Archive for the ‘Charles Seife’ Category

The Golden Ratio

February 23, 2008

Divide a line in two, such that the ratio of the small part to the large part is equal to the ratio of the large part to the whole line.

For the sake of simplicity, let’s say that the small part is 1 foot long. If the small part is 1 foot long, and the large part is x feet long, then the length of the whole line is obviously 1 + x feet long. The ratio of the small part to the large part is 1/x while the ratio of the large part to the whole thing is x/(1+x)

The Golden Ratio is where the ratio of the small to the large is equal to the ratio of the large to the whole, so we set the two ratios equal to each other: x/(1+x) = 1/x

Multiply both sides by x to get: x2/(1+x) = 1

Then multiply both sides by (1+x) to get: x2 = 1 + x

Subtract 1 + x from both sides to get: x2x – 1 = 0

This is a quadratic equation. The value of x which meets this equation is: (1 + √5)/2

This is about 1.618

Thus, take a line of length 2.618 feet and divide it up into two parts, the short part is 1 foot in length, the second part is 1.618 feet. The two parts have a Golden Ratio.

Zero, The Biography of a Dangerous Idea by Charles Seife

Cardinality and Ordinality: life was so easy until zero came along

December 6, 2007

When we are dealing with the counting numbers — 1, 2, 3, and so on — it is easy to rank them in order.   One is the first counting number, two is the second counting number, and three is the third.  We don’t have to worry about mixing up the value of a number — its cardinality — with the order in which it arrives — its ordinality — since they are essentially the same thing.  For years, this was the state of affairs, and everybody was happy.  But as zero came into the fold, the neat relationship between a number’s cardinality and its ordinality was ruined.  The numbers went 0, 1, 2, 3: zero came first, one was second in line, and two was in third place (i.e. 0 has a cardinal value of 0 and an ordinal value of 1; 1 has a cardinal value of 1 and an ordinal value of 2; 2 has a cardinal value of 2 and an ordinal value of 3).  No longer were cardinality and ordinality interchangeable.

Zero, the Biography of a Dangerous Idea by Charles Seife

Underneath every revolution lay a zero — and an infinity

September 12, 2007

“Zero is powerful because it is infinity’s twin.  They are equal and opposite, yin and yang.  They are equally paradoxical and troubling.  The biggest questions in science and religion are about nothingness and eternity; the void and the infinite, zero and infinity.  The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion.  Underneath every revolution lay a zero — and an infinity.”

Zero, The Biography of a Dangerous Idea by Charles Seife