Riemann prose, part 2


The Greeks were really good at solving problems using pictures. These were geometric problems, and involved determining the relationships between the sides and angles of a triangle, dividing angles in half, and squaring the circle. One result of this process was the realization that you could have fractions.

A fraction represents some unit object that is evenly subdivided. Dividing something doesn’t change the fundamental fact that the subunits still all add up to the unit object. That is, if you have a cake (mmm, cake, gurgle) and you cut it into 4 pieces of equal size, you have 4 pieces that, if pushed back together again, are still the same size as the original cake.

Divide the cake in half, you get 2 halves (2/2).
Divide it in thirds, 3 thirds (3/3).
Divide it in fourths, 4 fourths (4/4).

In all cases, an integer divided by itself equals 1.

If you take only one subunit (1/2, 1/3, 1/4), you get the building block step size for recreating that cake.

It’s easy to see why people want to believe that this is where you stop with the rational numbers.

Interestingly, if you look closer at these fractions, they are telling you to divide 1 by the integer below it. It’s right there, in the number – “divide me! Do it! Do it! Do it!”

Unfortunately, the results aren’t always very clean, and this brings up the issue of decimal values, which has an unclear history. The wiki entry implies that fractional decimals were used as part of the decimal system in India, but that it was marked by putting a line over the units digit of the number. Arab mathematicians eventually used a comma (“,”) as a decimal separator, while the period (“.”) was adopted in France.

So, 1/2 is equal to 0.5.

This fraction is a non-repeating decimal number. People like these, because they’re easy to understand.

1/3 = 0.33333

This is a repeating decimal, and is not beloved quite so much because, if you look at it really, really closely, IT NEVER ENDS! But still, it can be represented as a fraction with integers in the numerator and denominator, so it’s still rational.

The numbers that really drove, and can still drive, people crazy are the ones that can’t be formed as fractions. Such as 1.20001213 + random non-repeating digits. There are a lot of these numbers, which is very irrational.

(To be continued.)

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