Three mysterious Math Objects:
- Pi,
- Complex Numbers,
- Prime Numbers
are hiding in circle.
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Tag Archives: Complex Number
Mystery of i
1. 17AD – Cardano: (stole it from other) for cubic equation.
2. 18AD – Lebniz: i is like “Holy Spirit” between ‘there’ and ‘not there’.
3. 19AD – Gauss: i as Geometry
Point (a,b) in plane RxR :
z=a+ib
a on x-axis, b on i-axis
4. 20AD – Hamilton:
i as ordered-pair (a,b) with unique operations:
(a, b)+(c,d)=(a+c, b+d)
(a, b).(c,d)=(ac-bd, bc+ad)
Artin Field Extension
Emile Artin’s very unique book “Galois Theory” (1971) on “Finite Field Extension” interpreted by Vector Space.
Let H a Field with subfield G
F is G’s subfield:
H ⊃ G ⊃ F
Example:
Let
F = Q = Rational Field
G = Q(√2) = Larger Extended Field Q with irrational root √2
H = Q(√2, √3) = Largest Extended Field Q with irrational roots (√2 & 3)
{1, √2} forms basis of Q(√2) over Q
{1, √3} basis of Q(√2, √3) over Q(√2)
[since √3 ≠ p+ q√2 , ∀p,q ∈ Q]
=> {1,√2, √3, √6} basis of Q(√2, √3) over Q
=> Q(√2, √3) is a 4-dimensional Vector Space over Q.
Isomorphism (≌)
Q(√2) ≌ Q[x] / {x² – 2}
Read as:
Q(√2) isomorphic to the quotient of the Polynomial ring Q[x] modulo the Principal Ideal {x² – 2}
Q[x] the Polynomial Ring
{x² – 2} is the Principal Ideal
Complex Number (C)
C = R[x] / {x² + 1}
R[x] the Polynomial Ring with coefficients in the Field R
{x² + 1} is the Principal Ideal
Questions:
Since R[x] / {x² + 1} is the Field C
Why below are not Fields ?
R[x] / {x³ + 1}
R[x] / {x^4 + 1}
C[x] / {x² + 1}
Hint: they are not irreducible in that particular Field, not a Principal Ideal.
Note: C[x] the Polynomial Ring with coefficients in the Field C
Structure Beyond C
Ask: NZQRC…X?
(Nine Zulu Queens Rule China …)
Is there a number system X beyond Complex C?
We know that Z is extended of N
because of solving equation like :
y+2=0
=> y= -2 ∈Z
Similarly Q extends Z:
3y-2 =0
=> y= 2/3 ∈ Q
R extends Q:
y² = 2
=> y = √2 ∈ R
C extends R:
y² = -1
=> y = √-1 ∈ C
What about something extends C ?
Answer: No such number system !
Why ?
Let the Polynomial equation
P(x) = xⁿ +…+ dx³+ cx² + bx + a
P(x) has n solutions in C (by Gauss Fundamental Law of Algebra): z0, z1, z2, …zn-1
P(x) can be factorized as:
P(x) = (x-z0).(x-z1).(x-z2)….(x -zn-1)
P(x)=0 has all n complex solutions still in C
=> closed in C or Complete in C
=> no need to extend C like the previous number systems (NZQR)
