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rediscover mathematics from 0 and 1


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i believe everything in mathematics can be rediscovered from the concepts of beauty, identity, division and symmetry

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rediscover mathematics from 0 and 1

  1. 1. The Series Editor Rediscover Mathematics From 0 And 1 Series 1. CONFIDENCE, INTEREST ; BEAUTY AND SYMMETRY – KEY TO UNDERSTANDING A.Confidence : Pin-pointing ‘Difficulty’ And Enclosing It With An Appropriate Symbol. L et us join the worldwide movement for making Mathematics simple and interesting and accessible to average student. In fact there is nothing genius-like in study of Mathematics, rather there is much idiot-like in it in the very sense as a computer is an intelligent idiot. What is required for study of Mathematics is not great IQ, but a little patience, that too, only until a little habit of study is developed. Then the habit would take care of further study. The aim of the teaching is to make “difficult concepts made easy” although many times it happens otherwise, i.e., “easy things made difficult”. ( It is bound to be so, for, there is no shortcut to labour. Either you think hard or work hard, or maybe, do something of both.). Take any example of any commonplace problem in Arithmetic ; and remember the “Village School Master” with his “ Little Head Could Contain All That He Knew!”. However, what your teacher taught you is, only to naming a symbol “x” for the unknown quantity, and proceed along the lines of the given problem and until you arrive at something like an “equation”. Then only remember that equal quantities added to, subtracted from, multiplied, or divided by equal quantities give rise to equal results only; and you proceeded mechanically with transposition, cross multiplication etc., etc., until automatically the “symbol” revealed its “value”. Then it has no other job to do, and poor that! That had to be thanklessly discarded thereafter. This is the magic of a name or symbol. After all what is there in a name? Perhaps anybody’s name represents the person in entirety as far as others understand him. So the symbol is power. Entire Mathematics is the discipline of symbols and their languages. Once the symbol is described, it carries its whole Algebra, and Grammar invisibly and silently bundled with it. Take, for example, Trigonometric ratios, functions, inverse circular functions, logarithms, determinants, and matrices, and just anything read in Mathematics, you name it. The examples will present themselves as we proceed. The spirit behind writing this book is not to teach Mathematics, nor to teach rigor, but only to show that confidence can be taught . We construct an entire language with a few symbols and they develop into topics, complete, compact and connected. In fact, Mathematics is a collection of languages of symbols and languages. To teach mathematics however, is not as easy as that. It appears so evident and natural to the teacher that he or she is baffled when the student does not understand a trifle . One should imagine the acceptable lowest level of understanding keeping oneself in the position of the student; then only it could be known where it pricks. Sometimes we encounter strange real issues, stranger that fiction although small or trivial they might be, and just circumvent it without letting the student notice it. To be a good student is one thing; you get the best out of your teachers, both good and bad as well. To be a good teacher is a challenge; one has to face the good students . Reality is more surprising than imagination ! A teacher has to face it. To be a good writer is still a bigger challenge. One cannot sacrifice rigor while putting things on pen and paper. One writer only knows, where she or he got around the jerks and stops while he or she was a teacher ( or failed to do so) . Another point is , what we are to teach is not exactly Mathematics, broadly not even rigor, but the exact role is, to teach “confidence”; yes, confidence is to be taught, taught to the extent it could be. Our great Headmaster told us once, if you find a problem does not match with its answer, then revise all the problems and examples preceding it, and all the chapters preceding it .If the mismatch still persists, then either the question is wrong , or the answer is wrong. In other words, a right question is made only to be answered by the reader. So one must be able to answer it with a minimum application of mind. This teaching of confidence, pushed many of us through our academic pursuits. A dog crouched under a truck, suddenly gets out as soon as the engine roars; for, it knows what Newton’s laws are. But it does not know that it already knew Newton’s laws. The same story is relevant for 1
  2. 2. The Series Editor Rediscover Mathematics From 0 And 1 Series all of us, though to different extents. The more we learn , the more we understand how much we knew . It builds confidence. The more we learn, we come to know, how much more is there to be learnt ! This ignites inquisitiveness. This makes us humble. Probably this is the reason the writers of The Vedas never left any trace of their identity and they assigned all the authorship to God.The movement now is, to know less and less about more and more areas ; so that the students could be quickly pushed to the frontiers of knowledge instead of being experts in elements. But in that case, teaching of confidence is all the more important than teaching the topics themselves. The great Albert Einstein perhaps said, if only you could know where exactly the problem lies, the solution lies there itself. In his times, half of the world believed there was the all pervading invisible ether and the other half believed there was no ether. He put an end to the controversy by telling everybody to raise only the questions which probably have answers, and discard all others. Did he take the cue from a small child he taught, who told, “there can be another way besides the Earth going around the Sun and the Sun going around the Earth ? ” Let us verify Einstein’s ‘get out of difficulty just by reaching it’ method by one example that follows; to solve a quadratic equation. We would follow this method throughout the book series by making small projects instead of problems as they are generally understood. Project lets one feel like rediscovering rather than being taught. Thus the mind gladly and quickly assigns a memory address for storing the learning and recalling is easy. Example 1 ; Quadratic Equations :an example in ‘difficulty identification’: A simple project to solve a quadratic equations met in high school. First , take the quadratic equation, ax2 + bx + c = 0 with rational coefficients. How easy it would have been, only if the ‘bx’ term had not been there! So the difficulty seems to be the linear term and let us try to remove it. Now, put bx or x, inside a symbol t, such as t = x - h or x = t + h, proceed mechanically working with the equation so as to get rid of the linear term bx , and discard the t and h when x reveals its value and those extraneous symbols have nothing to do any more. Our equation becomes, a(t +h)2 +b(t +h) + c = 0  at2+ (2ah+b)t + ( ah2+bh +c)=0 If we choose, h = -b/2a, to make the second term 0, the equation reduces to , at2+ 0+(b2/4a-b2/2a+c)=0 ,(an eqn. in t having no first degree term in t) b 2  4ac b 2  4ac b or, at2 = b2/4a – c  t =  x =  2a 2a 2a Now the poor things, h and t have been thrown out mercilessly ! The roots of quadratic equation t = f(x) = ax2 + bx + c = 0, - b  b 2  4ac - b  b 2  4ac are   and   . 2a 2a We know in high school that this solution has been got by completing square method, either by dividing by a through out or multiplying by 4a throughout. The ‘difficulty’ identified in this way is to make ax2 + bx , a part of a complete square, a(x2 + bx/a+ b2/4a2) and then subtracting the term b2/4a , which was added extra. The roots are real if discriminant of the equation , b  4ac is positive, i.e., b2 – 4ac  0 2 and imaginary otherwise( by imaginary we mean, the roots of a negative number, if at all we agree for its existence). They are rational if b2 – 4ac is a perfect square and irrational otherwise . The roots are equal if the determinant is 0. The sum of the roots is - b /a, and their product is c / a (verify actually taking the sum and product and by calculating). The expression under the square root is called discriminant because it determines the very nature of roots of the quadratic. If the roots are irrational or 2
