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Mathematics Dictionary
Dr. K. G. Shih
Reviews
Subjects

  • AN 01 00 | - Outlines
  • AN 01 01 | - Numbers
  • AN 01 02 | - Inequality
  • AN 01 03 | - Absolute
  • AN 01 04 | - Quadratic equation
  • AN 01 05 | - Trigonometry : Identities
  • AN 01 06 | - Trigonometry : Equations
  • AN 01 07 | - Conic sections
  • AN 01 08 | - Law and theory
  • AN 01 09 | - Translation and rotation
  • AN 01 10 | - References
    • Answers


    AN 01 01. Numbers

    Integers
    • Negative integers : -1, -2, -3, ...
    • Positive integers : +1, +2, +3, ...
    Rational numbers
    • Rational number : r = n/m and m and n are integers
    Irrational number
    • It can not be expressed as ratio of two integers.
    • Examples : Sqr(2), Sqr(3), pi, ....
    • Rationalize irrational number : Change denominator to rational number
      • Example : Rationaize n = 1/(2 - Sqr(3))
      • Multiply numerator and denominator by (2 + Sqr(3))
      • n = (2 + Sqr(3))/((2 + Sqr(3))*(2 - Sqr(3))
      • n = (2 + Sqr(3))/(4 - 3)
      • n = (2 + Sqr(3))
    Real numbers
    • Real numbers contains integers, rational numbers and irrational numbers
    • All real numbers have a position on the number line
    • Real number does not include a/0 or 0/0
    • Real number does not include Sqr(x) if x is less than zero
    • Real number does not include log(x) if x is less or equal zero
    Decimal numbers
    • Repeating decimals can be as expressed ratio of two integers
    • Example : 0.33333 ..... = 1/3
    Complex number
    • Complex number is defined as a + b*i
      • a and b are real numbers
      • i = Sqr(-1)
      • Conjugate complex of a + b*i is a - b*i
        • Sum of them is real 2*a
        • Product of them is real a^2 + b^2
      • Rationalize an expression is to make denominator a real number
        • Rationalize z = 1/(2 + i)
        • Multiply numerator and denominator by (2 - i)
        • z = (2 - i)/((2 + i)*(2 - i))
        • z = (2 - i)/(2^2 - i^2)
        • z = (2 - i)/(4 - (-1))
        • z = (2 - i)/5
    Diagrams
    Go to Begin

    AN 01 02. Inequalties

    Rules of inequality
    • 1. if a < b then a + c < b + c
    • 2. if a < b and c < d then a + c < b + d
    • 3. if a < b and c > 0 then a*c < b*c
    • 4. If a < b and c < 0 then a*c > b*c
    • 5. If 0 < a < b then (1/a) > (1/b)

    Go to Begin

    AN 01 03. Absolute

    Absolute
    • If |x| = 1, then x = 1 and x = -1
    On number line : |x| < 2
    • It means -2 GT x LT 2
    • It is line section : Open interval (-2,2)
    • It is line section on number line : At ends of section use a small circle
    • Diagram : 21 01
    On two number line : |x| < 2
    • It means values are between x = -2 and x = 2
    • It is area bounded by lines x = -2 and x = 2 : boundary use dashed lines
    • Diagram : 21 01
    More definitions
    • On number line : -2 GE x LE 2
      • It is closed interval on number line
      • It is [-2,2]
      • The ends of line sections are small solid circles
    • On two number lines : -2 GE x LE 2
      • It is the area between x = -2 ans x = 2
      • The boundary of the area solid lines
    Diagrams
  • Graphs in Analytic geometry Section 1 in anlytic geometry

