- Go to college.
- Study math.
- There is more to math than calculus.
- There are many infinities.
- Arithmetic is deep.
- Graph theory is universal.
- Go to office hours.
- Find a quiet place to think.
- Read everything.
- Be kind.

# Category: Miscellaneous

# On the nature of math — abstraction

Abstraction is such a central part of modern mathematics that one forgets that it was not until Frechet’s 1906 thesis that sets of points with no a priori underlying structure (not assumed points in or functions on ) are considered and given a structure a posteriori (Frechet first defined abstract metric spaces). And after its success in analysis, abstraction took over significant parts of algebra, geometry, topology and logic.

Cheers,

Nalin Pithwa.

# Beginning steps to intuitive physics

Reference: Newtonian Mechanics by A. P. French, The M.I.T. Introductory Physics Series

In the beginning was Mechanics. — Max Von Laue, History of Physics (1950)

I offer this work as the mathematical principles of philosophy, for the whole burden of philosophy seems to consist in this — from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena. —- Newton, Preface to the Principia (1686).

- What is the order of magnitude of the number of times that the earth has rotated on its axis since the solar system was formed?
- During the average lifetime of a human being, how many heart beats are there? How many breaths?
- Make reasoned estimates of (a) the total number of ancestors you would have (ignoring inbreeding) since the beginning of the human race, and (b) the number of hairs on your head.
- Assume the present world population to be .(a) How many square kilometers of land are there per person? How many feet long is the side of a square of that area? (b) If one assumes that the population has been doubling every 50 years, through out the existence of the human race, when did Adam and Eve start it all? If the doubling every 50 years were to continue, how long will it be before people were standing shoulder to shoulder over all the land area of the world?
- Estimate the order of magnitude of the mass if (a) a speck of dust (b) a grain of salt (or sugar, or sand) (c) a mouse (d) an elephant (e) the water corresponding to 1 inch of waterfall over 1 square mile, (f) a small hill, 500 ft high; and (g) Mount Everest.
- Estimate the order of magnitude of the number of atoms in (a) a pin’s head (b) a human being (c) the earth’s atmosphere and (d) the whole earth
- Estimate the fraction of the total mass of earth that is now in the form of living things.
- Estimate (a) the total volume of ocean water on the earth and (b) the total mass of salt in all the oceans.
- It is estimated that there are about protons in the known universe. If all these were lumped into a sphere so that they were just touching, what would be the radius of the sphere? Ignore the spaces left when spherical objects are packed and take the radius of a proton to be about m.
- The sun is losing mass (in the form of radiant energy) at the rate of about 4 million tons per second. What fraction of its mass has it lost during the lifetime of the solar system?
- Estimate the time in minutes that it would take for a theatre audience of about 1000 people to use up 10 % of the available oxygen if the building were scaled. The average adult absorbs about one sixth of the oxygen that he or she inhales at each breath.
- Solar energy falls on the earth at the rate of about 2 cal/cm^{2}/minute. Estimate the total amount of power, in megawatts or horsepower, represented by the solar energy falling on an area of 100 square miles — about the area of a good-sized city. How would this compare with the total power requirements of such a city? (1 cal = 4.2Joules; 1W = 1 joule/sec; 1 HP is 746 W).
- Starting from an estimate of the total mileage that an automobile tire will give before wearing out, estimate what thickness of rubber is worn off during one revolution of the wheel. Consider the possible physical significance of the result.
- An inexpensive wristwatch is found to lose 2 minutes per day. (a) what is the fractional deviation from the correct rate? (b) By how much could the length of the ruler (nominally 1 feet long) differ from exactly 12 inches and still be fractionally as accurate as the watch?
- The astronomer Tycho Brahe made observations on the angular positions of stars and planets by using a quadrant, with one peephole at its centre of curvature and peephole mounted on the arc. One such quadrant had a radius of about 2 inches, and Tycho’s measurements could usually be treated to 1 minute of arc. What diameter of peepholes would have been needed for him to attain this accuracy?
- Jupiter has a mass about 300 times that of the earth, but its mean density is only about one-fifth that of the earth. (a) What radius would a planet of Jupiter’s mass and earth’s density would have? (b) What radius would a planet of earth’s mass and Jupiter’s density have?
- Identical spheres of material are tightly packed in a given volume of space. (a) Consider why one does not need to know the radius of the spheres, but only the density of the material, in order to calculate the total mass contained in the volume, provided that the linear dimensions of the volume are large compared to the radius of the individual spheres. (b) Consider the possibility of packing more material if two sizes of spheres may be chosen and used. (c) Show that the total surface of the spheres of part (a) does depend on the radius of the spheres (an important consideration in the design of such things as filters, which absorb in proportion to the total exposed surface area within a given volume)
