Mathematics Dictionary

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Mathematics Dictionary
Dr. K. G. Shih

Mathematics by Subjects
Mathematics by Keywords

Glossary by Keywords

Kewords in Alphabetic Order

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 |

A. Keywords

Absolute values

|x| = a means x = a or x = -a
|x| > a means x > a or x < -a
|x| < a means x < a or x > -a

Amicable number pairs

Sum of factors of n1 = n2, sum of factors of n2 = n1.
Then n1 and n2 are amicable pairs

Arithmetic series

a + (a+d) + (a+2*d) + (a+3*d) + .... (a+(n-1)*d)

Associative axiom

Addition : (a + b) + c = a + (b + c)
Multiplication : (a*b)*c = a*(b*c)

B. Keywords

Binary number

It is the base 2 number system.
It has digit from 0 to 1. Hence 1 + 1 = 10 in binary system.

Binomial distribution

In a sequence of n Bernoulli trails with probability of success p.
The probability of exactly k successes in n trials is
C(n,k)*(p^k)*(1-p)^(n-k)

Binomial theorem

(x+y)^n = Sum[C(n,r)*(x^(n-r))*(y^r)]
C(n,r) = n*(n-1)*(n-2)*...*(n-r+1)/r!

C. Keywords

Circle

Circumferance is 2*pi*r where pi = 3.1416
Area = pi*r^2
Equation : (x-h)^2 + (y-k)^2 = r^2
Locus 1 : Moving point P to fixed point is constant.
Locus 2 : Fixed point A and B. Moving point P which makes angle APB = 90.
Locus 3 : Fixed point A and B. Moving point P which makes AB^2 = AC^2 + BC^2.

z = a + b*i is complex nmuber. Where a and b are real number.
Imaginary number : i = Sqr(-1) and i^2 = -1.
The symbols e, i, pi were introuced into mathematics by Euler.
Conjugate of a + b*i is a - b*i

A subset of r elements chosen from a set of n element is C(n,r).
Take 3 elements from 3 elements is C(3,3) = 1.
Take 3 elements from 3 elements for permutation is P(3,3) = 6.

Commutative axiom

Addition : a + b = b + a
Multiplication : a*b = b*a

Conjugate complex numbers

Conjugate complex numbers are (a + b*i) and (a - b*i)
Sum of conjugate is real and product of conjugate is real

A numerical constant is a fixed number.

Coordiante system

The foundation of algebraic geometry was laid by Rene Descates (1596-1650).
It is used to locate a point in a plane.
The coordiantes are the abscissa and ordinate of point (x,y).

Cosine function and cosine Law

Cosine function is the ratio of adjacent and hypotheses of right triangle
a^2 = b^2 + c^2 - 2*b*c*cos(A)
b^2 = c^2 + a^2 - 2*c*a*cos(B)
c^2 = a^2 + b^2 - 2*a*b*cos(C)

D. Keywords

DeMoivre's theorem

Let cis(A) = cos(A) + i*sin(A)
Theorem 1 : cis(A)^n = cis(n*A) where n is positive integer.
Theorem 2 : cis(A)^(1/n) = cis((2*k*pi+A)/n) where k = 0,1,2....(n-1).

Each square matrix is associated with a number is called determinat

b^2 - 4*a*c is descriminant of y = a*x^2 + b*x + c
b^2 - 4*a*c is descriminant of a*x^2 + b*x*y + *y^2 + d*x + e*y + f = 0

Distance between two points

D = Sqr((x2-x1)^2 + (y2-y1)^2)

If y = F(x), x is the domain and y is the range.

E. Keywords

Ellipse

Equation : ((x-h)/a)^2 + ((y-k)/b)^2 = 1
Locus : fixed point G and F. Moving point P so that PG + PF = 2*a > GF.

Even and odd function

If F(-x) = +F(x), the function is even
If F(-x) = -F(x), the function is odd

The numeral which indicates the number of times the base (as factor).
Exponential function : y = a^x with a^0 = 1.
Exponential function : y = e^x with e = 2.71828182.....
The symbol e was introduced by Euler.

F. Keywords

Factor theory

If F(a) = 0, then (x-a) is a factor of F(x).

0! = 1
n! = n*(n-1)*(n-2)*(n-3)*....*3*2*1
Trailor zeros = n/5 + n/(5^2) + n/(5^3) + ....

Fibonacc's sequence

Sequence : 1, 1, 2, 3, 5, 8, 13, 21, ....
Recursion formula : T(0) = 0, T(1) = 1, T(n+2) = T(n+1) +T(n).

To each value of a variable there is a unique value of a second variable.
If the function is y = F(x), then x is independent and y is dependent variable.
The domain is x and the range is y.

G. Keywords

Geometric series

a + a*r + a*(r^2) + a*(r^3) + ..... + a*(r^(n-1))

H. Keywords

Hexadecimal number

It is the base 16 number system.
It has digit from 0 to 9 plus A,B,C,D,E,F. Hence 1 + F = 10 in hexa system.

Equation : ((x-h)/a)^2 - ((y-k)/b)^2 = 1
Locus : fixed point G and F. Moving point P so that |PG - PF} = 2*a < GF.

I. Keywords

Induction method

A set of numbers contains m, it also contains n.
If the set contains (m+n), then the set contains all positive integers.

Inverse function

The inverse of polynomial function y = F(x) is x = F(y).
The inverse of y = e^(x) is y = ln(x). Hence e^(ln(x)) = x.
The inverse of sin(x) is arcsin(x). Hence sin(arcsin(x)) = x.