  3. 3. The Series Editor Rediscover Mathematics From 0 And 1 Series imaginary, one is the conjugate of the other, i.e, if one is p+ √q or p + iq, the other is p - √q or p – iq; ( This is evident from the structure of the roots , they are already in p+ √q and p - √q form. To prove it, assume m+ √n is another root, so their product (p+ √q)(m+ √n) = c/a, a rational number .This is possible only when m+ √n is a multiple of conjugate of p+ √q, say k(p - √q). Now, the sum of the roots is p + √q + k(p - √q) = p(1 + k) + (1 - k )q = - b/a. Solving this for q we see that either 1 – k = 0 or q = {- b/a – p(1 + k)}/(1 – k), a rational number, which is a contradiction. So 1 – k must be 0 or k = 1 or the other root must be p - √q . ). In another way, we observe that the sum of the roots is – b/a and the product of the roots is c/a; both rational numbers by assumption. Then , if the roots themselves do not occur in conjugate pairs, how can their sum and product be rational ? No matter if the reader does not understand imaginary numbers . We would return to the subjects, quadratic equation and imaginary numbers in detail in later chapters at appropriate places and this is merely one example how to apply the “Difficulty Pin-pointing Method”. Actually this is a method how second term in a polynomial equation could be removed. (See further the chapters on quadratic equations, theory of equations, Cardan’s solution of cubic equation etc.) Exercise : Remove the second term in the eqn. ax3 + bx2 + cx + d = 0.(hint : just follow the steps in the example above) Not only we use symbols to ‘avoid difficulties’, often we use them to take their advantage. Take the example of Trigonometric ratios; these symbols have been aimed at measurement of heights and distances, but have gone a long way in development of complex numbers, theory of equations and so on. In a similar manner Calculus is developed to deal with infinitesimal numbers (infinitely small numbers) by assigning symbols to them . The process or method is, to identify the ‘difficulty’ or pinpoint it, adopt some symbol to enclose the difficulty, then proceed with known and standard methods until the symbol reveals itself or until the difficulty vanishes otherwise and toss away the symbol mercilessly. Wait. We can throw the symbol out in a particular problem, or well keep the symbol for future use if it has general importance, like ( - 1) = i or like log2 8 = 3 for 23 = 8 or sin – 1 ½ = 300 for sin 300 = ½ ., or a symbol for eliminant of equations such as matrices and determinants which find much use elsewhere. We shall return to the subject. Throughout the book series, we have adopted this method to develop a topic, adopting a symbol tacitly carrying its entire properties and characteristics; may it be Matrices and Determinants, Trigonometric functions, Inverse Circular Functions, Logarithms, Limits, Conic sections, Differential Coefficients, Integrals etc. etc. Due to this approach the topics appear in a manner how they were discovered and developed rather than a formal presentation. B. Beauty : The Way Nature Manifests Herself. A thing of beauty is a joy for ever. We are not going to present you a book on literature. But this is the fact. All mathematics is nothing but the four fundamental operations, Addition, Subtraction, Multiplication and Division. Subtraction is , in a way, addition of negatives; Multiplication is nothing but repeated addition and Division is multiplication of an inverse number. As such all the four fundamental operations are derived from Addition – The Yoga as in spiritual context. A literary truth is, ‘Beauty lies in the eyes of the beholder’. Ask any one to show you something without any beauty. The Nature being so beautiful in every detail, what may be a better key to understanding Nature, other than beauty. The truth is what exists. What did not exist in the past or shall not exist in future is no Truth. What exists here, and does not exist in USA is no Truth. And the Truth is our Consciousness. What else our consciousness might be ! Already we have cited the example of a dog under a truck knowing Newton’s laws, without knowing merely what they call it. The last one in the chain of Existence-Consciousness is Bliss, beauty , or 3
  4. 4. The Series Editor Rediscover Mathematics From 0 And 1 Series manifestation of everything perceptible or conceivable. Beauty , and manifestation are just synonyms ! Only one has to develop an eye for observing beauty. Really enough the necessary qualification is simplicity at heart. Thus the trio of Existence-Consciousness- Bliss blend into the One .- The Sat, The Chit And the Anand, Sacchidanand. Everything resides in ‘One’ along with its reciprocal; with one exception perhaps – Nothing or Zero. Thus we have ‘Two’ instead of ‘One’. Everything resides in Zero, paired off along with its negative. In Psychology they say that the act of seeing is a part of the object seen. In Indian Philosophy, the seer, the object seen and the process of seeing , all are parts of one generalization. And thus ‘Beauty lies in the eyes of the beholder’ – this is what establishes a connection between the seer and the seen; we never lay an eye upon something having no beauty. This book is nothing but a humble attempt to derive Mathematics from 1 or 0, the identities of multiplication and addition respectively by worshiping beauty. 0/0 forms take us to the concepts of differentiation and all the utilities that follow it. Every number can be thought of as a ratio of two functions each leading to the limit of 0. Even the vectors were discovered as a ratio, though they are used as much simple concepts powerful enough to derive complicated results.. One may try to rephrase Einstein’s words – the problem and the solution lie together inside something like identity or invariant or conservation, something unchanging. By locating the problem as far accurately as possible and giving some convenient symbol there, one may arrive at the solution just by mechanical operations on the symbol. Two examples revealing power of 1 and 0 follows immediately below. Example 2; The beauty and power of identities: sum of some +ve numbers = 0  each of them = 0. An identity is an equation which holds ( = holds true) for any admissible value of the variable(s). For holds for any value of x and y except 0. Also x  y x  y   x  y 1 1 xy 2 2  example, the equation xy xy is another example. Thus the identity stands on its own and does not depend upon the symbols though it is convenient to express them with the help of these symbols. One can say ‘ the sum of reciprocals of two numbers is the ratio of their sum to their product’.; and in like manner. In other words, an identity never affects , or depends upon the symbols in it, whatever number they may stand for. This is just in the same manner how the multiplicative identity does not affect any number to which it is multiplies or to which it divides; similarly the additive identity 0 never affects any number to which it is added or from which it is subtracted. Instead of identities that are evident at sight, like (a – b ) + (b – c ) +(c – a ) , we can assume or construct identities . If we assume (X– x )2 + (Y – y )2 +(Z – z)2 = 0 for all values of the variables X, Y, Z, x, y, and z ; or in other words if we assume it to be an identity,( call it a conditional identity) we immediately have the three equations X= x ,Y= y , Z = z. We dwell on this point again and again to prove beautiful results of use like equality of conjugate surds when given surds are equal; equality of complex numbers when given complex numbers are equal; proving that conjugate of a root of a polynomial is also another root of it; so on and so forth. An identity operation does not change or modify the argument (on which it operates) or the operand. For example, k = (k)2 ; an operation like squaring the square root of any number does not change that number. In other words it preserves the identity of the operand and it is therefore only that the operator is called so. Please note that the compound operation of first taking square root and then squaring is not same as first squaring and then taking the square root; simply because, it does not result in identity. Example 3; equating coefficients of similar powers in either side of an identity. One interesting thing about identities is that , If, ax 3  bx 2  cx  d  px 3  qx 2  rx  t is taken to be an identity, we must have, a = p, b = q, c = r , and d = t .(coefficients of corresponding powers of a particular variable from both sides shall be equal) This can be easily proved as under : 4
