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The set C of complex numbers is the set R2 of ordered pairs of real numbers. a+bi = [a,b]
 
Multiplication, Reciprocal & Division
For 2 dimensional numbers,
x = [p∠φ] = [p·→φ,p·↑φ] = [a,b]
y = [q∠ψ] = [q·→ψ,q·↑ψ] = [c,d]

Multiplication x·y = [(p·q)∠(φ+ψ)]
= [(p·q)·→(φ+ψ),(p·q)·↑(φ+ψ)]
= [p·q·(→φ·→ψ−↑φ·↑ψ), p·q·(↑φ·→ψ+→φ·↑ψ)]
= [p·→φ·q·→ψ−p·↑φ·q·↑ψ, p·↑φ·q·→ψ+p·→φ·q·↑ψ]
= [a·c−b·d,  ...
9628
 
Brian
9606
 
Complex Number
Hamilton introduced the approach to define the set C of complex numbers as the set R2 of ordered pairs [a,b] of real numbers and [a,b] + [c,d] = [a+c,b+d] [a,b] · [c,d] = [ac-bd,bc+ad] It is then just a matter of notation to express [a,b] as a+bi.
9609
 
Dave
dreaded, discriminant
9608
 
David Darling
9661
 
e, mathematical constant
e = (1+1/n)⩘n「n⪫∞」= Σ(1/n!)「n⪫∞」
9668
 
Eugene Kutoryansky
9569
 
Euler's formula
9669
 
Kirk Weiler
9617
 
Krista King
9647
 
Operations, Conjugates, and the Linear Factorization Theorem
3695
 
PBS Infinie Series
9648

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