Mathematics Dictionary

Mathematics Dictionary
Dr. K. G. Shih

Algebraic Topic Index

Symbol Defintion

...... Example : x^2 = square of x

Keywords .............. Find given keyword by numbers

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

21 05 Absolute : Solve Abs(x^2 - 6*Abs(x) + 8) = 0.5
03 03 Abundant number
03 11 Amicable number pairs : First amicale pairs are 220 and n. Find n
03 11 Amicable number pairs : First amicale pairs are 1184 and n. Find n
04 13 Angle between two lines
04 15 Angle between y = 2*x +3 and its inverse
08 14 AP : Angles of triangle ABC are 3 consecutive GP terms. Find angles
27 03 AP : a^2,b^2,c^2 in AP then 1/(a+b), 1/(a+c),1/(b+c) in AP
27 04 AP
- Sum[t(i)] = A for i=1 to a
- Sum[t(i)] = B for i=1 to b
- Sum[t(i)] = C for i=1 to c
- Find A*a*b*c*(b-c) + B*a*b*c*(c-a) + C*a*b*c*(a-b) if t(i) in AP
08 01 Arithemetic seires
17 00 Arrangement : n different things
03 13 Arrangement : An integer n is expressed as the sum of 3 integers
Asymptotes
- AL 21 03 : y = x + 2/x
- AL 27 05 : y = (x^2 + x - 12)/(x^2 - 4)
- AL 27 03 : y = x^2 + 1/x
- AL 27 12 : y = x^3 + 1/x
- AL 16 05 : Asymptote in parabola form

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Q02. B

17 08 Binomial distribution : B(x) = C(n,x)*(p^x)*(q^n-x))
07 02 Binomial expansion : Coeff of (r+1)th term = C(n,r)
03 13 Binomila expansion : Prove that 5^(2*n) - 24*n - 1 is divisible by 576
07 03 Binomial expansion and Pascal triangle
- 07 04 Sum[C(n+1),2] = C(n+2,3)
- 07 05 Sum[C(n+2),3] = C(n+3,4)
- 07 06 Sum[C(n+3),4] = C(n+4,5)
- 07 04 Sum[C(n+4),5] = C(n+5,6)
07 02 Binomial expansion coefficient
- C(n,0) + C(n,2) + ... = C(n,1) + C(n,3) + .... for n is odd
- C(n,r) = C(n,n-r)
07 01 Binomial theorem : T(n) = C(n,r)*(x^(n-r))*(y^r)
07 08 Binomial theorem : Find Sum[n^4] usimg Sum[C(n+3,4)] = C(n+4,5)
07 09 Binomial theorem : Fibonacci's sequence in Pascal triangle
07 10 Binomial theorem : Series from C(n,r)
07 11 Binomial theorem : Constant oefficient in (x + 1/(x^2))^n
07 12 Binomial theorem : Expand Sqr(1 + x^2) in series form
07 13 Binomial theorem : Use Pascal triangle find coefficients of (x+y)^7
07 14 Binomial theorem : Expand (x+1)^n
07 15 Binomial theorem : Find coefficient (x^3)*(y^5) in expansion of (x+y)^n
07 16 Binomial theorem : Expand 1/(1 + x^2)
07 17 Binomial theorem : C(n,r-1) + C(n,r) = C(n+1,r)
07 18 Binomial theorem : Coeff of x^(r-1), x^r and x^(r+1) are 3 consecutive AP
07 19 Binomial theorem : Coefficients of (x^p)*(y^q)*(z^r) in (x+y+z)^n

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Q03. C

17 03 Combination : C(n,r)
03 02 Complex number system
08 16 Compound annually Interest
12 02 Conjugate complex
- If (a+b*i) and (c+d*i) are conjugate, then c=a and d=-b
- Sum is real 2*a and product is real a^2 + b^2
11 02 Cubic equations
21 01 Cubic functions : Graphic solution