    Go to Begin

    AN 01 04. Quadratic equation and function

    Definition
    • Quadratic equation : It is 2nd degree equation a*x^2 + b*x + c = 0
    • Quadratic formula
      • x1 = (-b + Sqr(b^2 - 4*a*c))/(2*a)
      • x2 = (-b - Sqr(b^2 - 4*a*c))/(2*a)
    • Discriminant : D = b^2 - 4*a*c
      • D GT 0 : it has two different real roots
      • D EQ 0 : it has two same real roots
      • D LT 0 : it has two different complex roots (roots are conjugate)
    • Equation theory : Relation of roots r,s with coefficients a,b,c
      • Since a*x^2 + b*x + c = (x - r)*(x - s)
      • Hence Sum of roots = r + s = -b/a
      • Product of roots = r*s = c/a
    Graph of quadratic functions
  • Graphs in Analytic geometry Section 1 in algebra
    Example 1 : Compare y = (x^2)/2 and y = -(x^2)/2
    • y = (x^2)/2
      • It is a parabola
      • Vertex at (xv,yv) : xv = 0 and yv = 0
      • It opens upward
      • Focus at (xf,yf) : xf = xv and yf = yv + D/2 where D =1/(1/2) = 2
      • Equation of directrix is y = yv - D/2 = -1
    • y = -(x^2)/2
      • It is a parabola
      • Vertex at (xv,yv) : xv = 0 and yv = 0
      • It opens downward
      • Focus at (xf,yf) : xf = xv and yf = yv - D/2 where D =1/(1/2) = 2
      • Equation of directrix is y = yv + D/2 = 1
    Example 2 : Compare x = (y^2)/2 and x = -(y)^2/2
    • x = (y^2)/2
      • It is a parabola
      • Vertex at (xv,yv) : xv = 0 and yv = 0
      • It opens to the right
      • Focus at (xf,yf) : xf = xv + D/2 and yf = yv where D =1/(1/2) = 2
      • Equation of directrix is x = xv - D/2 = -1
    • y = -(x^2)/2
      • It is a parabola
      • Vertex at (xv,yv) : xv = 0 and yv = 0
      • It opens to the left
      • Focus at (xf,yf) : xf = xv - D/2 and yf = yv where D =1/(1/2) = 2
      • Equation of directrix is x = xv + D/2 = 1

    Go to Begin

    AN 01 05. Trigonometry : Identieis

    Trigonometric ratio : Right triangle ABC and angle C = 90
    • sin(A) = Opp/Hyp and csc(A) = 1/sin(A)
    • cos(A) = ADj/Hyp and sec(A) = 1/cos(A)
    • tan(A) = Opp/Adj and cot(A) = 1/tan(A)
    Pythagorean relation
    • cos(A)^2 + sin(A)^2 = 1. It is unit circle
    • tan(A)^2 + 1 = sec(A)^2. It is unit hyperbola
    • cot(A)^2 + 1 = csc(A)^2. It is unit hyperbola
    Sum and different angles
    • sin(A+B) = sin(A)*cos(B) + cos(A)*sin(B)
    • sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B)
    • cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B)
    • cos(A-B) = cos(A)*cos(B) + sin(A)*sin(B)
    Half angle
    • sin(A/2) = Sqr((1 - cos(A)/2)
    • cos(A/2) = Sqr((1 + cos(A)/2)
    • tan(A/2) = Sqr((1 - cos(A)/(1 + cos(A))
    Multiple angles
    • sin(2*x) = 2*sin(x)*cos(x)
    • cos(2*x) = 2*cos(x)^2 - 1 = 1 - 2*sin(x)^2
    • tan(2*x) = 2*tan(x)/(1 - tan(x)^2)

    Go to Begin

    AN 01 06. Trigonometry : Equations

    Signs of function
    • sin(x) = (+) | sin(x) = (+)
    • cos(x) = (-) | cos(x) = (+)
    • tan(x) = (-) | tan(x) = (+)
    • -------------|-------------
    • sin(x) = (-) | sin(x) = (-)
    • cos(x) = (-) | cos(x) = (+)
    • tan(x) = (+) | tan(x) = (-)
    General solution : A is the answer in 1st quadrant
    • In 1st quadrant : x = 2*n*pi + A
    • In 2nd quadrant : x = (2*n + 1)*pi - A
    • In 3rd quadrant : x = (2*n + 1)*pi + A
    • In 4th quadrant : x = 2*n*pi - A
    Equations
    • sin(x) = a and -1 GE x LE 1. Principal solution in 1st quadrant is arcsin(Abs(a))
      • Solution for a GT 0
        • x = arcsin(a) in 1st quadrant
        • x = pi - arcsin(a) in 2nd quadrant
      • Solution for a LT 0
        • x = pi + arcsin(a) in 3rd quadrant
        • x = 2*pi - arcsin(a) in 4th quadrant
  • General solution
    • If a is positive
      • x = 2*n*pi + arcsin(x) in 1st quadrant
      • x = (2*n + 1)*pi - arcsin(x) in 2nd quadrant
    • If a is positive
      • x = (2*n + 1)*pi + arcsin(x) in 3rd quadrant
      • x = 2*n*pi - arcsin(x) in 4th quadrant
    • If a is positive and negative
      • x = n*pi - arcsin(x) in 2nd and 4th quadrants
      • x = n*pi + arcsin(x) in 1st and 3rd quadrant

