The range A < x < B for a statistic x is important. 0 Number with no name for most of human history 1,2,3 Numbers worth names and honorific titles. 6 Arrangements of three objects Small numbers rare diseases, collectors items. 10-99 Tens. Whites facing death penalty in USA. 2^6 64 Number of I-Ching Hexagrams 100-999 Hundreds. Black people facing death penalty in USA. 1000-9999 Thousands. Bosnian men massacred at Srebenica. 10000 Moon (Thai). Brique (French). 100000 Lak (Urdu, Arabic). Millions people, dollars, animals, diseases. Jews killed by Nazis. Prison population of USA. 504475641 Surface of earth. km^2 Billions hyperinflation, micro-organisms, science figures. Telephone numbers 2^32 4294967296. Number of Internet IP addresses. Arrangements of 13 objects. Size of PC Hard disk in bytes. Six Billion World population in October, 1999. 18 Digits 278237540975596768. Surface of Dyson sphere. km^2 20-21 Digits c^2 in metric. Bang for your Gram. Einstein. Arrangements of 21 objects. 68 digits Shuffles of a standard 52 card pack. 80 digits Protons in Universe. 3/2*136*2^256. Eddington. > 80 Digits Probabilities, possibilities. 10^10 digits Possibilities for human genome. Alef-zero Cantor's name for countable infinity. Here c^2 refers to the square of the speed of light. Excercise: Try learning the counting numbers from 0 to 10 in a language which you do not know yet. Generally you cannot tell how the number is going to be stored in a computer. Privileged numbers such as Pi or friend's phone numbers are often remembered by people. Certain other numbers are better not stored but given generator functions. You don't care whether a right angle is ninety degrees or Pi/2 radians. Just keep three, four, five and a triangle. Euclid described prime numbers, which are numbers with no factors apart from one and themselves, and irrational numbers which are numbers which cannot be expressed as fractions. The English mathematian Wallis is said to have invented a symbol like a figure eight laying on its side to represent infinity, a number bigger than any ordinary counting number. Many numbers used to represent reality on computers are the result of sampling where a range of values is split into segments and a name or value is assigned to each segment. Colour displays and images are the most common example of this. Fan Dee, Fan Lotteree Thai saying. Numbers games are another aspect of numbers. 'Running Numbers' has been dramatized by Hollywood films and folk-rock ballads. These emphasise a sort of struggle against 'state-control' of numbers. Buddhist monks in Thailand risk getting killed if they say too much about the next lottery number. There are cultures where almost everyone seems to bet on numbers. Next week's imaginary number becomes yesterday's real number. Just like an alcohol hangover. For many the lottery number is like the dreams of opium. One could say that Marx got it wrong and in place of the statement:- Religion is the Opium of the Masses Imaginary Number is the Opium of the Masses Q: What's that special number used in the Theory of Everything ? A: You mean numbers like 'forty two' which occur in the Hitch Hiker's Guide to the Universe series. There are many fundamental equations of physics, starting with Newton's equation for gravitation. All of these have values expressed in metric units although some organisations in the Anglo-Saxon world try to cling on to imperial units of weights and measures. Physical constants are usually given in 'floating point notation': X= M E EXP or M E -EXP where 'M' is a decimimal number, normally between 1 and 10 and the 'E' stands for a power of 10, so that 27.5 could be written 2.75 E1 and 1/8 can be written 1.25 E-1. A selection of physical constants is given. The first group is taken from Appendix 2 of 'The Chemical Bond' by Linus Pauling. Velocity of light c 2.99793 E10 cm/sec Electronic charge e 4.8029 E-10 statcoulomb Mass of electron m 9.1083 E-28 gram Planck's constant h 6.6252 E-27 erg sec Avagadro's number N 0.60229 E24 /mole Energy of 1 ev 1.60206 E-12 erg Wavelength of 1 ev quantum 12397.67 Angstrom Mass of proton M(p) 1.67239 E-24 gram Mass of neutron M(n) 1.67470 E-24 gram Boltzmann's constant k 1.3805 E-16 erg/deg Gas constant R 1.9872 cal/(deg * mole) Energy of 1 gram mass 5.6100 E32 ev Q: What is Algebra ? A: Algebra is a way of anticipating problems in calculations by letting symbols stand for unknown numbers. It is said to originate with the Arabs. There are many Arabic words similar to Algebra. These often mean force or coercion. The power form is also a name for God. 