks were built in Europe.

The mechanism is the oldest known complex scientific calculator. It contains many gears, and is sometimes called the first known analog computer, although its flawless manufacturing suggests that it may have had a number of undiscovered predecessors during the Hellenistic Period. It appears to be constructed upon theories of astronomy and mathematics developed by Greek astronomers and it is estimated that it was made around 150-100 BC.

## Non-differentiable continuous functions exist

This statement, if understood, should shock you out of your socks. It basically means that there are continuous curves that have no tangent! Can you imagine what such a curve looks like? Hint: are you good at fractions?

http://en.wikipedia.org/wiki/Weierstrass_function

http://www.math.cmu.edu/%7Ebobpego/21132/nowhdiff.pdf

## Intuition Failure

x = 0.999…

10x = 9.99…

10x-x = 9.99… – 0.999…

9x = 9

x = 1

0.999… = 1

If you are like me, you look at this equation and scoff. Common sense and years of math intuition tell you that what is on the left is not the same as what is on the right. If you are like me, you are wrong. But if you are like me, you will look it up on wikipedia or at math wolfram, you will see that there are rigorous proofs… but you will yet still have doubts, unsatisfied. You are not alone. This is one of those bedeviling problems that has worn many a thinker – from Pythagoras forward – bald with head scratching.

Welcome to wonderland.

## What is Math?

From the Good Math / Bad Math blog: Throughout elementary and high school, I got awful marks in math. I always assumed I was just stupid in that way, which is perfectly possible. I also hated my teacher, so that didn’t help. A friend of mine got his PhD in math from Harvard before he [more …]

Gauss said that if the truth of this formula is not immediately apparent to you, you will never be a top notch mathematician.

Even for those of us who do not immediately see the truth of this formula, we can nevertheless recognize the breathtaking beauty of an equation that relates so many fundamentals: zero, one, addition, multiplication, exponents, pi, i, and e. Truly staggering! And here’s a video explaining it…

## Why the area of a circle = area of triangle

I can’t explain it any better than this

In the preface of his book, Everything and More: A Compact History of Infinity, the late David Foster Wallace talks about the power of abstraction to suck you into an abyss of insanity. Mathematicians, who deal in pure abstraction, are more prone to insanity than poets and artists, he claims. Trying to define the mathematical [more …]

## Euler’s Identity Formula

What is it? It is the single most beautiful equation in all of mathematics. As this blogger points out: It relates five of the Most Important Numbers in the World (0, 1, e, i, and ) using three of the Most Important Operations in the World (addition, multiplication, and exponentiation) and nothing else. According to [more …]

## Understanding Multiplication

… Our understanding of multiplication changed over time: With integers (3 × 4), multiplication is repeated addition With real numbers (3.12 x sqrt(2)), multiplication is scaling With negative numbers (-2.3 * 4.3), multiplication is flipping and scaling With complex numbers (3 * 3i), multiplication is rotating and scaling We’re evolving towards a general notion of [more …]