  5. 5. The Series Editor Rediscover Mathematics From 0 And 1 Series Since the equation is true for any value of x, putting x = 0, we get, d = t and they obviously cancel out from both sides. Then we could assume values of x as 1, -1 and 2 say, and get three equations involving a - p, b - q, and c - r ; and on solving them , we can get, each of a – p etc., each = 0. Alternatively, after canceling out d and t from both sides, we can get another identity, or equation which holds for all values of x. Putting x = 0 again in this eqn., we get c = r . Repeating the process, we get the desired result. Example 4; equating coefficients of similar trigonometric ratio’s in either side of an identity. If p cos x + q sin x + r = ( a cos x + b sin x + c ) +  (b cos x - a sin x) +  is taken to be an identity, (in other words, if a given expression in sin x and cos x is changed to another expression in sin x and cos x , for some desired convenience) then we must have the coefficients of sin x to be equal in both sides and so also the coefficients of cos x. We have, p cos x + q sin x + r = ( a cos x + b sin x + c ) +  (b cos x - a sin x )+  for all x p cos x + q sin x + r = (a + b) cos x + (b - a)sin x + (c + ) for all x, ………………(a) Or, Putting cos x = 1 throughout, ( so that sin x = 0), we get, p + r = (a + b) + (c + ) ……………………………………………….……(b) Putting cos x = - 1 throughout, ( so that sin x = 0), we get, - p + r = - (a + b) + (c + ) ……………………………………….……(c) Adding (b) and (c) and dividing throughout by 2 , we get, p = (a + b) and r = (c + ) ………………………..…………………...(d) Putting these values in (a) we get, q = b - a ………………………..……………………….....(e) So we get three independent equations in , , and  from (d) and (e) which may be easily solved to find , , and  in terms of p, q and r. ( the result has many uses in integration chapter) It seems at first sight that identities are trivially true and are of little utility. No. Just like 0 is a trivial nothing , but embodies a world of secrets like a black hole, identities contain a world of secrets inside them. We give below an example how the concept of identity is used to reveal the sum of an infinite series hidden in the symbols we define ,to get rid of a difficulty. Example 5; Infinite Sequences and Series : Another example in ‘difficulty identification’ or use of ‘equating of coefficients’ : to find out the sum of a series like the following;  t n  1x 2  2 x 3  3x 4.......... upto..n ( n  1)....n  terms.......... .......... .( A ) Difficulty here is that we do not know the sum of the series. But we think that it must involve n and its powers and some constant, of course, independent of n . So let us assume (like defining symbols as we often do) ,  t n  A  Bn  Cn 2  Dn 3 ......................................................(B) tr, the r-th term, or the general term for that matter, may be written as r(r + 1). (Had all of them been equal, we could have simply multiplied ‘n’ with any term to get the sum. That makes the sum one degree higher in ‘n’, than the degree of ‘n’ in tn . So it is safe to assume that, if n = 6, we don’t require more than 7 terms in the series for tn . So we have taken terms upto n3 in (B) when t n has no other power of n larger than 2. Similar must be the things for n+1 terms too; then,  t n 1  A  B(n  1)  C(n  1) 2  D(n  1)3.................................(C)  t n 1  1x 2  2x3  3x 4..........upto..(n  1)(n  2)................................(D) 5
  6. 6. The Series Editor Rediscover Mathematics From 0 And 1 Series Subtracting(A) from (B), and (C) from (D) , we get,  t n 1   t n  t n 1  (n  1)(n  2)  B  C(2n  1)  D(3n 2  3n  1)  2  3n  n 2  (B  C  D)  n (2C  3D)  n 2 (3D).............................(E) Equating co-efficients of similar powers of n from both sides, we get, B  C  D  2,2C  3D  3,...and..3D  1  D  1 3 ,...C  1,...and...B  2 3 With these values and putting n = 1 (or any integer you like) in (B), which is also an identity, we get, t  t1  1x 2  A  2 3  1  1 3,..Or..A  0 1  2  3n  n 2  n ( n  1)(n  2) n3 2n  tn   n 2  ..   t n  n     3 3 3 3   Review Exercises : Try for expression for sum of these serieses: 1) Sum of 1st n natural numbers, tn, tn =n, or in other words n 2) Sum of squares of 1st n natural numbers, tn, tn =n2 or in other words n2. 3) Sum of cubes of 1st n natural numbers, tn, tn =n3 or n3. There are numerous examples throughout the book where we would be using this technique to rediscover and redevelop many topics from identities. Before that the reader may try some identities from high school some of which are given below. The reader can try as many of them as possible and at ease. Use the fact that if a = b put throughout the expression makes it 0, then a – b is a factor; similarly if a = - b put throughout the expression makes it 0, then a + b is a factor. 4) (a – b) + (b – c) + (c – a) = 0; 5) c(a – b) + a(b – c) + b(c – a) = 0 6) c(a – b)3 + a(b – c)3 + b(c – a)3 = (a + b + c)(a – b)(b – c)(c – a) 7) c(a – b)2 + a(b – c)2 + b(c – a)2 + 8abc = (a + b)(b + c)(c + a) 8) c4 (a2 – b2) + a4 (b2 – c2) + b4 (c2 – a2) = - (a – b)(b – c)(c – a)(a + b)(b + c)(c + a) 9) (b - c)3(b + c – 2a) + (c - a)3(c + a – 2b) + (c - a)3(c + a – 2b) = 0 10) (b - c)(b + c – 2a)3 + (c - a)(c + a – 2b)3 + (c - a)(c + a – 2b)3 = 0 11) (ab – c2)(ac – b2) + (bc – a2)(ba – c2) +(bc – a2)(ba – c2) = bc(bc – a2) + ca(ca – b2) + ca(ca – b2) 12) bc(b –c) + ca(c –a) + ab(a –b) = - (b –c)(c –a)(a –b) 13) a2 (b –c) + b2 (c –a) + c2 (a –b) = - (b –c)(c –a)(a –b) 14) a(b2 –c2) + b(c2 –a2) + c(a2 –b2) = - (b –c)(c –a)(a –b) 15) a3 (b –c) + b3 (c –a) + c3 (a –b) = - (b –c)(c –a)(a –b)(a + b + c) 16) a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – bc – ca – ab) 17) a3 + b3 + c3 – 3abc = (a + b + c)(b2 – ca + c2 – ab + a2 – bc) 18) a3 + b3 + c3 – 3abc = ½ (a + b + c)[(b – c)2 + (c – a)2 + (a – b)2] 19) (b – c)3 + (c – a)3 + (a – b) 3 – 3(b – c)(c – a)(a – b) = 0 20) (a + b + c)3 = a3 + 3a2b + 6abc 21) (a + b + c + d)3 = a3 + 3a2b + 6abc 22) bc(b +c) + ca(c +a) + ab(a +b) + 2abc = (b + c)(c + a)(a + b) 23) a2 (b +c) + b2 (c +a) + c2 (a +b) + 2abc = (b + c)(c + a)(a + b) 24) (b + c)(c + a)(a + b) + abc = (a + b + c)( bc + ca + ab) 6
  7. 7. The Series Editor Rediscover Mathematics From 0 And 1 Series 25) (a + b + c) (-a + b + c) (a - b + c) (a + b - c) = 2b2 c2 + 2c2 a2 + 2a2 b2 – a4 – b4 – c4 26) c (a4 – b4) + a (b4 – c4) + b (c4 – a4) = (b –c)(c –a)(a –b)(a2 + b2 + c2 – bc – ca – ab) 27) (a + b)5 = a5 + b5 + 5ab(a + b)( a2 + ab + b2) 28) (a + b + c)5 = a5 + b5 + c5 + 5(a + b) (b + c)(c + a)( a2 + b2 + c2 + bc + ca + ab) 29) c2 (a3 – b3) + a2 (b3 – c3) + b2 (c3 – a3) = (ab + bc + ca)(a – b)(b – c)(c – a) 30) bc(c2 – b2) + ca(a2 – c2) + ab(b2 – a2) = (b – c)(c – a)(a – b)(a + b + c) 31) (b +c){(r + p)(x + y) – (p + q)(z + x)}+ (c +a){(p + q)(y + z) – (q + r)(x + y)} 32) + (c +a){(p + q)(y + z) – (q + r)(x + y)} = 2[a(qz – ry) + b(rx – pz) + c(py – qx)] 33) (a – x)2{(b – y)2(c – z)2 - (b – z)2(c – y)2}+(a – x)2{(b – y)2(c – z)2 - (b – z)2(c – y)2} +(a – x)2{(b – y)2(c – z)2 - (b – z)2(c – y)2} = 2(b – c) (c – a) (a – b) (y – z) (z – x) (x – y) and so on and so forth. All Mathematical formulae read in high school are identities or conditional identities and we know their utility. As examples of some conditional identities put a + b + c = 0 in expressions above where a+b + c appears: the result may be taken as a conditional identity. If a + b + c = 0, prove the following : 34) 2bc = a2 – b2 – c2 35) 8a2b2c2 = (a2 – b2 – c2)( b2 – c2 – a2)( c2 – a2 –b2) 36) a3 + b3 + c3 = 3abc 37) 2(a4 + b4 + c4) = (a2 + b2 + c2)2 38) 3a2b2c2 – 2(bc + ca + ab)3 = a6 + b6 + c6 39) a5 + b5 + c5 + 5abc(bc + ca + ab) a 2  b 2  c2 a 5  b5  c5 a 7  b7  c7   40) 2 5 7 bc ca a b a c b    / 3  3 /   41)   bc ca a b a b c Symmetry , Anti-symmetry And Asymmetry Are some Aspects Of Beauty. A look at the previous examples of identities makes us think about symmetry anti-symmetry and cyclic symmetry of the expressions. Discoverer of electrical generator must have observed the change in electric field due to motion of chares, i.e., electric current causes a magnetic field; and it must have occurred to him that a change in magnetic field may generate electric current. Symmetry reveals the things that are not explicit. It also helps us to write expressions in brief. For example, a stands for a + b if two elements are taken; it stands for a + b + c or a + b + c + d if 3 or 4 elements are taken. Similarly a2 stands for a2 + b2 or a2 + b2 + c2 or a2 + b2 + c2 + d2 if 2 or 3 or 4 elements are taken. The expression a2 + b2 is symmetrical with respect to a and b in a sense that a and b can be replaced with each other without affecting the value of the expression. The feature may be termed bilateral symmetry. The expressions such as a2 + b2 + c2 or bc + ca + ab are bilaterally symmetrical, as any two of them can be interchanged without changing the value of the expression. In addition, the latter expressions are of cyclic symmetry; i.e., if a is replaced by b, b is replaced by c and c is replaced by a simultaneously, the expression is unchanged. To illustrate the method of applying symmetry concept in working out problems, consider factorizing the expression (a + b + c)5 - a5 - b5 - c5 . The value of the expression is unchanged if we put b in place of a , c in place of b and a in place of c. But the expression becomes 0 if b = - a throughout. Hence b + a or a + b must be a factor of it. Similarly b + c and c + a must be factors of it. As such the expression contains (b + c)(c + a)(a + b) as a factor. The latter factor is of third degree whereas the expression to be factorised is of 5th degree. So it contains another factor of 2nd degree. As the expression is symmetric in a, b and c; so also all its factors must be symmetric cyclically. A general expression in three elements and in 2nd degree would be 7