J. Keywords

Go to Begin

K. Keywords

Go to Begin

L. Keywords

Law of cosine : a^2 = b^2 + c^2 - 2*b*c*cos(A)
Law of sine : a = 2*R*sin(A)
LE = Less than and Equal to
LT = Less than
Logarithmic function

The logarithm function the the base e is ln(x).
If y = e^x, the x = ln(y).
Change base : log10(x) = ln(x)/ln(10). Where log10 is base 10.

M. Keywords

Mantissa

The fractional part of log10(n)

A matrix is a rectangular array of numbers enclosed by parentheses.

N. Keywords

Negative number

It is used as solution for equation.
It was introduced by Girolamo Cardon (1501-1576).
It is a set of numbers less than zero.

O. Keywords

Octal number

It is the base 8 number system.
It has digit from 0 to 7. Hence 1 + 7 = 10 in octal system.

(2,8) is numbers between 2 and 8. 2 and 8 are not included
It is 2 GT N LT 8

P. Keywords

Parabola

Equation : (y - k) = (x - h)/(2*D). D is distance between directrix and focus.
Locus : Moving point to fixed point F and directrix has same distance.
Directrix : a line perpendicular to the principal axis
The focus and directrix of y = a*x^2 + b*x + c
- D = 1/(2*a)
- Vertex : xv = -b/(2*a) yv = F(xv)
- Focus : xf = xv and yf = yv + D/2
- Directrix : equation of directrix is y = yv - D/2

Pascal Triangle : Coefficients of binomial expansion

1 1 ........................ (x+y)^1
1 2 1 ...................... (x+y)^2
1 3 3 1 .................... (x+y)^3
1 4 6 4 1 .................. (x+y)^4

Join the feet of three alitudes of triangle ABC forming a pedal triangle JKL.

An arrangement of symbols in which repition is not allowed.
P(n,r) = n*(n-1)*(n-2)*....*(n-r+1)
P(n,n) = n!

F(x) = a0 + a1*x + a2*(x^2) + a3*(x^3) + ..... + an*(x^n)
Where a0, a1, a2, .... an are coefficnets which can be real or complex.

Polynomial : Product

(a + b)^2 = a^2 + 2*a*b + b^2
(a - b)^2 = a^2 - 2*a*b + b^2
(a + b)^3 = a^3 + 3*(a^2)*b + 3*a*(b^2) + b^3
(a - b)^3 = a^3 - 3*(a^2)*b + 3*a*(b^2) - b^3

Polynomial : Factoring

a^2 - b^2 = (a + b)*(a - b)
a^3 + b^3 = (a + b)*(a^2 - a*b + b^2)
a^3 - b^3 = (a - b)*(a^2 + a*b + b^2)
a^4 - b^4 = (a + b)*(a - b)*(a^2 + 1)

Pythagorean Law : Right triangle ABC

Deinition 1 : a^2 + b^2 = c^2
Defintion 2
- In a circle, Triangle ABC is right triangle if BC is diameter.
- Other rwo sides related with sine and cosine function
- Hence sine and cosine functions are called chord functions.

Pythagorean triple

Example : 3^2 + 4^2 = 5^2

Q. Keywords

Quadratic equation : A second degree polynomial equation
Quadratic formula

Equation : a*x^2 + b*x + c = 0
Root 1 : x = (-b + Sqr(b^2 - 4*a*c))/(2*a)
Root 2 : x = (-b - Sqr(b^2 - 4*a*c))/(2*a)

R. Keywords

Radians

The length of an arc measured by angle A on unit circle.
One radian = (180 degrees)/pi

If y = F(x), y is the range and x is the domain

Rational exponent : x^(m/n) = nth root of (x^m)
Re-arrangement

Addition : Addends of sum can be rearranged in any order
Multiplication : Multiplicants of produc can be rearranged in any order

Remainder theory

If F(x) is divided by (x - a), the remainder is F(a).

S. Keywords

Sector

Arc length = r*A where A is angle and r is radius
Area of sector = A*(r^2)/2

Significant figure

The number of digits in a number has meaning.
For example, for 7 significant figure the 8th digit has no meaning.

Sine function and sine Law

It is based on relation between chord of circle and and the central angle.
Hence sine function is also called as chord function.
Now sine function is the ratio of the opposite and the hypotheses.
Sine Law : a/sin(A) = b/sin(b) = c/sin(C) = 2*R where R is the radius of ex-circle.

Sloe between two points

Slope = (y2-y1)/(x2-x1)
Slope = 0, two points on horizontal line.
Slope = infinite, two points on vertical line.

Symmetric matrix

The elements of matrix give A(i,j) = A(j,i).

T. Keywords

Transitive law : If x GT y and y GT z then x GT z
Trichotomy axiom : Every real number is either positvie or negative of zero
Trigonometric Ratios

sin(A) = Opp/Hyp and csc(A) = 1/sin(A)
cos(A) = Adj/Hyp and sec(A) = 1/cos(A)
tan(A) = Opp/Hyp and cot(A) = 1/tan(A)

U. Keywords

Union : Elements of set A are also in set B
Unit circle

x = cos(t) and y = sin(t) is a unit circle
This is based on cos(t)^2 + sin(t)^2.

x = tan(t) and y = sec(t) is a unit hyperbola
This is based on tan(t)^2 + 1 = sec(t)^2.

V. Keywords

Variable

Dependent variable : If y = F(x), y is the dependent variable.
Independent variable : If y = F(x), x is the independent variable.

A line segment having an initial point A, a terminate point B.
The direction is from A to B.

Vertex of y = a*x^2 + b*x + c

xv = -b/(2*a)
yv = F(xv) where y = F(x) = b^2 - 4*a*c/(4*a)

W. Keywords

Go to Begin

X. Keywords

Go to Begin

Y. Keywords

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Z. Keywords

Go to Begin

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