MHF 3404  – HISTORY OF MATHEMATICS                     FLORIDA INT'L UNIV.
      Review for Test #2 (April 9th)                                                                    FALL  2009
      

     REMEMBER TO BRING AN 8’’x11”  BLUE EXAM BOOKLET FOR THE TEST


             KEY CONCEPTS AND MAIN DEFINITIONS: (Test #2: Ch.8-Ch.12), Ch13

     Binomial theorem, Pascal's Indian-Chinese triangle, Pascal's identity, Unitary problems I,
     Chinese Shortfall & Excess method, Simultaneous equations involving 2 or 3 unknowns,
     Quadratic equations, Unitary Problems II, multiply the fruit by the desire and divide by
     the measure, Brahmagupta's Rule, Positive and Negative numbers as profits and debts,
     Solving Linear & Quadratic Diophantine equations, The Pell Equation: ax2+b = y2,  ibn

     Mun'im solutions of combinatorial problems by Dynamic Programming, Counting paths
     of length (i+j) from (0,0) to (i,j) by using Dynamic Programming & by using permutations
     of multi-sets, The six Trigonometric Functions,  The Law of Sines, Leonardo Bigollo of
     Pisa (aka Leonardo Fibonacci) and the introduction of the Hindu-Arabic numerals, The

     Rabbit problem and Difference Equations (Recursive Relations), Complex numbers, The
     Cube roots of Unity, The Renaissance, Solutions of Polynomial Equations of degree less
     than 5, the discriminant D, the nature of the solutions of a Cubic Equation, del Ferro and
     Fontana's (Tartaglia) solutions to the cubic, Ferrari's solution to the quartic, Cardano's
     exposition of the methods in his "Great Arts". Prosthaphaeresis (plus&minus) of sinAsinB,
     The origins of logarithms; Napier, Burgi, & Briggs, Prime numbers, divisibility of numbers
     by a give integer, Descartes, Fermat, sums of powers, Fermat's Last Conjecture.
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      Kepler, Galileo, Tangent & Quadrature Problems, The origin of the Calculus, Barrow,
      Newton's method of fluxions, Leibnitz's infinitesimals, Robinson's modern theory of
      infinitesimals, The Hyper-real numbers, The Bernoulli family, Bernoulli's 0/0 Rule (aka
      L'Hospital's 0/0 Rule), The Brachistochrone & Tautochrone Problems, Euler - el maestro
      de infinitesimales.

 
               MAIN PROBLEM SOLVING TECHNIQUES:      

1.      Proving various Binomial identities using Algebraic and Combinatorial methods (Liu Hui)
2.   Solving problems using the shortfall & excess method and simultaneous equations.
3.   Solving unitary problems involving three variables by using Brahmagupta's rule.

4.   Finding more integer sol. of  ax2 +1 = y2   from a given solution by Bhaskara's method.
5.   Finding solutions to Combinatorial Problems by Dynamic Programming  & other methods
6.   Finding relations between the six trigometric functions:  e.g., cos(x)  =  1/ [1+ tan2(x)]1/2.
7.   Finding solutions to Linear Difference Equations (with init.cond.) by using the E-method.

8.   Finding solutions to Reduced Cubic Equations with non-negative & negative discriminants.
9.   Determining the nature of the solutions of a Reduced Cubic Equation without solving it.
10. Finding the cubic resolvant of a Reduced Quartic Equation and the nature of its solutions.
11. Proving properties of the trig. functions by using the identity  eix = cos(x) +  i sin(x).
12. Proving properties of logarithmic functions by using inverses and the laws of exponents. 
13. Determining if a number is divisble by 2, 3, 4, ...,11; and doing Arithmetic (mod k).

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14. Using Robinson's infinitesimals to find limits of functions & dervatives of functions
15. Using infinitesimals and sums to find integrals of functions and to solve other problems.


