One Two Three . . . Infinity: Facts and Speculations of Science (Dover Books on Mathematics)

by George Gamow

in fact there is on the average only about 1 atom per cubic meter of space.

Georg Cantor, who can be truly named the founder of the “arithmetics of infinity.”

In fact in the world of infinity a part may be equal to the whole!

Thus it is impossible to establish a one-to-one correspondence between the points on a line and the integer numbers, which means that the infinity of points on a line is larger, or stronger, than the infinity of all integer or fractional numbers.

the number of all points on a plane is equal to the number of all points on a line.

Thus according to the criterion of Cantor, the infinity of all the points within a square is equal to the infinity of all the points on a line.

Goldbach conjecture, proposed in 1742, which states that each even number can be represented as the sum of two primes.

each even number is the sum of not more than 300,000 primes.

Vinogradoff, who was able to reduce it to “the sum of four primes.”

that the percentage of primes within an interval from 1 to any larger number N, is approximately stated by the natural logarithm of N.

a triangle with three sides in the ratio of 3:4:5 must include one right angle. In fact the ancient Egyptians used such a tri-angle, now known as an Egyptian triangle, as a carpenter’s square.

we now have proofs that no solution of the Fermat equation is possible when n has any value smaller than 269.

the multiplication by i is geometrically equivalent to a counterclockwise rotation by a right angle.

V+F = E+2.

V=no of vertices F=no of faces E=no of edges

The relationship we have just found between the number of vertices, edges, and faces in a polyhedron was first noticed by the famous French mathematician of the seventeenth century, René Descartes, and its strict proof was demonstrated somewhat later by another mathematical genius, Leonard Euler, whose name it now carries.

“Absolute space, in its own nature, without relation to anything external, remains always similar and immovable,” and “Absolute, true, and mathematical time, of itself, and from its own nature, flows equally without relation to anything external.”

The idea of Michelson’s experiment is very simple and is based on a physical picture according to which light represents some kind of wave motion traveling through the so called “light-carrying ether,” a hypothetical substance uniformly filling up interstellar space as well as the intervals between the atoms in all material bodies.

all common mechanical properties of various substances result from their atomic structure, but this rule makes no sense whatsoever when applied to an absolutely continuous substance such as that which light ether is considered to be.

we do not feel the “ether wind,” since it is supposed to penetrate without any difficulty between the atoms forming our body, but we should be able to detect its presence by measuring the velocity of light in different directions in relation to our motion.

one often calls the lines representing the shortest distances on curved surfaces geodesical lines or geodesics, because this notion was first introduced in geodesy, that is, the science of measurements on the surface of the Earth.

the phenomenon of gravity is merely the effect of the curvature of the four-dimensional space-time world.

the sum of angles of a triangle drawn on the surface of a sphere is always larger than 180°. If you try to draw a triangle on the saddle surface you will find that the sum of its angles is always less than 180°.

any moving material particle occupies in any given moment a definite position in space, and that the consecutive positions of this particle form a continuous line known as the trajectory

the atoms of all different chemical elements have a common nature representing only various degrees of “concentration” of hydrogen atoms.

the nuclei of different atomic species are composed of various numbers of elementary hydrogen nuclei, which, because of their role in the structure of matter, were given the special name of “protons.”

some of the protons forming complex atomic nuclei have lost their original positive charges and are electrically neutral.

protons and neutrons should not be considered as two entirely different kinds of particles, but rather as two different electrical states of the same basic particle known now under the name of a “nucleon.”

it is known that protons can turn into neutrons by losing their positive charge, and neutrons can turn into protons by acquiring it.

If there are too many neutrons some of them will turn into protons by ejecting a negative electron, which leaves the nucleus.

If there are too many protons, some of them will turn into neutrons emitting a positive electron.

electrons emitted from the nucleus are known as beta (β)-particles.

beam of neutrinos would go without much difficulty through the thickness of several light-years of lead!

along with the electron the nucleus always emits a neutrino, which carries the balance of energy.

the nuclei of all elements heavier than silver are principally unstable, and, under the action of a sufficiently strong fillip from outside, would break up into two or more parts, with the liberation of a considerable amount of internal nuclear energy (Figure 65a). On the contrary, we should expect a spontaneous fusion process whenever two light nuclei with a combined atomic weight less than that of silver come close together (

This state of affairs, in which a certain process will not get under way without initial excitation, is generally known in science as the state of metastability,