1. Penrose continues his exploration of the intersection between math and reality as he looks at several different types of number and asks if they have any basis in the physical world. As he proceeds through different categories of number, he interacts with Euclid, the Greek mathematicians and these questions as the Greeks first posed them.
2. So what about natural numbers, the counting numbers? Obviously they have a basis in my fingers. I can count distinct items. (Zero was of course added by mathematicians from India since the time of Christ.The natural numbers plus zero constitute whole numbers.) In the end, Richard Dedekind (1831-1916) and Georg Cantor (1845-1918) created set theory to be able to define numbers abstractly and not by way of physical objects.
3. Brahmagupta also came up with the notion of negative numbers. The whole numbers and their negatives together form the set of integers. Penrose asks if negative numbers exist in the physical world. After all, I can define a certain direction as negative (in space or time) but that's just a matter of definition. If I treat my distance as a "scalar" (directionless quantity), then all directions are positive.
Penrose suggests that the discovery of the electron and other negative entities in physics (e.g., antimatter) suggest that there is in fact a physical correspondent to negative numbers.
4. The chapter starts with the Greek quest to see if all numbers can be expressed in a rational form. That is, can all numbers be expressed as a ratio of two numbers, one number divided by another. As it turns out, they cannot. The length of the hypotenuse of most right triangles usually cannot be expressed as a fraction.
One question he addresses briefly in the chapter is the scale of the infinite. The invention of the calculus theoretically divides numbers up into infinitely small quantities. Similarly, the scale of the universe as we know it is quite large. There is debate to which he will return later in the book about whether there is some ultimately small unit of length (the Planck length), smaller than which you cannot go. I personally am sympathetic to this idea.
5. So the Greeks conceded that there are irrational numbers. These are numbers that cannot be expressed in the form of a ratio. Together, the set of rational and irrational numbers constitute the set of real numbers. The chapter that follows deals with imaginary or complex numbers, which it turns out also have a basis in the physical world.
The irrational numbers are basically numbers that when expressed in decimal form (decimals were also invented in India and based on to Europe through Arab mathematicians) go on forever without a simple repetition. Intriguing to me is that the Greeks were able to express these irrational numbers in terms of "continued fractions." These are fractions like the picture above.
The Greeks were able to find a repeatable pattern in these continued fractions. "Quadratic irrationals" are irrational numbers that can be put into the form of a whole number plus the square root of a number. (Pi would be an example of an irrational number that does not fit this form.) In decimal form, the numbers right after the first number of the square root form a chiasm (a, b, c, ... c, b, a). Then the next decimal is always twice the first one. Then this pattern repeats itself forever.
Here is the square root of 14: 3. 121 6 121 6 121 6 121 6...