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Mathematics Dictionary
Dr. K. G. Shih
Algebraic Formula

A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P |
Q | R | S | T | U | V | W | X | Y | Z |
Topics by Keywords


Q01. A

  • Amicable pairs 220 and 284
    • Sum of factors of 220 : 1 + 2 + 4 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
    • Sum of factors of 284 : 1 + 2 + 4 + 71 + 142 = 220
  • Asymptote of y = F(x)
    • Horizontal asymptote : y = a when x goes to infinite
    • Vertical asymptote : x = a when y goes to infinite
    • Slant asymptote : y = a*x + b when x goes to infinite

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    Q02. B


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    Q03. C


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    Q04. D

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    Q05. E
  • Equation Theory of a*x^2 + b*x + c = 0
    • Roots are r and s
    • Sum of roots : r + s = -b/a
    • Product of roots : r*s = c/a
  • Equation Theory of a*x^3 + b*x^2 + c*x + d = 0
    • Roots are r s and t
    • Sum of roots : r + s + t = -b/a
    • Combination of two roots : r*s + r*t + s*t = c/a
    • Product of roots : r*s*t = -d/a
    • Coefficient of x : 1/r + 1/s + 1/t = c/(a*r*s*t)

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    Q06. F

    • Factor theory : (x-a) is a factor of F(x) if F(a) = 0
    • Factors of expression
      • a^2 - b^2 = (a - b)*(a + b)
      • a^3 - b^3 = (a - b)*(a^2 + a*b + b^3)
      • a^3 + b^3 = (a + b)*(a^2 - a*b + b^3)
      • a^4 - b^4 = (a - b)*(a + b)*(a^2 + b^2)
      • x^5 - 1 = (x - 1)*(x^4 + x^3 + x^2 + x + 1)
      • x^5 + 1 = (x + 1)*(x^4 - x^3 + x^2 - x + 1)
    • Factorial
      • n! = n*(n-1)*(n-2)*....*3*2*1
      • Trailor zeros of n! : Int(n/5) + Int(n/25) + Int(n/125) + Int(n/625) + ...
    • Fibonacci's sequence
      • T(0) = 0 and T(1) = 1
      • T(n) = T(n-1) + T(n-2) and n GT 1
      • Sequence : 1, 1, 2, 3, 5, 8, 13, 21, ....
    • 05 04 Focus of parabola

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    Q07. G

    • 08 02 Geometric series

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    Q08. H
  • Hyperbolic functions
    • cosh(x) = ((e^x) + (e^(-x)))/2
    • sinh(x) = ((e^x) - (e^(-x)))/2
    • tanh(x) = sinh(x)/cosh(x)
  • Hyperbolic functions : Identities
    • cosh(x)^2 - sinh(x)^2 = 1
    • sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)
    • cosh(x+y) = cosh(x)*cosh(y) + sinh(x)*sinh(y)
    • sinh(x-y) = sinh(x)*cosh(y) - cosh(x)*sinh(y)
    • cosh(x-y) = cosh(x)*cosh(y) - sinh(x)*sinh(y)
    • sinh(2*x) = 2*sinh(x)*cosh(y)
    • cosh(2*x) = cosh(x)^2 + sinh(x)^2
  • Hyperbolic functions : Compare with trigonometric function in PM section 14
  • Hypergeometric probability
    • H(x) = C(n,x)*C(N-n,n-x)/C(N,n)
    • H(x) = C(n,x)*C(N-n,s-x)/C(N,s)