'Modern Algebra' was invented in the 1800s, and one of the main advances was the interchange between numbers and functions or operators. The French mathematicians Fourier and Galois pioneered this line of thought. Algebra as tought in schools came from much earlier times. Italy saw the earliest European developments when Cardano and others came up with a method for solving cubic and quartic equations. Formal algebra came with the attempts to build mathematics around a system of axioms in the style of Euclid. David Hilbert and Bertrand Russell made these attempts fashionable in the early 1900s, but Kurt Godel showed that this approach had limitations during the 1930s. For most computer science applications the formal approach is quite good enough. SET AXIOMS In the 1960s and 1970s the English started to teach 'set' theory in schools. This was often done in an ideological vacuum. The kids were not always confronted by such basic questions as race and class in this context. Nor were they necessarily drilled in hard questions on set theory which had previously been in the syllabus: calculations about permutations and combinations. The success of the British National Lottery shows just how far things have fallen. Sets are like packs of playing cards, boxes of chess pieces, selections of lottery numbers, or even populations loving, living and dying. Early mathematics teaching concentrated on the computational aspect of set theory, nowadays called Combinatorial Analysis. Horse races are an excellent example of sets, as are dictionaries. The maintenance of knowledge about particular sets is of great economic importance. A goldrush mentality fuels speculation on world stockmarkets as search engine companies go public. The ignorance of next week's Lottery Numbers is also important for profits of the operator. The order of winners in a horse race or the subsets of football matches with draws or high scores provides the paydirt for a global money extraction industry. A set of numbers can be written in brackets: {1 3 5 7 9} is the set of odd numbers less than ten. A set containing no elements at all is called the empty set and it is written {}. The set consisting of {0 1} is special, because 0 and 1 can be made to correspond to values 'true' and 'false'. Sets are connected with logic by using the phrase 'a is a member of A', or 'b belongs to class B'. Such statements are always either true or false in classical set theory. Fuzzy logic is a new form of set theory allowing for intermediate values between true and false. Modern Algebra uses certain conventions such as upper case for sets, and lower case for members. Many statements of logic can be translated to theorems in the algebra of sets. The most common are De Morgan's Laws. A union B or 'A u B' is defined as elements x which are members of A or B. A ^ B is defined as elements x which are in both A and B. A-B is the set of x where x is in A but not B. The number of elements in a set X may be written as c(X). For any set X and subsets A and B the following identities hold. Here '<=' stands for less than or equal to. If the intersection of two sets is empty then they are said to be disjoint. c(A^B) <= c(A) <= c(A u B) <= c(X) c(A^B) <= c(B) <= c(A u B) <= c(X) c(A u B) <= c(A) + c(B) c(A^B) <= Min(c(A), c(B)) Here Min means minimum. c(A u B) = c(A) + c(B) - c(A^B) Inclusion and Exclusion (A u B)^C = (A^C) u (B^C) (A^B) u C = (A u C)^(B u C) X - A u B = (X-A) ^ (X-B) De Morgan's Laws X - A ^ B = (X-A) u (X-B) If A is a subset of a set X then it is possible to define a post-fix function called % with A%=100.0*c(A)/c(X) where '/' stands for divide. The '%' function is merely the ordinary percent calculation. Exercise: Q: 60% of housholds have a car and 95% have a TV. How many have neither ? A: At most 5%. Excercise: Q: 47.5% voted Democrat and 46.7% voted Republican so how many voted for neither ? A: At least 18.8%. You can't vote twice, but many did not vote at all. Divide the population into voters and non-voters. Let U be the proportion of non-voters. When U is close to 1 then a dictator selected by popular acclaim could be seen as the most democratic option. N=U+(1-U)*0.188 None R=(1-U)*0.467 Republican D=(1-U)*0.475 Democrat MILLION DOLLAR REWARD Excercise: Goldbach's conjecture. Let P={3 5 7 11 13 17 ...} be the set of odd primes and let E={6 8 10 ..} be the set of even numbers greater than four. Then is E=P+P? Here P+P denotes the set formed by the sums of all pairs p+q with p,q in P. of P+P. It is already known that E=P+P*P*..P where P*P*P..P is some product of prime numbers. It is also known that every sufficiently large odd number is the sum of three odd primes. Vinogradov [1] proved this in 1937. It is also known, by computer search that any even number n is the sum of two primes for n < 400,000,000,000,000 (four hundred trillion in the American style). Goldbach outlined this conjecture in a letter to Euler in 1742. Now the publisher FABER is offering a million dollar reward for anyone who submits a proof of the conjecture, or a counter example before March 15, 2002 [2]. Another interesting problem arises with functions which generate sequences of primes. For low limits the sequence x=199+210*j gives 10 consecutive primes: 199 409 619 829 1039 1249 1459 1669 1879 2089 Euler knew that the function f(x)=41+x+x^2 gives prime values for x=0 to 39. This quadratic sequence is far better than any linear function. The cubic function 29+117t-20t^2+t^3 gives primes for its first 19 values. 