    Go to Begin

    AN 01 07. Conic sections

    Conic sections
    • Equation of circle
      • (x - h)^2 + (y - k)^2 = r^2
      • (h, k) is center and r is radius
    • Equation of ellipse
      • ((x - h)/a)^2 + ((y - k)/b)^2 = 1
      • (h, k) is center and a and b are semi-axis
      • Focal length = f = Sqr(a^2 - b^2)
    • Equation of hyperbola
      • ((x - h)/a)^2 - ((y - k)/b)^2 = 1
      • (h, k) is center and a and b are semi-axis
      • Focal length = f = Sqr(a^2 + b^2)
      • Asymptotes
        • y - k = +b*(x - h)/a
        • y - k = -b*(x - h)/a
    • Equation of parabola
      • y - k = (x - h)/(2*D)
      • (h, k) is center and D is the distance from focus to directrix
      • Focal length = f = Sqr(a^2 + b^2)
      • Vertex is at (h, k-D/2)
      • If y = a*x^2 + b*x + c, then D = 1/(2*a)

    Go to Begin

    AN 01 08. Theory and law

    Geometry
    • Triangle ABC inscribed in circle : Angle ACB = 90 degrees if AB is diameter
    • Pythagorean law : c^2 = a^2 + b^2
    Trignometry
    • Cosine law of triangle ABC : a^2 = b^2 + c^2 - 2*b*c*cos(A)
    • Sine law of Triangle ABC : R is the circum center
      • a = 2*R*sin(A)
      • b = 2*R*sin(B)
      • c = 2*R*sin(C)
    Algebra
    • Binomial theory
      • (x + y)^n = Sum[C(n,r)*(x^(n-r))*(y^r)
      • C(n,r) = n*(n-1)*(n-2)*....*(n-r+1)/(r!)
    • Permutation and combination
      • P(n,r) = n*(n-1)*(n-2)*....*(n-r+1)
      • P(n,n) = n*(n-1)*(n-2)*....*3*2*1
      • C(n,r) = n*(n-1)*(n-2)*....*(n-r+1)/(r!)
      • C(n,n) = 1 = C(n,0)
      • C(n,1) = n = C(n, n-1)
      • C(n,r) = n = C(n, n-r)
      • n! = P(n,n) = n*(n-1)*(n-2)*....*3*2*1
      • Trailor aeros of n! : Int(n/5) + Int(n/(5^2) + Int(n/(5^3) + Int(n/(5^4)
    • Reminder theory : F(x) divide by (x - a), remainder is F(a)
    • Factor theory : F(x) is divisible by (x - a), then (x - a) is a factor of F(x)