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    Q09. I

  • Induction method
    • 14 00 Induction method
    • 14 06 Induction : S(n) = 1/(1*3) + 1/(3*5) + .... = (1 - 1/(2*n+1))/2
    • 14 07 Induction : S(n) = 1/(1*2*3) + 1/(2*3*4) + .... = n*(n+3)/(4*(n+1)*(n+2)
    • 14 08 Induction : S(n) = 1^2 + 3^2 + 5^2 + ... = n*(4*n^2 - 1)/3
    • 14 09 Induction : S(n) = 1*(2^2) + 2*(3^2) + ... = n*(n+1)*(n+2)*(3*n+5)/12
    • 14 10 Induction : S(n) = (1 -4/1)*(1 - 4/9)*(1 - 4/25)*(1 - 4/49)*......
    • 14 11 Induction : S(n) = (1 + 1)*(1 + 1/2)*(1 + 1/3)*.....*(1 + 1/n) = n + 1
    • 14 12 Induction : S(n) = (1^2)/(1*3) + (2^2)/(3*5) + .... = (n*(n+1))/(2*(2*n+1))
    • 14 13 Induction : S(n) = 2 + 4 + 6 + ..... + 2*n = (n + 1/2)^2. Is it true
    • 14 14 Induction : S(n) = n^2 + n + 41. Is it always a prime number ?
    • 14 17 Induction : Prove that 1! + 2! + 3! + 4! + ..... + n! = 3^(n-1)
  • 15 00 Inequality
    • 15 02 Solve x^2 - 6*x + 8 < 0
    • 15 08 Solve (x-1)/(x+1) > 1
    • 15 09 Solve (x-1)/(x+1) > 2
  • Inverse of y = e^x is y = ln(x)
    • e^(ln(x)) = x
    • ln(e^x) = x
  • Inverse of y = ln(x) is y = e^x
    • e^(ln(x)) = x
    • ln(e^x) = x
  • 05 03 Inverse of y = a*x^2 + b*x + c is x = a*y^2 + b*y + c
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    Q10. J


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    Q11. K


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    Q12. L

    • 11 01 Linear equations
    • 09 02 Logarithmic laws

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    Q13. M

    • 03 04 Magic number patterns
    • 03 05 Matrix pattern : Find row number and column number of 100
      • 1, 03, 06, 10, 15, 21, ...
      • 2, 05, 09, 14, 20, .......
      • 4, 08, 13, 19, ...........
      • 7, 12, 18, ...............
    • 03 10 Multiple of number whose digits are same.

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    Q14. N

    • 17 12 Normal distribution : N(z LT a)
    • 03 11 Number : Amicable number pairs.
    • 03 05 Number : Numbers arranged in matrix. 1st row sequence 1,3,6,10, ...
    • 03 12 Number : Perfect numbers.
    • 03 03 Number : Properties based on factors of number.
    • 14 15 Number : Prove that ((n^3) + 3*(n^2) + n)/3 is an integer
    • 14 16 Number : Prove that (2*(n^3) + 3*(n^2) + n) is divisible by 6
    • 07 01 n! : definition and trailor zeros in n!

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    Q15. O


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    Q16. P

    • 05 04 Parabola : Locus, Focus and directrix
    • 08 05 Pascal triangle : Sequence and series
    • 03 12 Perfect number : How to find the third perfect number ?
      • 1st perfect number : 1 + 2 + 3 = 6
      • 2nd perfect number : 1 + 2 + 4 + 7 + 14 = 28
    • 03 07 Perfect square numbers.
    • 17 01 Permutation : P(n,r)
      • 17 02 Take r from N symbols for arrangement without duplicate : P(n,r)
      • 17 02 Take r from N symbols for arrangement with duplicate : n^r
    • 08 13 Pi in series
    • 21 00 Polynomial equation
      • 21 11 Solve x^7 + 2*x^6 - 5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + x + 1 = 0

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    Q17. Q

    • 11 04 Quaint equation
    • 21 01 Quaint equation : Graphic solution
    • 12 06 Quartic equation : (x-p)*(x-q)*(x-r)*(x-s) = t
    • 11 03 Quartic equation : (x-5)*(x-7)*(x+4)*(x+6) = 504
    • 21 01 Quartic function : y = a*x^4 + b*x^3 + ....
    • 21 01 Quartic function : y = (x-p)*(x-q)*(x-r)*(x-s) - t
    • 11 01 Quadratic equations
    • 11 10 Quadratic equations : x^2 -5*x + 2*Sqr(x^2 - 5*x + 3) = 12
    • 05 01 Quadratic functions : Defintion and application
    • 05 02 Quadratic functions : With absolute operation
    • 05 03 Quadratic functions : Inverse
    • 05 04 Quadratic functions : Parabola

