Mathematics Dictionary

What is Axiom ?
- Mathematical statements are obviously true in any cases without proof.
- Example for addition 1 + 2 = 2 + 1.
What is Law ?
- Mathematical statements are true in any cases but need proof.
- Example : Pythagorean Law in right angle triangle c^2 = a^2 + b^2.
What is Theory ?
- Mathematical statement has been tested & confirmed to be true for related case.
- Example : Factor theory of functions

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AL 01 02. Axioms in algebra

Additive Axiom

If a = b abd c = d, then (a + c) = (b + d).

Multiplicative Axiom

If a = b abd c = d, then a*c = b*d.

Commutative Axiom

For addition : x + y = y + x.
For multiplication : x*y = y*x.
Note : This is not applied to subtraction and division.

Associative Axiom

For addition : (x + y) + z = x + (y + z).
For multiplication : (x*y)*z = x*(y*z).
Note : This is not applied to subtraction and division.

Distributive Axiom

For addition : (x + y)*z = x*z + y*z.
For multiplication : x*y = y*x.
Note : This is not applied to subtraction and division.

Identity

For addition :
- x + 0 = x.
- x - x = 0
For multiplication :
- x*1 = x.
- x*(1/x) = 1.

Negative sign

(-x)*(-y) = +x*y
(-x)*(+y) = -x*y
Where * stands for multiplication

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AL 01 03. Laws in algebra

1. Laws of exponent : It is in Lesson 06 05
2. Laws of logarithm : It is in Lesson 09 02
3. Pythagorean law : In right triangle c^2 = a^2 + b^2
4. Triangle inscribed circle : Angle C = 90 degrees if AB is diameter

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AL 01 04. Theories in algebra

1. Factor thoery of functions F(x).

If F(a) = 0, then (x-a) is a factor of F(x)
Hence F(x) = (x-a)*G(x).
If F(-a) = 0, then (x+a) is a factor of F(x)
Hence F(x) = (x+a)*G(x).
Example : Prove that x = 2 is a factor of F(x) = x^2 - 6*x + 8
- F(2) = 2^2 - 6*2 + 8 = 0
- Hence (x-2) is a factor of F(x)
- Hence F(x) = (x-2)*(x-4).

2. Remainder theory

If F(x) is divided by (x-a), then remainder is F(a).
If F(x) is divided by (x+a), then remainder is F(-a).
Example : F(x) = x^2 - 6*x + 8 is divided by (x-3), find remainder.
- F(3) = 3^2 - 6*3 + 8 = -1.
- F(x) divide by (x-3), the remainder is -1.

3. Binomial theory

(x+y)^n = x^n + C(n,1)*x^(n-1)*y + C(n,2)*(x^(n-2))*(y^2) + ..... + y^n.

Where C(n,r) = n*(n-1)*(n-2)*...*(n-r+1)/n!

Study subject | Binomial theorem.
Where C(n,r) = n*(n-1)*(n-2)*...*(n-r+1)/n!
Study subject | Binomial expansion and number sequences.

4. Equation theory of quadratic equation

Let roots be r and s
Then a*x^2 + b*x + c = 0
Then x^2 + (b/a)*x + c/a = (x - r)*(x - s) = 0
Then x^2 + (b/a)*x + c/a = x^2 - (r + s)*x + r*s = 0
Equation theory
- Sum of roots = r + s = -b/a
- Product of roots = r*s = c/a

5. DeMoivre's theory : It is used in complex expression

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AL 01 05. Special numbers

Common used numbers

Power : x^0 = 1
Exponent e^0 = 1
Exponent e^1 = 2.71828182....
Logarithm : Log(1) = 0 where Log is sign of logarithm with any base
Natural logarithm : ln(1) = 0 where ln(x) is logrithm of x with base e
Factorial : 0! = 1 and 1! = 1
Iminary number : Sqr(-1) = i where Sqr = square root sign
pi = 3.141592....

Logarithmic numbers (Base 10)

Log10(10) = 1
Log10(100) = 2
Log10(1000) = 3
Log10(10000) = 4
Log10(0.1) = -1
Log10(0.01) = -2
Log10(0.001) = -3
Log10(0.0001) = -4

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AL 01 06. Product formula

Basic common used formua

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

Basic common used formua

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

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AL 01 07. Absolute value

Abs(-1) = |-1| = 1 where Abs is the sign for absolute
Abs(x) = 1
- x = 1
- x = -1

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AL 01 08. In-equality

Sign used in the web page

EQ = Equal
GE = Greater than and equal
GT = Greater than
LE = Less then and equal to
LT = Less than

Interval of set of number

Open interval from 1 to 4
- It is expressed as (1,4)
- It also expressed as 1 GT x LT 4
- On number line : Use small circle at x = 1 and x = 4
Close interval from 1 to 4
- It is expressed as [1,4]
- It also expressed as 1 GE x LE 4
- On number line : Use small solid circle at x = 1 and x = 4
Open and close
- (1,4] means 1 GT x LE 4
- [1,4) means 1 GE x LT 4
- On number line : Use small circle for open and solid circle for close

Rules about in-equality

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)

Diagrams

Diagrams AN 21 01 : In-equality and absolute

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AL 01 09. The real number system

Number line : All real number has position on number line

One number line
Two number line

Not allowed cases

1. 1/0 is not allowed
2. 0/0 is not allowed
3. Sqr(x) and x LT 0 is not allowed
3. Log(x) and x LE 0 is not allowed

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Q10. Answer

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Q00. Outlines

Axiom

Additive Axiom : If a = b abd c = d, then (a + c) = (b + d).
Multiplicative Axiom : If a = b abd c = d, then a*c = b*d.
Commutative Axiom
- For addition : x + y = y + x.
- For multiplication : x*y = y*x.
Associative Axiom
- For addition : (x + y) + z = x + (y + z).
- For multiplication : (x*y)*z = x*(y*z).
Distributive Axiom
- For addition : (x + y)*z = x*z + y*z.
- For multiplication : x*y = y*x.
Identity
- For addition :
  - x + 0 = x.
  - x - x = 0
- For multiplication :
  - x*1 = x.
  - x*(1/x) = 1.

Theory

Factor theory : If F(a) = 0 then (x-a) is a factor of F(x)
Remainder theory : If F(x) is divided by (x-a), the remaider id F(a)

Laws

Laws of exponent
Laws of logarithms
Pythagorean law of right triagle : c^2 = a^2 + b^2

Product formula

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

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