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Q04. D

03 03 Deficient number
12 10 DeMoivre's theory : Solve X^4 + X^3 + x^2 + x + 1 = 0
12 10 DeMoivre's theory : Solve X^5 - 1 = 0
13 00 Determinant
05 04 Directrix of parabola
05 01 Discriminant of y = a*x^2 + b*x + c
02 16 Domain and range

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Q05. E

06 07 Euler function
11 08 Equations : Solve 2 linear equations
11 09 Equations : Solve 3 linear equations
11 05 Equations : Solve |x^2 - 6*|x| + 8| = 0.5
18 11 Equations : Solve x^4 - i = 0
18 12 Equations : Solve x^4 + i = 0
11 07 Equations : Solve x^4 - x^3 + x^2 - x + 1 = 0
11 06 Equations : Solve x^4 + x^3 + x^2 + x + 1 = 0
11 07 Equations : Solve x^n - 1 = 0
11 06 Equations : Solve x^n + 1 = 0
11 07 Equations : Solve x^n - 1 = 0
18 04 Equations : Solve x^5 - 1 = 0
18 07 Equations : Solve x^5 + 1 = 0
18 10 Equations : Solve x^4 - x^3 + x^2 - x + 1 = 0
18 09 Equations : Solve x^4 + x^3 + x^2 + x + 1 = 0
06 09 Exponent : e^x + e^(2*x) + e^y + e^(2*y) = 12
06 11 Exponent : Sketch e^x + e^(2*x) + e^y + e^(2*y) = 12
06 16 Exponential Family
06 04 Exponential Law

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Q06. F

02 11 Factor of expressions
01 04 Factor theory
07 01 Factorial n!
07 09 Fibonacci's sequence in Pascal triangle
05 04 Focus of parabola
02 16 Function : Domain and range
16 09 Function : Domain and range
02 08 Function : Names
02 09 Function : Properties

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Q07. G

08 02 Geometric series
08 15 GP : Angles of triangle ABC are 3 consecutive GP terms. Find angles if r=3

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Q08. H

06 16 Hyperbolic functions : e.g. y = sinh(x)
06 17 Hyperbolic functions : Identities
- Prove that cosh(x)^2 - sinh(x) = 1
- Prove that sinh(2*x) = 2*sinh(x)*cosh(x)
- Prove that sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y)
- Prove that d/dx(sinh(x)) = cosh(x)
17 09 Hypergeometric probability H(x) = C(n,x)*C(N-n,n-x)/C(N,n)
17 10 Hypergeometric probability H(x) = C(n,x)*C(N-n,s-x)/C(N,s)

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Q09. I

19 02 Indefinite equation : One people and one hundred cake
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
08 16 Interest : Compound annually
08 17 Interest : Deposit with fixed amount, rate and period
04 16 Intersection between y = 2*x +3 and its inverse
06 08 Inverse of y = e^x
09 12 Inverse of hyperbolic functions
04 03 Inverse of linear functions
09 03 Inverse of y = ln(x)
05 03 Inverse of quadratic functions

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Q10. J

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Q11. K

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Q12. L

04 02 Linear functions
04 03 Linear equations
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 49 whose digits are same
03 10 Multiple of number whose digits are same

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Q14. N

17 12 Normal distribution : N(z LT a)
03 13 Number : 5^(2*n) - 24*n - 1 is divisible by 576
03 11 Number : Amicable number pairs.
03 13 Number : An integer n is expressed as the sum of 3 integers
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 ?
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
27 11 Quaint equation : y = x^5 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12
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
27 10 Quartic function : y = x^4 + x^3 -4*x^2 + x + 1
27 11 Quaint equation : y = x^5 3*x^4 - 5*x^3 - 15*x^2 + 4*x + 12
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)
27 12 Rational function : y = x^3 + 1/x
27 13 Rational function : y = x^2 + 1/x
27 14 Rational function : y = x^3 + 1/x and y = x^3 - 1/x (compare)
16 04 Rational function : y = (x^2 -2*x + 1)/(x)
16 05 Rational function : y = ((x - 1)^3)/(2*x)
27 05 Rational function : y = (x^2 + x - 12)/(x^2 - 4)
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) ?
02 06 Sythetic division