  8. 8. The Series Editor Rediscover Mathematics From 0 And 1 Series A(a2 + b2 + c2) + B(bc + ca + ab) where A and B have to be determined. So we have complete factorization as (a + b + c)5 - a5 - b5 - c5 = (b + c)(c + a)(a + b)[A (a2 + b2 + c2)+ B(bc + ca + ab)] Now since this is an identity, the expression holds for any value of a, b and c. Putting each equal to 1 and each equal to 2 in turn we get, A + B = 10 and 5A + 2B = 35. Solving the equations, we get, the values of A and B , both equal to 5 and the complete factorization becomes, (a + b + c)5 - a5 - b5 - c5 =5 (b + c)(c + a)(a + b)(a2 + b2 + c2 + bc + ca + ab) An expression such as a2 (b –c) + b2 (c –a) + c2 (a –b) is changed to –[ a2 (b –c) + b2 (c –a) + c2 (a –b)], i.e., its own negative when any two of its variables are interchanged with each other. Such an expression is said to be alternating or anti-symmetric. More about symmetry and its uses shall be discussed from topic to topic later on; especially in transformation of graphs. Some other aspects of beauty are continuity, completeness, compactness, connectedness, convergence and uniformity. Examples shall follow throughout the book. The concept of continuity of functions shall be discussed in Calculus in a later chapter. Convergence concept shall be discussed in the chapters for sequences a, serieses and limits. While watching a movie or drama we note a touching sequence of events and keep guessing what should happen at last. If the end comes of our expectation we feel continuity in the story line. If the end of the story keeps us guessing still , we must feel something lacking in the story, e.g., there may not be an end to the drama and it may not be said to be complete. What happens in this case is a sequence of chosen events leads to a limiting event, a point which is not included in the story; As such the sequence of events is not continuous and the story is not complete. The concepts as such, are better illustrated in Topology, which is a set of some subsets called open subsets with a structure – closed under arbitrary unions and finite intersections. The topic of topology is an attempt to provide a common platform to Algebra, Analysis, Differential equations, etc. etc. The principle of equivalence in mechanics , as propounded by Newton, states that the laws of mechanics are symmetric with respect to all inertial frames; i.e. , they do not change if we change frames of reference with a new one moving at a constant velocity with the old one. A kind of proof or illustration would be given at the appropriate place . It would be shown that acceleration of somebody measured in one frame of reference will just be same in be same as measured in a different frame of reference moving at a constant velocity from the initial frame of reference taken. It would be just child’s play and the reader even might have done the derivation is high school. Einstein derived the epoch making theory of relativity only from two assumptions; one : the principle of equivalence with only one word changed ; he wrote Physics in place of Mechanics. The second assumption is that the velocity of light in empty space does not depend on ( not added to nor subtracted from) the velocity of its source. The latter assumption is nothing but wise acceptance of failure to observe the expected result in the famous Michelson Morley experiment to measure absolute velocity of earth . ( Doesn’t it seem that the theory of relativity was derived from the very antithesis of relativity ?)  for those who refuse to wait until then. Let us measure acceleration a of some object while we stand still on the v1  v 0 a ground. Acceleration is , where v1 , v0 are its final and initial velocities and t is time taken for this change t of velocity. If we observe the same from object from a train with velocity u, do not observe the initial and final velocities, but observe the initial and final relative velocities instead , v1 – u and v0 – u . Now acceleration observed ( v 1  u )  ( v 0  u ) v1  v 0   a , again, which proves the proposition. One can change all these from the train is t t symbols except for time to vectors and prove the proposition in case of vector velocities too. 8
  9. 9. The Series Editor Rediscover Mathematics From 0 And 1 Series The conservation laws in Physics like conservation of mass, conservation of energy, conservation of momentum, conservation of angular momentum, conservation of spin etc. tell us that the totalities of quantities like mass, momentum, energy, spin etc before an event like collision or explosion remains unchanged after the collision or explosion etc., as the event may be. So total mass, momentum, energy etc. are invariants . So the totals of these quantities in a system do not change i.e., they are symmetrical about a point of event like collision or explosion etc. Total mass or energy or momentum in a system of particles before a collision taking place in the system is conserved after the process of collision. Similar is the case after a chemical reaction is completed. The new theory of relativity has combined the laws of conservation of mass and conservation of energy into one law, conservation of mass and energy together, by showing equivalence of mass and energy. Not a single instance has been observed violating the these principles of conservation. We would give an expression to illustrate the principle of conservation of momentum and conservation of energy at appropriate place .The principle of equivalence in theory of relativity has led to Lorenz transformation of coordinates which in turn, has led to the result of equivalence of mass and energy. Some of the invariants in transformation of coordinates we shall discuss later on in the chapter for transformation of graphs. The starting point in solving problems involving equations of motion is these conservation laws which give the differential equations of motion in a particular situation; the latter is then solved by applying standard Mathematical techniques to get the equations of motion. By differential equation we mean an equation involving physical quantities such as velocity etc. and their rates of change; the latter called differential coefficients. A differential equation is solved when we get equations involving the physical quantities only and not involving their differential coefficients. Do the phrases “The principle of equivalence”, “The conservation laws in Physics” and “invariants in Physics or in Mathematics” sound like the concept of identity we just discussed. Decide for yourself. A special mention may be made of Schrödinger’s uncertainty principle which enunciates that the product of errors of measurements of complementary physical quantities like position and momentum shall be at least equal to Plank’s constant; s.p  h ; This is a completely theoretical fact having nothing to do with precision of measuring instruments. If we set s = 0 to know s or position completely precisely, it requires p to become infinite; i.e., the momentum p shall have infinite error in its measurement and thus cannot be determined at all. This is nothing but symmetry; just in the same sense as 4/1 and ¼ are symmetrical. Symmetrical conjugates combine with each other to result in identity ! The way they combine or associate with each other may be different i.e. (+4) adds up with ( - 4 ) to result in 0, the identity of addition and 4/1 and ¼ have to be multiplied with each other to result in multiplication identity 1. break open the identity and you get symmetrical parts; again fit the parts together, you get the identity. The beauty of the statement lies in the philosophy of quantum theory stating that no physical quantity can ever be measured deterministically but only probabilities can be expected. When the notion of exactitude of measurements is lost, the philosophy of cause and effect, the basic philosophy of all experimental sciences is put at stake. It turns out that a bigger particle has some chance of tunneling through a smaller particle or simply you could just pass right through a wall for that matter. The concept of continuum of cause and  Suppose two bodies of masses m1 and m2 with velocities u1 and u2 respectively collide and their velocities get changed to v1 and v2 respectively. By Newton’s third law the force exerted by one body on the other should be equal m1 (v 1  u1 ) m 2 (u 2  v 2 )  and opposite. Let them be F and – F respectively. Equivalently, where t is the brief time t t for which the two bodies are in contact while colliding, or , m1v1 + m2v2 = m1u1 + m2u2 ; i.e., the total momentum before collision is the total momentum after collision , as expected.( vectors may replace scalars throughout, if you please) Similarly , from the principle of conservation of energy, we can derive an expression of kinetic energy of a body of mass m, the energy, W say, possessed by it by virtue of its velocity. Surely it would be equal to the work done by it against a force opposing its motion until it comes to rest. If the body travels a distance s in the process, 2 then we have, the work done W = Fs = mas = ½ m 2as = ½ m1v which is the expression of kinetic energy we sought after. 9