- Calculate the ratio of surface area to volume for (a) a sphere of radius r (b) a cube of edge a, and (c) a right circular cylinder of diameter and height both equal to d. For a given value of the volume, which of these shapes has the greatest surface area? The least surface area ?
- How many seconds of arc does the diameter of the earth subtend at the sun? At what distance from an observer should a football be placed to subtend an equal angle?
- From the time the lower limb of the sun touches the horizon it takes approximately 2 min for the sun to disappear beneath the horizon. (a) Approximately what angle (expressed both in degrees and in radians) does the diameter of the sun subtend at the earth? (b) At what distance from your eye does a coin of about 0.75 inches diameter (e.g. a dime or a nickel) just block out the disk of the sun?
- What solid angle in steridians does the sun subtend at the earth?
- How many inches per mile does a terrestrial great circle (e.g. a meridian of longitude) deviate from a straight line?
- A crude measure of the roughness of a nearly spherical surface could be defined by , where is the height or depth of local irregularities. Estimate this ratio for an orange, a ping-pong ball, and the earth.
- What is the probability (expressed as 1 chance in ) that a good-sized meteorite falling to the earth would strike a man-made structure? A human?
- Two students want to measure the speed of sound by the following procedure. One of them, positioned some distance away from the other, sets off a firecracker. The second student starts a stopwatch when he sees the flash and stops it when he hears the bang. The speed of sound in air is roughly 300m per second, and the students must admit the possibility of an error (of undetermined sign) of perhaps 0.3 seconds inthe elapsed time recorded. If they wish to keep the error in the measured speed of sound to within 5% what is the minimum distance over which they can perform the experiment?
- A right triangle has sides of length 5m, 1 m adjoining the right angle. Calculate the length of the hypotenuse from the binomial expansion to two terms only, and estimate the fractional error in this approximate result.
- The radius of a sphere is measured with an uncertainty of 1 percent. What is the percentage uncertainty in the volume?
- Construct a piece of semilogarithmic graph paper by using the graduations on your slide rule to mark off the ordinates and a normal ruler to mark off the abscissa. On this piece of paper draw a graph of the function .
- The subjective sensations of loudness or brightness have been judged to be approximately proportional to the logarithm of the intensity, so that equal multiples of intensity are associated with equal arithmetic increases in sensation (For example, intensities proportional to 2, 4, 8, and 16 would correspond to equal increases in sensation). In acoustics, this has led to the measurement of sound intensities in decibels. Taking as a reference value the intensity of the faintest audible sound, the decibel level of a sound of intensity I is defined by the equation: . (a) An intolerable noise level is represented by about 120 dB. By what factor does the intensity of such a sound exceed the threshold intensity ? (b) A simple logarithmic scale is used to describe the relative brightness of stars (as seen from the earth) in terms of magnitudes. Stars differing by “one magnitude” have a ratio of apparent brightness equal to about 2.5 Thus, a “first magnitude” (very bright) star is 2.5 times brighter than a second magnitude star times brighter than a third-magnitude star, and so on. (These differences are due to largely differences of distance). The faintest stars detectable with the 200 inches Palomar telescope are of about the twenty-fourth magnitude. By what factor is the amount of light reaching us from such a star less than we receive from a first magnitude star?
- The universe appears to be undergoing a general expansion in which the galaxies are receding from us at speeds proportional to their distances. This is described by Hubble’s Law, where the constant corresponds to v becoming equal to speed of light, meters per second at metres. This would imply that the mean mass per unit volume in the universe is decreasing with time. (a) Suppose that the universe is represented by a sphere of volume V at any instant. Show that the fractional incfease of volume per unit time is given by (b) Calculate the fractional decrease of mean density per second and per century.
- The table lists the mean orbit radii of successive planets expressed in terms of the earth’s orbit radius. The planets are numbered in order (n)

(a) Make a graph in which is ordinate and the number n is abscissa. (Or, alternatively, plot values of against n on semilogarithmic paper). On this same graph, replot the points for Jupiter, Saturn, and Uranus at values of n increased by unity (that is, at n=6, 7, and 8). The points representing the seven planets can then be reasonably well fitted by a straight line.