                MAIN FORMULAS AND THEOREMS:
1.    (a+b)n  =  SUM k = 0 to n {C(n,k) . an-k. bk},     C(n,k) =  C(n-1,k-1) + C(n-1,k)
2.    Answer = Cross-product /(Shortfall + Excess),       (Pi)Brahmagupta  = 3.1416
3.    Transpose the fruits, L = product of larger set, S= product of smaller set, Answer=L/S
4.    (x0,y0)= given solution to ax2+b=y2 & b0=b;  x1 = 2.x0.y0, y1=a.(x0)2+(y0)2, b1=(b0)2
        In general,  xn+2 = xn.yn+1 +  xn+1.yn ,  yn+2 = a.xn.xn+1 +  yn.yn+1 ,  bn+2 = bn.bn+1
5.     No. of paths of length (i+j) from (0,0) to (i,j) =  No. of  perm. of the multi-set [i.R, j.U] 
6.     sin2(x)+cos2(x)=1, tan(x)=sin(x)/cos(x), sec(x)=1/cos(x), csc(x)=1/sin(x), cot(x)=1/tan(x)

7.    (a)  If r1 and r2 are distinct roots of the auxiliary equations, then an =  A.(r1)n + B.(r2)n . 
         (b)  If r1 and r2 are the same roots of the auxiliary equations, then an =  (A+Bn).(r1)n.
8.    Reduced Cubic equation:  y3 + p.y + q = 0;  Discriminant: D =  [(p/3)3 + (q/2)2].

      (a) If D>0, then  y0 = u+v,  y1 = -(u+v)/2  +  i.31/2.(u-v)/2,  y2 = -(u+v)/2  -  i.31/2.(u-v)/2
            where u = [(-q/2) + D1/2 ]1/3 and  v = [(-q/2) - D1/2 ]1/3 .

      (b) If D<0, then yk = 2.r1/3.cos [(A+ 2.PI.k)/3], where r = (-p/3)3/2 and cos(A) = (-q/2r)

9.    (a)  If p = 0 and D = 0, then q = 0 and there is one real root, y = 0, of multiplicity 3.
       (b)  If p is not zero and D = 0, then there are two real roots, with one of multiplicity 2.        

       (c)  If D>0, then there is one real root and two conjugate complex roots.  

       (d)  If D<0, then there are three distinct real roots.

 
10.  Reduced Quartic Eq.: y4 + p.y2 + q.y + r = 0, Resolvant: z3 + 2p.z2 + (p2-4r).z - q2 = 0,
               2y1 =  (z1)1/2 + (z2)1/2 + (z3)1/2         2y2 =  (z1)1/2 - (z2)1/2 - (z3)1/2  
               2y3 = - (z1)1/2 + (z2)1/2 - (z3)1/2         2y4 = - (z1)1/2 - (z2)1/2 + (z3)1/2   
       (a) If the resolvant has 3 non-negative real solutions, then the quartic has 4 real solutions
       (b) If the resolvant has 1 non-negative & 2 negative real solutions, then the quartic has 2
             pairs of conjugate complex solutions.
       (c) If the resolvant has 1 real root (non-neg or neg.) and  2 conjugate complex solutions,
            then the quartic has 2 real solutions and one pair of conjugate complex solutions.
       (d) Resolvant cannot have 2 positive. & one neg. real roots because prod. of roots =q2

11.   cos(2x) + i.sin(2x) = e2ix = (eix)2= [cos(x) + i.sin(x)]2 = (cos2x -sin2x) + i.2cos(x)sin(x)

        So by equating real & imag. parts:  cos(2x)  = (cos2x -sin2x) & sin(2x) = 2cos(x)sin(x)

12.   logb(x.y) = logb(x) + logb((y),   logb(x/y) = logb(x) - logb(y),   logb(xr) = r.logb(x)   
13.   Testing a number for divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, and11 - for example,

        abcd is divisible by 7   if and only if   abc - 2(d) is divisible by 7.
14.   Fact 1:  x = y (mod k) if and only if  x-y = an integer multiple of k.

        Fact 2:  If hcf(q,r)=1, then k divides q.r  ifand only if  k divides q or k divides r.
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15.   (a) lim x_>a f(x) = stp[f(x+*)]  (b) f '(x) = stp{[f(x+*) - f(x)]/*}, where * is infinitesimal

16.    Integral of f(x) from a to b  =  stp{SUMk= 1 to N  f[a + k.(b-a)/N]. [(b-a)/N]} ,
         where N is an infinite positive integer.