  10. 10. The Series Editor Rediscover Mathematics From 0 And 1 Series effect still thrives, albeit in terms of probabilities though not in terms of exactitude as was in the good old days of classical physics. Behind these conservation laws or invariances, there is a grand principle of nature. Economy of action is the grand principle behind. Newton was a great integrator. He integrated the works of Keppler, Hookes and Copernicus and declared the beginning of the era of scientific progress giving science a foundation; his three laws , appearing almost axiomatic. Another great integration or synthesis was achieved with the advent of a new era of Quantum Mechanics, with its harbingers, Max Plank, Neil Bohr and others. Big controversies about particle nature of radiation and wave nature of matter were settled by giving matter and energy a common platform , the wave equation due to Schrödinger. Radiation was accepted to be effected in quanta, or ‘in packets’ instead of in continuous fashion. Classical Mechanics predicted that a charged particle shall continuously radiate energy resulting in shrinking of its orbit and shall fall into the nucleus within very short time and stable atoms were incomprehensible as such. Neil Bohr used the ‘quantum’ idea of Max Plank and told to the world to accept that the electrons simply do not radiate while being in their orbits and only radiate in lump sums while jumping from orbit to orbit. The structures of orbits became complicated to comprehend of course, but the dual nature of radiation and matter were explained. Classical mechanics which beautifully predicted celestial phenomena, could be viewed as a limiting case of Quantum Mechanics and the two domains were unified. Another great integration or synthesis was brought about , at about the same time by Albert Einstein for explanation of behaviour of matter approaching the speed of light and the concept of matter, energy, space and time had to be redefined. He brought about the Special Theory of Relativity , the single most outstanding discovery and philosophy of the twenty first century, which identified the century, together with Quantum Mechanics. The simple idea behind was, that the speed light does not depend on the speed of its source. Mechanics at lower velocities was viewed as a limiting case of the theory of relativity. Necessary Mathematics was already at hand , developed by Maxwell in his electrodynamics theory and the simple idea of constancy of velocity of light in empty space was acceptance of failure of the famous Michelson & Morley experiment aimed to use difference of velocities of light in moving back and forth to measure absolute velocity of earth. These great integrators like Newton and Einstein were motivated by the idea of sort of identity, symmetry and invariance etc. What was fundamental belief driving behind these principles of invariance and symmetry – it was the principle of economy – the way of Nature’s doing things. It was as back as the days of Ptolemy and Aristotle when observations were made regarding angles of incidence and angles of refraction when path of light changed mediums; but the relationship between them eluded until some three hundred years until the days of Fresnel who correlated them in the laws of refraction. About a hundred years later, Format gave the law a firm footing; the law was deduced from the belief that light travels through the path in which minimum time is taken. It is only through this principle of economy that Hamilton ‘deduced’ the Newton’s laws again. Euler went further through the problems of maximization or minimization and created a new discipline in Mathematics which is known as Calculus of Variation today. Now any idiot can swear that if the grand unification theory is going to be complete, it would be only through courtesy of belief in nature’s principle of economy – from which the relative external features like symmetry and identity emerge. . The scientific world today is striving towards grand unification theory of the four fundamental forces of nature or looking for a common origin of the four forces; namely, electromagnetism, gravitation, the strong and the weak nuclear forces. It is the sense of beauty only which enables integration of apparently contradicting theories such as the Classical and the Quantum Mechanics, the wave theory and corpuscle theories of light, the classical and the relativistic Mechanics and so on. It is the art and science of Mathematics only which formulates and guides such endeavours. This is the dream idea of Einstein and we are nearing the idea year by year. Newton has given us a comprehensive concept for force, mass, energy, velocity etc. and explained Mm FG the gravitational force of universal application with the formula for interaction between any r2 10
  11. 11. The Series Editor Rediscover Mathematics From 0 And 1 Series two (point masses or spherical bodies of) masses M and m at a distance r from each other. So gravitational force has been recognized as a fundamental force in nature. This force is responsible to hold together solar systems and nebulae as well as pieces of any matter together to have some shape or size. it goes without saying that solids have definite size and shape only because of small intermolecular distance between them; liquids have only definite ‘size’ or volume as intermolecular distance is more than that in the solid state and gasses have no definite size nor shape but for the large intermolecular distances. Similar is the Coulomb’s law of electrostatic attraction or repulsion, describing the force between two point charges Q Qq and q at a distance r apart from each other to be F  k . Electric currents or moving charges showed r2 magnetization as natural magnets and as such, all magnetism was ultimately attributed to electric currents. Faraday imagined that , if a motion of electric charges gives rise to magnetic poles in effect, a motion of magnetic poles must produce some electric current and it was very much true. This led to the fact that electricity and magnetism are not two different entities , but are manifestation of one thing, m1m 2 electromagnetism which also included the Coulomb’s law for magnetic force F   between two r2 magnetic poles m1 and m2 at a distance r apart.. So electromagnetic forces are fundamental forces responsible for binding electrons to the nucleus to form atoms and are sufficient to explain away chemical properties of matter. Though the electrostatic attraction and gravitational forces both vary inversely as square of distances, the magnitude of the latter is 1036 times the former. Calculate the forces between a proton and an electron in an hydrogen atom given r  0.53 x 10 – 10 m, charge of one electron as well as that of proton being e = 1.6022 x 10- 19 Coul. in magnitude; mass of electron m = 9.1093 x 10 – 31 kg, mass of proton M being some 1836 times that of electron; G = 6.673 x 10- 11 Nm2/kg2, k = (1/ 40) = 9.1 x 10 – 19 Nm2/Coul2. So much of data is absolutely not necessary to compare the forces; neither the distance of electron from nucleus, nor its mass and charge both; only e/m is sufficient for the purpose , it being 1.76 x 1011Coul/kg. A single electromagnetic force can be understood to act upon a charged particle, whether stationary or moving, accounting for electrostatic and electromagnetic forces. This is Lorentz force given by F = qE + qv x B where q is the charge with velocity v, E is the electrostatic field, and B is the magnetic induction at the point where the charge lies instantaneously. What makes protons stay together inside the nucleus when they should have repelled each other away ? It is the strong nuclear force which should be great enough to overcome the electrostatic repulsion and in fact it is of the order of 1038 times the gravitational force between two protons . or of the order of 100 times the electromagnetic force so that it overcomes repulsive Coulomb forces between protons in the nucleus. But it is short ranged and almost ineffective beyond a distance of 10-15m which is in the order of nuclear radius. Beyond the nucleus the electromagnetic force reigns supreme , but it dies out fast beyond atomic or molecular radius though theoretically it acts upto infinite distance. Beyond the atomic radius of 10-10m only gravitational force is visible as there are no charged particles to masquerade it. We see it acts through great distances as known from its formula. The strong nuclear force is said to act upon quarks and gluons of which the protons and neutrons are made up of, and thus accounts for formation of nucleus and disintegration of nucleus too, for that matter. Radiation of -particles has been successfully explained in terms of this force. An -particle is equal to Helium nucleus consisting of two protons and two neutrons. Crudely speaking neutrons are necessary to bind protons together in the nucleus on the face of electrostatic repulsion and more and more of them is necessary as atomic number is increased, until the nucleus becomes unstable and becomes radioactive to emit an -particle to become stable. For protons and neutrons cannot be shed off unless they are in an unit of two plus two of each, i.e., an -particle. More correct picture is of course, the one involving quarks and gluons, believed to be ultimate particle of every matter . Protons and neutrons are believed to be made up of 3 quarks each of different ‘colour’. Colour is a property of quarks and gluons analogous to charge , are of three types, ‘red’ ,‘blue’ and ‘green’ and quarks with different colour attract each other and quarks of same colour repel ach other. Gluons carry a colour 11
