Successfully reported this slideshow.
Upcoming SlideShare
×

# rediscover mathematics from 0 and 1

986 views

Published on

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

Published in: Education, Technology
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

### rediscover mathematics from 0 and 1

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