    Go to Begin

    AN 01 09. Translation and rotation

    Translation
    • Change the origin to (h,k) from (0,0)
    • The curve does not change.
    • The curve after translation is congruent to the original curve with same direction
    • Example : Translate y = x^2 from (0,0) to (-1,-2)
      • After translation, the curve is y + 2 = (x + 1)^2
      • Sketch the graph : Use AN 21 13
    Rotation
    • Rotation matrix
      • | +cos(A) +sin(A) |
      • | -sin(A) +cos(A) |
    • Rotate from Oxy to Opq
      • p = +x*cos(A) + y*sin(A)
      • q = -x*sin(A) + y*cos(A)
    Example : Compare x^2 - y^2 = -1 and x*y = 1
    • x^2 - x^2 = -1
      • It is a hyperbola with a = 1 and b = 1
      • The princial axis is x = 0
      • The vertex is (xv,yv) : xv = 0 and yv = 0
      • The focal length is f = Sqr(a^2 + b^2) = Sqr(2)
      • Find focus : See AN 08
      • Find equation of driectrix : See AN 08
      • Asymptotes : y = x and y = -x
    • x*y = 1
      • Rotate x*y = 1 to Opq system with origin 45 degrees
        • p = +x*cos(45) + y*sin(45) = Sqr(2)*(+x + y)/2
        • q = -x*sin(45) + y*cos(45) = Sqr(2)*(-x + y)/2
        • Solve for x and y, we have
        • x = Sqr(2)*(p - q)/2 and y = Sqr(2)*(p + q)/2
      • Substitute x and y into x*y = 1
        • (1/2)*(p - q)*(p + q) = 1
        • p^2 - q^2 = 2
      • It is a hyperbola with a = Sqr(2) and b = Sqr(2) after rotation 45 degrees
      • The princial axis is p = q
      • The vertex is (pv,qv) : pv = 0 and pv = 0
      • The focal length is f = Sqr(a^2 + b^2) = 2
      • Find focus : See AN 08
      • Find equation of driectrix : See AN 08
      • Asymptotes : p = q and p = -q
      • Rotate back, we have x*y = 1
        • Asymptotes is x = 0 and y = 0
        • Principal axis is y = x and focal length is 2
        • Equation of directrix is perpendicular to y = x
    • Diagram : See AN 21 01

    Go to Begin

    AN 01 10. References

    Algebra     Geometry Trigonometry
    Go to Begin

    AN 01 00. Outlines

    Algebra
    • a^2 - b^2 = (a - b)*(a + b)
    • (a + b)^2 = a^2 + 2*a*b + b^2
    • (a - b)^2 = a^2 - 2*a*b + b^2
    • Binomial theory
      • (x + y)^n = Sum[C(n,r)*(x^(n-r))*(y^r)
      • C(n,r) = n*(n-1)*(n-2)*....*(n-r+1)/(r!)
    • Permutation and combination
      • P(n,r) = n*(n-1)*(n-2)*....*(n-r+1)
      • P(n,n) = n*(n-1)*(n-2)*....*3*2*1
      • C(n,r) = n*(n-1)*(n-2)*....*(n-r+1)/(r!)
      • C(n,n) = 1
      • C(n,0) = 1
      • n! = P(n,n) = n*(n-1)*(n-2)*....*3*2*1
      • Trailor aeros of n! : Int(n/5) + Int(n/(5^2) + Int(n/(5^3) + Int(n/(5^4)
    • Reminder theory : F(x) divide by (x - a), remainder is F(a)
    • Factor theory : F(x) is divisible by (x - a), then (x - a) is a factor of F(x)
    Geometry
    • Triangle ABC inscribed in circle : Angle ACB = 90 degrees if AB is diameter
    • Pythagorean law : c^2 = a^2 + b^2
    Trignometry
    • Cosine law of triangle ABC : a^2 = b^2 + c^2 - 2*b*c*cos(A)
    • Sine law of Triangle ABC : R is the circum center
      • a = 2*R*sin(A)
      • b = 2*R*sin(B)
      • c = 2*R*sin(C)
    Conic sections
    • Equation of circle
      • (x - h)^2 + (y - k)^2 = r^2
      • (h, k) is center and r is radius
    • Equation of ellipse
      • ((x - h)/a)^2 + ((y - k)/b)^2 = 1
      • (h, k) is center and a and b are semi-axis
      • Focal length = f = Sqr(a^2 - b^2)
    • Equation of hyperbola
      • ((x - h)/a)^2 - ((y - k)/b)^2 = 1
      • (h, k) is center and a and b are semi-axis
      • Focal length = f = Sqr(a^2 + b^2)
    • Equation of parabola
      • y - k = (x - h)/(2*D)
      • (h, k) is center and D is the distance from focus to directrix
      • Focal length = f = Sqr(a^2 + b^2)
      • Vertex is at (h, k-D/2)
    Rotation
    • Rotation matrix
      • | +cos(A) +sin(A) |
      • | -sin(A) +cos(A) |
    • Rotate from Oxy to Opq
      • p = +x*cos(A) + y*sin(A)
      • q = -x*sin(A) + y*cos(A)

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