181+205t-28t^2+t^3 gives 20 consecutive primes, with repetitions. 12983 -2440t+107t^2+t^3 t=0 to 16 643 -231t +26t^2+t^3 t=0 to 13 5099 -143t +10t^2+t^3 t=0 to 13 14771 +480t -53t^2+t^3 t=0 to 13 1063 -87t -10t^2+t^3 t=0 to 12 6869 +376t -53t^2+t^3 t=0 to 11 MORE SET THEORY Mathematicians invented sets long before computer lanuages evolved. Sets are easy to copy. As a tool of thought, they are unprotected by legalese or copyright. They are public domain stuff. Mathematical terminology alone seems a sufficient deterrent. Given two sets A and B it is possible imagine a table of pairs (a,b). The sets may be people and cars, or stock market shares and prices. The set of all possible products is written A x B and a typical element of the set A x B is the pair (a,b) with a in A and b in B. The set A x B is called the Cartesian Product. The number of elements of A x B is c(A) times c(B). If A and B are different horse races, then the prediction of a winner from each race is called a double, and is simply a member of the cartesian product. It is also possible to define cartesian products on more than two sets. The product formula can be generalised. In particular the size of the cartesian product of k copies of A is c(A) to the power k. For any given set X it is possible to define a 'relation' as a subset R of the cartesian product A x A. An equivalence relation satisfies three simple rules: Reflexive If x in X then (x,x) in R. Symmetric If (x,y) is in R then so is (y,x). Transitive If (x,y) and (y,z) are in R then so is (x,z). Given an equivalence relation then it is possible any x in X to define a unique subset of X containg x: class(x)={y | (x,y) in R}. This is called the equivalence class of x with respect to R. Any two equivalence classes are disjoint, for if a in class(x)^class(y) then (a,x) and (a,y) are both in R, so (x,a) in R by symmetry, and therefore (x,y) in R and in fact class(x)=class(y). Since the reflexive rule implies that class(x) contains at least x as an element it follows that every element of x is in at least one equivalence class. The set of equivalence classes is often written X/R. Since the classes are disjoint, and every element is in just one equivalence class we can write X = union {C(x) with x in X/R}. Example: A person cannot exist in a modern state without being slotted into a cartesian product of some pair of sets. If a person is treated for a disease the bureaucracy maps that person into the cartesian product of (people x diseases). A relationship can be defined defined on the set of people by saying that (a,b) in R if a and b suffer from a given disease. Not only AIDS, but also multiple injuries will appear in clusters. Needless to say modern computers are incapable of doing justice to these types of databases. Analysis of these relationships is crucial in discussions on public financing. ELITE SCHOOLS Example: Consider education, with the relation R being determined by school. Two people a,b are in this relation if there is a finite set of people a=x[0],x[1],x[2], .. x[n]=b where x[i],x[i+1] both went to the same school. If a person x did not go to school at all, then let (x,x) in R. Then the population can be broken into a disjoint union of equivalence classes. When inter-school transfers are common there may be just a single class. When the education of women is forbidden then there are at least two classes. In a global world these classes are unlikely to correspond to geography. A classless society could be seen as a world where everyone participated in the same distance learning scheme. Attempts to make everyone participate in the same school _system_ have been tried many times. When schools have been dominated by competing religions there have occassionally been problems. Northern Ireland is seen by many as a classic example. If teaching becomes a less attractive profession, and the cost of internet connectivity falls drastically then the uneducated could become replaced by the 'media-educated'. Whoever dominates the media effictively controls education for large numbers of people. When a set is finite the class equation can be written: c(X) = Sum c(Xi) where X1,X2,.. are the equivalence classes. Besides relations the subsets of A x B include the graphs of functions. A function from A to B written f:A->B is simply a subset G of A x B such that for each a in A there exists a single element b in B with (a,b) in G. For shares and prices the most common subset of the cartesian product is the price list, but other functions are possible: the price in two weeks time, for example, or the previous years low. DNA AND PROTEINS

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