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The set C of complex numbers is the set R2 of ordered pairs of real numbers.
a+bi = [a,b]

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, ...

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, ...

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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.

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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