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    Q18. R

    • 16 01 Rational function : y = 1/(x + 1)
    • 16 02 Rational function : y = 1/(x^2 - 1)
    • 16 03 Rational function : y = 1/(x^3 -2*x - x + 2)
    • 16 06 Rational function : y = x + 4/(x^2)
    • 16 04 Rational function : y = (x^2 -2*x + 1)/(x)
    • 16 05 Rational function : y = ((x - 1)^3)/(2*x)
    • 03 01 Real number system
    • 01 04 Remainder theory

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    Q19. S

    • 07 04 Sequence in Pascal triangle.
      • T(1)
      • T(n)
      • T(n*(n+1)/2!)
      • T(n*(n+1)*(n+2)/3!)
      • T(n*(n+1)*(n+2)*(n+3)/4!)
    • 08 01 Series : A.P.
    • 08 02 Series : G.P.
    • 08 13 Series : pi
    • 08 03 Series : Sum[n^2] = n*(n+1)*(2*n+1)/6.
    • 08 04 Series : Sum[n^3] = (n*(n+1)/2)^2.
    • 07 08 Series : Sum[n^4] = ?
    • 08 05 Series : Series and sequence in Pascal triangle.
      • Sum[n*(n+1)/2!] = n*(n+1)*(n+2)/3!
      • Sum[n*(n+1)*(n+2)/3!] = n*(n+1)*(n+2)*(n+3)/4!
      • Sum[n*(n+1)*(n+2)*(n+3)/4!] = n*(n+1)*(n+2)*(n+3)*(n+4)/5!
    • 08 05 Series : Series and sequence in Pascal triangle.
      • Sum[C(n+1),2)] = C(n+2,3)
      • Sum[C(n+2),3)] = C(n+3,4)
      • Sum[C(n+3),4)] = C(n+4,5)
    • 08 06 Series : Special
      • S(n) = 1/(1*2) + 1/(2*3) + 1/(3*4) + ........ + 1/((n-1)*n) = ?.
      • S(n) = 1/(1*2*3) + 1/(2*3*4) + ... + 1/((n*(n+1)*(n+2)) = ?
      • S(n) = 1/3 + 1/15 + 1/35 + ...... + 1/((2*n-1)*(2*n+1)) = ?
    • 08 08 Series : 1^3 + 3^3 + 5^3 + ..... (2*n-1)^3 = ?
    • 08 09 Series : 1*(2^2) + 2*(3^2) + 3*(4^2) + ..... n*((n+1)^2) = ?
    • 08 07 Series : S(n) = 1 - 2 + 3 - 4 + 5 - 6 + ....... - n if n is even
    • 08 08 Series : 1^3 + 3^3 + 5^3 + .... = ?
    • 08 09 Series : 1*(2^2) + 2*(3^2) + 3*(4^2) + .... = ?
    • 08 10 Series : 1^3 + 2^3 + ... + n^3 GT (n^4)/4 GT 1^3 + 2^3 + ... +(n-1)^3
    • 08 11 Series : (1-1/4)*(1-1/9)*(1-1/16)*....*(1-1/((n+1)^2 = (n+1)/2*n. n GT 1
    • 08 12 Series : 1 - 2 + 3 - 4 + ...... + n and n is odd
    • 11 10 Simultaneous equations
      • x + y = 3 (line)
      • x*y = 2 (Hyperbola)
    • 11 10 Simultaneous equations
      • x - y = 1
      • x*y = 2 (Hyperbola)
    • 11 10 Simultaneous equations
      • x^2 + y^2 = 4 (Circle)
      • x*y = 1 (Hyperbola)
    • 16 06 Slant asymptote in y = x + 4/(x^2)
    • 03 07 Square free numbers.
    • 03 08 Square root : How to find Sqr(3) ?

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    Q20. T


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    Q21. U


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    Q22. V

    • 05 05 Vertex of quadratic function
    • 16 06 Vertical asymptote in y = x + 4/(x^2)

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    Q23. W


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    Q24. X


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    Q25. Y


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    Q26. Z


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