de-CH
utf-8
math math-format polynomials
Multiplikation komplexer Zahlen
l-02-04
multiple
A_REAL, A_IMAG, B_REAL, B_IMAG, ANSWER_REAL, ANSWER_IMAG
16
100
randRangeNonZero(-5, 5) randRangeNonZero(-5, 5) randRangeNonZero(-5, 5) randRangeNonZero(-5, 5) "\\pink{" + A_REAL + "}" "\\pink{" + coefficient(A_IMAG) + "i}" "\\blue{" + B_REAL + "}" "\\blue{" + coefficient(B_IMAG) + "i}" "\\pink{" + complexNumber(A_REAL, A_IMAG) + "}" "\\blue{" + complexNumber(B_REAL, B_IMAG) + "}" (A_REAL * B_REAL) - (A_IMAG * B_IMAG) (A_REAL * B_IMAG) + (A_IMAG * B_REAL)

Bestimmen Sie (A_REP) \cdot (B_REP) = ?

ANSWER_REAL + ANSWER_IMAGi

Komplexe Zahlen werden wie Binome ausmultipliziert.

Zuerst benutze die Distributivität:

(A_REP) \cdot (B_REP) =
(A_REAL_COLORED \cdot B_REAL_COLORED) + (A_REAL_COLORED \cdot B_IMAG_COLORED) + (A_IMAG_COLORED \cdot B_REAL_COLORED) + (A_IMAG_COLORED \cdot B_IMAG_COLORED)

Vereinfachen ergibt:

(A_REAL * B_REAL) + (coefficient(A_REAL * B_IMAG)i) + (coefficient(A_IMAG * B_REAL)i) + (coefficient(A_IMAG * B_IMAG)i^2)

Imaginärteile können zusammen gruppiert werden.

A_REAL * B_REAL + (A_REAL * B_IMAG + A_IMAG * B_REAL)i + coefficient(A_IMAG * B_IMAG) i^2

Nachdem wir i^2 = -1 einfügen, wird die Lösung

A_REAL * B_REAL + (A_REAL * B_IMAG + A_IMAG * B_REAL)i - negParens( A_IMAG * B_IMAG )

Dies kann man vereinfachen zu: (A_REAL * B_REAL - A_IMAG * B_IMAG) + (ANSWER_IMAGi) = complexNumber( ANSWER_REAL, ANSWER_IMAG)