(b) If n=5 in the revised plot is taken to represent the asteroid belt between the orbits of Mars and Jupiter, what value of would your graph imply for this ? Compare with the actual mean radius of the asteroid belt.

(c) If n=9 is taken to suggest an orbit radius for the next planet (Neptune) beyond Uranus, what value of would your graph imply? Compare with the observed value.

(d) Consider whether in the light of (b) and (c) your graph can be regarded as the expression of a physical law with predictive value. (As a matter of history, it was so used.)

Approximations:

Binomial Theorem:

for ,

For example,

For example,

For ,

Other expansions:

For radians, tends to

for radians, tends to 1

For

For ,

Cheers, cheers, cheers,

Nalin Pithwa

# Is Math really abstract? I N Herstein answers…

Reference: Chapter 1: Abstract Algebra Third Edition, I. N. Herstein, Prentice Hall International Edition:

For many readers/students of pure mathematics, such a book will be their first contact with abstract mathematics. The subject to be discussed is usually called “abstract algebra,” but the difficulties that the reader may encounter are not so much due to the “algebra” part as they are to the “abstract” part.

On seeing some area of abstract mathematics for the first time,be it in analysis, topology, or what not, there seems to be a common reaction for the novice. This can best be described by a feeling of being adrift, of not having something solid to hang on to. This is not too surprising, for while many of the ideas are fundamentally quite simple, they are subtle and seem to elude one’s grasp the first time around. One way to mitigate this feeling of limbo, or asking oneself “What is the point of all this?” is to take the concept at hand and see what it says in particular concrete cases. In other words, the best road to good understanding of the notions introduced is to look at examples. This is true in all of mathematics.

Can one, with a few strokes, quickly describe the essence, purpose, and background for abstract algebra, for example?

We start with some collection of objects S and endow this collection with an algebraic structure by assuming that we can combine, in one or several ways (usually two) elements of this set S to obtain, once more, elements of this set S. These ways of combining elements of S we call operations on S. Then we try to condition or regulate the nature of S by imposing rules on how these operations behave on S. These rules are usually called axioms defining the particular structure on S. These axioms are for us to define, but the choice made comes, historically in mathematics from noticing that there are many concrete mathematical systems that satisfy these rules or axioms. In algebra, we study algebraic objects or structures called groups, rings, fields.

Of course, one could try many sets of axioms to define new structures. What would we require of such a structure? Certainly we would want that the axioms be consistent, that is, that we should not be led to some nonsensical contradiction computing within the framework of the allowable things the axioms permit us to do. But that is not enough. We can easily set up such algebraic structures by imposing a set of rules on a set S that lead to a pathological or weird system. Furthermore, there may be very few examples of something obeying the rules we have laid down.

Time has shown that certain structures defined by “axioms” play an important role in mathematics (and other areas as well) and that certain others are of no interest. The ones we mentioned earlier, namely, groups, rings, fields, and vector spaces have stood the test of time.

A word about the use of “axioms.” In everyday language, “an axiom means a self-evident truth”. But we are not using every day language; we are dealing with mathematics. An axiom is not a universal truth — but one of several rules spelling out a given mathematical structure. The axiom is true in the system we are studying because we forced it to be true by “force” or “our choice” or “hypothesis”. It is a licence, in that particular structure to do certain things.

We return to something we said earlier about the reaction that many students have on their first encounter with this kind of algebra, namely, a lack of feeling that the material is something they can get their teeth into. Do not be discouraged if the initial exposure leaves you in a bit of a fog.Stick with it, try to understand what a given concept says and most importantly, look at particular, concrete examples of the concept under discussion.

Follow the same approach in linear algebra, analysis and topology.

Cheers, cheers, cheers,

Nalin Pithwa