  12. 12. The Series Editor Rediscover Mathematics From 0 And 1 Series and an anti-colour charge.. A proton consists of three quarks of each different colour. Within a distance of less than 10-15m (a femtometer)or nuclear radius the strong force is repulsive and keeps the nucleons (protons and neutrons) apart. Within a distance of 1 to 2 femtometers the strong force is attractive force overcomes the electrostatic repulsion being of the order of 100 times the electromagnetic force Quarks and leptons were believed to be ultimate building blocks of all matter. The leptons, which include the electron, do not “feel” the strong force. However, quarks and leptons both experience a second nuclear force, the weak force. This force, which is responsible for certain types of radioactivity classed together as beta decay, is feeble in comparison with electromagnetism , of the order of 10-15 compared to electromagnetic forces and act in very short range of 10-15m, within nuclear radius. In -decay, the nucleus emits an electron with high energy and an antineutrino of no rest mass and a neutron is converted into a proton in the process. Similar is the process of electron capture by a nucleus from the innermost shell and similar is the process of positive electron or positron emission ; in all the three cases the week nuclear force is involved. Much is to be done in the area of unification of forces or to have closer look for symmetries and similarities though much has been done. To remind of history, it was Sir Isaac Newton around 1687 who ‘unified’ celestial forces with ‘terrestrial’ forces, i.e., made it evident that the acceleration due to gravity is nothing but a manifestation of universal gravitation. All magnetism was traced back to moving electric charges , thanks to the works of Hans Christian Oersted and Michel Faraday , around 1825. It is J. Clark Maxwell who showed that visible light was an ‘electromagnetic’ radiation; which paved the way to understand that heat waves, ultraviolet, microwave, x-rays, -rays, radio waves , cosmic rays all are electromagnetic radiations. Such unification brought about great confusion about wave nature of particles and particle nature of waves and confusion about ether, the thinnest wall in the empty space to lean on. So developed the quantum theory to unify waves and particles; a number of scientists were involved , over many decades and ultimately the concept of ether was discarded. Mathematics required it to be of very high density and nobody was prepared to swallow the idea along the gut anyway. Nuclear physics and radio activity brought newer phenomena to observation and again we were in the mesh of particles, particles and more particles. Force, which was not due to contact, or action at a distance, whether gravitational, electromagnetic, strong or week nuclear forces, were seen to be understood with ‘carrier particles’ of forces; most successfully electromagnetic forces were explained, with exchange of ‘photon’ as carrier of electromagnetic forces; just as a play ball influencing the behaviour of a pack of playing children. ( When two children run toward the ball to catch it, they seem to attract each other). Albert Einstein’s general theory of relativity postulated ‘graviton’ , photon-like particle as carrier of gravitational force although quantum theory for it is yet to be fully developed. ‘Gluon’ was postulated to be carrier of strong nuclear force or quarks exchanged gluons on interaction under influence of strong nuclear forces. W- particles and Z-particles were thought to be the exchange particles for week nuclear forces though big particles they were indeed, the former with + or – charges and the latter neutral. A single ‘electro-week theory’ was postulated by Sheldon Glashow, Abdus Salam, Steven Weinberg around 1980 in order to look for a common origin of electromagnetic and week nuclear forces and it worked very well. It was verified experimentally by Cario Rubia and Simon Vander Meer around 1984 but the grand unification is still a far cry. But the efforts in this direction are not discouraging ; Abdus Salam had said that the unification is expected if proton decays at all, and it was found to be decaying, with experiments carried out in the deepest mines e.g., the Kolar gold fields in India. A significant improvement has been achieved in 2004 towards The Grand Unification Theory or The Theory of Everything and Nobel prize for Physics has been shared by three physicists; for the first time for theoretical research. Interest : It Can Remove Hurdles Even As Big As Mountains in your way. Dedication is the sincerity of interest taken in the subject. When a big experiment fails, trying to prove or disprove something, it is never a disappointment for a scientist. Rather it harbingers a new era, and is  Three people shared the 2004 Nobel prize for Physics for a major development in the Grand Unification Theory (GUT), or Theory Of Everything (TOE), or Quantum Chromodynamics (QC), by successfully including gravitation into the common fold of the field theories. 12
  13. 13. The Series Editor Rediscover Mathematics From 0 And 1 Series epoch making. When Michelson & Marley failed in their famous experiment designed to measure absolute velocity of earth in ether, Albert Einstein formulated The Special Theory Of Relativity from this failure. The Mathematics needed by him had already been developed more than two hundred years before. He used Maxwelian transformation of coordinates in his theory encompassing all branches of Physics which was used only in the field of Electromagnetism. In the area of Mathematics, it would be shown how endeavours to prove the fundamental theorem of Algebra ( every polynomial has at least a root, real or complex) opened up the systematic study of various algebraic structures like groups, rings, integral domains, fields and vector spaces. C.The Open Mind : We Learn Until We Are Dead; The process of induction; scientific methods. Simplicity is a necessary mindset for deciphering apparently complicated mysteries of Nature. Failed midway in some endeavours, one must do-it-all-over-again readily; and in the process, more often than not, new directions are envisioned. A story about Einstein goes like this : a small girl discussed with him about the Earth going round the Sun . it was certainly difficult for her to understand, for she perceived the Sun going round the Earth. When Einstein tried to explain her that the astronomical measurements would be same in both ways; no matter whether the Earth went round the Sun or otherwise, she interrupted asking whether it might just be possible that nobody goes round the other. Simple enough is the question; but it made the big head meditate and the General Theory of Relativity was ushered in. Simplicity is the way upto the truth. What truth is, nobody knows. But it has no double standards. What tastes sweat to you must do so to me or anybody else. All religions and ethics endorse this. That is the proper way to live and let live; to protect the environment, to prosper and to save the planet in brief. A grain of rice may be small and negligible for a rice merchant but never so for spectroscopic study and never so for an ant either. But Mathematics describes smallness for everybody, applicable everywhere and at all times. What Calculus studies is only this smallness or largeness. Development of number system in the following articles displays one example open mind or simplicity of thought or ability to learn until death – call it whatever you please. Can there be some number whose square is negative ; when we know that squares of both positive and negative numbers both are positive only? Why should one accept that numbers could only be either positive or negative? Doesn’t that sound simple enough ? Yes, it is. Great geniuses who foresee formidable results are sometimes unable to prove them. Many examples may be cited, like the four colour theorem due to Euler ; it states that four colours are required at most for clouring any number of regions in a map. But the theorem was proved only after half a century later. More fundamental the result is, more difficult is becomes for proving; and is accepted as an axiom for a long period of time until somebody formally proves it. But such situation does not arise too often; though it is not rare. The reason is; we have a powerful logic or method of Mathematical induction to verify if something is true for as many number of cases as one may count. In Mathematics, if a million examples are given to support validity of some statement; the statement is not necessarily proved; but a single example to the contrary is sufficient to disprove the statement. Very true. The method of Mathematical induction ensures that a single exception shall not be found. And thus it is a proof, full-fledged , rigorousely mathematical in spirit We try to formally state what it means : a) Principle Of Mathematical Induction : If a statement is true for n = 1, or equivalently, if P(1) is true, and it is true for n = m + 1 P(m + 1) is true whenever it is true for n = m; whenever P(m) is true, then it is true for all n  N. then it is true for all natural numbers In short, to accept the truth of some statement, we must exclude possibility of even a single exception !The reader must clearly see the mathematical rigor in the principle of mathematical induction ; in contrast to the case of the following example : 13
