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 arithmetic_structure_of_complex_numbers [2014/01/29 19:35]nikolaj arithmetic_structure_of_complex_numbers [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2014/01/29 19:35 nikolaj 2014/01/29 19:22 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:11 nikolaj 2013/12/21 19:57 external edit2013/09/03 01:14 nikolaj created 2014/01/29 19:35 nikolaj 2014/01/29 19:22 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:11 nikolaj 2013/12/21 19:57 external edit2013/09/03 01:14 nikolaj created Line 1: Line 1: ===== Arithmetic structure of complex numbers ===== ===== Arithmetic structure of complex numbers ===== ==== Set ==== ==== Set ==== - | @#FFBB00: $\langle \mathbb C,​+_\mathbb{C},​\cdot_\mathbb{C} \rangle$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\langle \mathbb C,​+_\mathbb{C},​\cdot_\mathbb{C} \rangle$ | - | @#55EE55: $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ | + | @#55EE55: postulate ​  | @#55EE55: $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ | - | @#55EE55: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ | + | @#55EE55: postulate ​  | @#55EE55: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ | As defined in [[complex number]], the pattern with $x+iy$ denotes $\langle x,y\rangle$ with $x,y\in \mathbb R$. The operations $+_\mathbb{R}$ and $\cdot_\mathbb{R}$ on the right hand sides are these of [[arithmetic structure of real numbers]]. As defined in [[complex number]], the pattern with $x+iy$ denotes $\langle x,y\rangle$ with $x,y\in \mathbb R$. The operations $+_\mathbb{R}$ and $\cdot_\mathbb{R}$ on the right hand sides are these of [[arithmetic structure of real numbers]]. Line 21: Line 21: ^ $|\sum_{k=1}^n z_k|\le \sum_k^n|z_k|$ ^ ^ $|\sum_{k=1}^n z_k|\le \sum_k^n|z_k|$ ^ ==== Parents ==== ==== Parents ==== - === Requirements ​=== + === Context ​=== [[Complex number]] [[Complex number]] === Element of === === Element of === [[Field]] [[Field]]