Math Problem Statement
Démontrer, grâce aux coordonnées cartésiennes, que |𝑧1𝑧2| = |𝑧1||𝑧2|
Solution
Pour démontrer que en utilisant les coordonnées cartésiennes, nous allons partir de la forme cartésienne des nombres complexes et .
Étape 1 : Représentation des nombres complexes
Soient et , où sont des réels et est l'unité imaginaire telle que .
La norme (ou module) d'un nombre complexe est donnée par : Ainsi :
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Cartesian Coordinates
Modulus of Complex Numbers
Multiplication of Complex Numbers
Formulas
Modulus formula: |z| = √(x² + y²) for z = x + iy
Multiplication of complex numbers: z1z2 = (x1 + iy1)(x2 + iy2)
Modulus of a product: |z1z2| = √((x1x2 - y1y2)² + (x1y2 + y1x2)²)
Theorems
Modulus of a Product Theorem: |z1z2| = |z1||z2|
Suitable Grade Level
Undergraduate level (or advanced high school for students familiar with complex numbers)
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