  14. 14. The Series Editor Rediscover Mathematics From 0 And 1 Series “Ram shall die one day. Hari shall die one day. Gopal shall die one day . Thus it is deduced that any man shall die one day.” The statement is very true. But we find it difficult to see whether the statement as such, excludes the possibility of even a single exception. b) Induction in another different way in which the principle of mathematical induction is elucidated is given below. If a statement is true for n = 1, and it can be proved to be true for n = m when it is true for all n < m, the statement is true for any n  N. The reader can understand the equivalence of the two different formulations. Evidently the principle of mathematical induction can only be applied to statements which admit of successive cases corresponding to the order of natural numbers 1, 2, 3, … etc. It is in fact, the basis acceptance of some rule after experimental verification . And this makes mathematics an experimental science. Example 6 : To prove 2 n  (n  1) 2 for all n ≥ 6 , n  N It is not correct to say that the statement has exceptions of 5 cases 1 ≤ n ≤ 5 . Rather the statement 2 n  (n  1) 2 for all n  N has 5 exceptions . Hence the former statement comes under jurisdiction of principle of mathematical induction. The steps of induction process are : Step 1 : To see whether 2 6  (6  1) 2 which is obviously true as 64 > 49. Step 2 : Assume 2 m  (m  1) 2 for some m ≥ 6 and to prove that 2 m 1  (m  1  1) 2 Now multiplying both sides of 2 m  (m  1) 2 by 2, we have, 2 m 1  2(m  1) 2  2m 2  4m  2 ; 2 m 1  m 2  4m  ( m 2  2)  m 2  4m  4  ( m 2  2)  m 2  4m  4  m  2  2 Or, As m2 – 2 > 0 as m > 6. Thus the proof is complete as the statement is proved for n = M + 1. Note : The statement 2 n  (n  1) 2 for all n ≥ 6 , n  N, enjoys the status of only a conjecture until it is proved; by way of induction or otherwise. Example 2 : Prove by induction (and otherwise) that the number of all the subsets of a set having n number of members is 2n. By induction : Let us assume it for n = m, where m is a particular natural number . Now take a set A with n = m + 1 members. To count its subsets let us do in two steps; one, let us pick up a particular member ‘b’ and count all the subsets containing b . In the second step let us count all the subsets which do not contain b and finish counting in this way. Let B = A – {b}, and it has only m number of members. In the first step, we can take any subset of B and then include b to it. Since B has only m number of members, we count 2m subsets in this way, by assumption. In the second step, we count all the subsets of A which do not include b. this can be done by taking all the subsets counted in the first step and excluding b from every one of them. Thus the total count in second step is 2m also. There is not a subset in common among the collections in two steps. So our total count in both steps should be 2 x 2m = 2m+1 . So we have proved the statement for n = m + 1, while we assumed it for n = m. Now we have to see if the statement is true for n = 1, 2 etc. Clearly for n = 0, A =  , it has only one subset  itself .the total number of subsets is 1 = 20, which satisfies the statement. For n = 1, A = {a}, say; and the subsets are  and A, a total 2 = 21 subsets, thus satisfying the statement. For n = 2, let A = {a, b}. the subsets are : , {a}, {b} and {a, b} , a total 4 or 22 of them, satisfying the statement. We have already seen that if the statement is true for n = 2, then it must be true for n = 2 + 1 = 3; and thus we proceed to n = 3, 4 5…. Etc as long as we please. Otherwise: If the set has n members, we make subsets by picking up 0 at a time, 1 at a time, two at a time, and so on upto all at a time. If we don’t pick up a member we get  and we can do it only once( Aren’t 14
  15. 15. The Series Editor Rediscover Mathematics From 0 And 1 Series all ’s the same). We can pick up one member out of n members in n different ways, 2 different members in n C2 ways, , 3 different members in nC3 ways and so on. So the total count would be n C0 + nC1 + nC2 + n C3 +………………. Cn different ways. By a result of Binomial theorem the expression comes upto 2n. n c) Why after all we require some sort of proof : As intelligent animals, we need verify that a particular statement applies in every case without exception before we accept it. The difficulty or impossibility of verifying every case of application of the statement forces us to look for a proof. For example, to prove that man is mortal, how many cases can we count or how long could we do so ? Hence we require a logical proof. The principle of mathematical induction gives us just as a shortcut to the process of counting and verifying the applicability of a certain statement case by case; assuming it for some case and deducing the validity for the next case. And ultimately, verifying validity of the statement for a particular case say for n = m, extends the validity of the statement to all the subsequent cases. The other statement of the principle of mathematical induction also essentially extends verification of the statement to the case n = m when the validity of it is assumed for all cases n < m. d) Exclusion of exceptions; or contradiction of contradiction is also a method of proof : Naturally. When ‘proof’ is ‘verification of a certain statement case by case’ and ‘without exception’, a method of proof can be given just by guarantying that not a single case of exception hold good. In other words, when contradiction of the statement is disproved, the statement stands automatically proved. Like the method of induction, which is more often applied to theorems of fundamental importance , the proof of which otherwise is difficult; in the same manner, the method of contradiction of contradiction is applied for proving theorems of fundamental importance ,the proof of which is otherwise difficult. Examples of proofs by excluding possibility of contradiction of the statement under consideration abound in Mathematics and shall be encountered as we proceed. Vide examples under section irrational numbers below for examples. Examples : e) Deduction – a method of proof : As we have stated above, logical deduction is a device or tool for verification of a statement and thus it is accepted as a method of proof. One example would suffice to illustrate the conjecture. A polynomial or a rational integral algebraic expression is like a0xn + a1xn - 1 + a2xn -2 +…… an-1x +an where n is a positive integer and the coefficients are real or complex numbers. The fundamental theorem of Algebra states that it has n roots, real or complex. The statement is equivalent to a polynomial of degree n has at least one root, real or complex. Either of these two statements can be logically deduced from the other. If we assume that a polynomial of degree has n roots, then it follows that it has at least a root and thus the second statement follows from the first. Conversely, if the polynomial has a root at all, say x =  then a0n + a1n - 1 + a2n -2 +…… an-1 +an = 0 so that a0xn + a1xn - 1 + a2xn -2 +…… an-1x +an = a0xn + a1xn - 1 + a2xn -2 +…… an-1x +an - a0n - a1n - 1 - a2n -2 -…… an-1 -an = a0 (xn -n ) + a1 (xn – 1 - n – 1)+ a2 (xn - 2 - n - 2 ) +…… an-1 (x - ) and it is easily seen that (x - ) is a factor of the polynomial as it is a factor of the expression on the right. Thus the two statements are deduced from each other and are equivalent. D. Scientific methods : a) Philosophy of science : 15
  16. 16. The Series Editor Rediscover Mathematics From 0 And 1 Series Scientific attitude has been different from non-scientific from time immemorial identifying itself with revolutionary thought process or thinking differently from established line of thought nurturing inquisitiveness and not to accept anything without experimental verification even if it is logically sound; a logically consistent statement may be accepted ad-hoc and it must be put to experimental test as soon as possible. It is a philosophical doctrine fundamental to the school of logical positivism holding that a statement is meaningful only if it is either empirically verifiable or else tautological (i.e., such that its truth arises entirely from the meanings of its terms).A metaphysical, aesthetic, religious or ethical statement either may be tautological (true by logic alone) or experimentally verifiable to be accepted. All sciences share the same language, laws, and method or at least one or two of these features . A unity-of-science movement arose in the Vienna Circle, a group of scientists and philosophers that met regularly in Vienna in the 1920s and '30s reiterated this view. It is well known that Socrates was poisoned for telling what was called heresy. But Mathematics shall accept whatever is logically true – whether endorsed by experiment or not; so that the logic may be reserved for future use. For example, the theory of tensors and Maxwell’s transformations were used by Einstein for formulating the theory of relativity. b) Experiments and observation All scientific terms could be restated as, or reduced to, a set of basic statements, or “protocol” sentences, describing immediate experience or perception. Unity of language has meant the reduction of all scientific terms to terms of physics. The unity of law means that the laws of the various sciences are to be deduced from some set of fundamental laws, often thought to be those of physics and that the procedures for testing and supporting statements in the various sciences are basically the same. ‘Observable ‘ element in a particular statement may be evident from the following example. A pebble is thrown into the air at a particular speed and with a particular inclination with the horizontal direction. A mere application of laws of gravitation would have been sufficient to describe the motion of the pebble thereafter; only if a bird suddenly appears from nowhere to collide with it. But that does not fail the laws of gravitation or mechanics or physics or science for that matter. If the experiment is conducted in a protected space, the results observed shall be repeated; at least be consistent with each other. So the failure of prediction in this example does not negate the physical laws; it only states that prediction of future is no business of science, no matter even if it is true 10 out of 10 times! We would prove in a latter chapter that Arithmetic average of a number of observations shall approximate the measurement of the observable more closely as the number of observations increases. Generally the number of observations is limited to 10 in any standard experimental procedure. c) Analysis and synthesis ; Analysis proceeds from one to many and synthesis proceeds from many to one. 1 (5)  (5)  0,...or....( 4) x    1 ; what does this mean ? Anything together with its antithesis is found 4 ‘inside’ the identity – the two placed symmetrically with respect to each other. So symmetry may be regarded as finer details or may find its synonym in ‘analysis’, whereas the identity may be regarded as a place holder for symmetry or may be regarded as ‘synthesis’ of all theses and their antitheses. The statement of the seer Einstein fits into these lines ; the ‘difficulty’ and the ‘solution’ are antitheses of each other and they coexist together so that it suffices to find out (or stumble upon) only one of them which would eventually lead to the other in order that it might have some worth or meaning at all. The writer is too small being to guarantee about finding antithesis of any problem so easily. However, to believe in the statement proves a good starting point. Analysis or being interested in small details is out of inquisitiveness, in elementary stage. After a good deal of analysis is done, we try to be mature and synthesize. A bright example is Vedic Mathematics by His Holiness Sri Sri Jagadguru Shankaracharya Bharathikrishna Tirthaji Maharaj – that reduces all mathematics to breeze ; with just a finger count of 19 sutras and a vedic code, of course. Without synthesis how anyone goes to know the end or at least the 16
  17. 17. The Series Editor Rediscover Mathematics From 0 And 1 Series frontiers of knowledge, when a full lifetime is not sufficient to study say, only Geometry ? So this book is an humble endeavour to synthesize complex numbers, vectors, coordinate geometry and matrices into one entity of study; and to synthesize uniform circular motion, simple harmonic motion, elasticity, planetary motion, electrostatics, and magnetostatics all under the banner of Central Forces. And the entire book may be regarded as either a book on Functions or one on Theory of Equations, or one on Calculus or one on Physics or one on Vector Analysis; with Algebra, Trigonometry, Coordinate Geometry, Probability, Statistics, and Differential Equations etc. regarded as limbs on the many parallel skeletons. The grand unification theory which is yet to be developed in fullness is an endeavour towards the aim of Physics to be able to derive explanations for the apparently diverse phenomena of the four fundamental forces of nature from one source; striving towards one identity for all the four forces – in view of the intrinsic similarities among themselves. This ‘reductionism’ is generalization of concepts of ‘identity’, ‘invariants’, ‘ conservation laws’ and the ‘ intrinsic similarities’ is generalization of symmetry or beauty leading to identity or invariance. ‘Pin pointing difficulty and enclosing it in a name’ – exemplified by solution of quadratic equation in the beginning of this chapter - how does the statement fit into the statement ‘ analyzing identity we arrive at symmetry and synthesizing symmetry we get at identity ? It is this way - ‘the difficulty’ and ‘the solution’ are symmetrical counterparts of the ‘whole picture’ characterised by its own identity. Further examples of symbolization include determinants, matrices, logarithms, inverse circular functions and so on. Only a definition of the symbol tacitly contains all its properties and the rules associated with it – just in a manner , as taking a pack of 52 cards, setting up a hierarchy of order and a set of rules; rest of the game is just a matter of calculations. Super-symmetry and anti-symmetry : d) Particles have broadly been divided into two types, fermions having spin quantum number ½ or – ½, and bosons are the other type, having spin numbers  1.The symmetry we have referred in connection with conservation of momentum etc. means that total momentum after a collision (say) is equal to total momentum before collision. A similar ‘super-symmetry’ is displayed in subatomic level. Fermions can be transformed into bosons without changing the structure of the underlying theory of the particles and their interactions and vice versa ( i.e., back again to fermions). Super-symmetry provides a connection between the known elementary particles of matter (quarks and leptons, which are all fermions) and the messenger particles that convey the fundamental forces (all bosons). Thus it shows that one type of particle is in effect a different facet of the other type. Super-symmetry reduces the number of basic types of particle from two to one. This feature has in fact encouraged to look for a grand unification theory. Another advantage of super-symmetry is that it requires a particular fermion must have a super-symmetric conjugate boson particle somewhere. This has doubled the number of types of fundamental particles known and compelled scientists to look for an anti-particle of every particle discovered or postulated. Super-string theory of 1970’s which assumes strings as ultimate extended particles to explain all the four fundamental forces has been supported by concept of super-symmetry, although it has problems to sort out. It works in a field of 10 dimensions, , 6 of which are assumed to be truncated to very short distances within which quarks have a free play. This is rather strange and asymmetrical considerations to explain the most recent ‘theory of everything’. But any asymmetry, as it were, promises of more subtle symmetry. For example, Higgs suggested ‘ breaking of symmetry’ for explaining week nuclear reactions where ‘parity’ did not seem to be conserved and this led to the concept of super-symmetry in course of development. In Mathematics, we come across non-commutative algebraic structures or operations more often than not. Not only it is tolerated with open mind, but is sincerely accepted as it gives philosophical insight and is meant for theoretical development after all. 17
  18. 18. The Series Editor Rediscover Mathematics From 0 And 1 Series E.Division: Symmetry may be considered as a division of identity. Subtraction may be viewed as a division into unequal parts and can be represented by ordered pair (a, b) standing for a – b. A rational number is a division and can be represented by an ordered pair. An irrational number is a continued fraction, again a division. A complex number can be represented by an ordered pair, so it is some sort of division. So vectors and matrices are. Quaternions were discovered as vector division. The sequences and series may be thought of as some sort of division . e.g, 1  1  x  x 2  x 3  x 4  ............ . The entire Calculus may be viewed as some sort of division, 1 x limits of division. In a locus in Coordinate Geometry the point varies but the scheme, or the equation represents the entire locus; So it is a concept of entirety ,similar to the concept of identity, invariants, constants fixed points etc. A function f(x) =0 may be written as x = φ(x); the roots of the former are fixed points of the later. In short, the writer believes Mathematics and Physics are visible with the telescope or microscope of identity–division–symmetry . The difficulty and its removal together constitute a division of knowledge. Symmetry is a division into symmetrical parts of a whole. Symmetry and identity are only different views of looking at the same thing. By saying momentum is conserved in a collision is same thing as saying total momentum before collision is equal to the total momentum after collision. It is the invariance laws or conservation laws of momentum and energy produce differential equations of motion which in turn, yield the equations of motion when solved. Thus identity – division – symmetry is the only formula for ultimate knowledge of Nature. 18