24
randFromArray( [ true, false ] )
randRange(1, 10)
randRange(1, 10)
A_RADIUS * B_RADIUS
"{" + A_RADIUS + "}"
randRange(1, DENOMINATOR - 1 )
A_ANGLE_NUMERATOR * PI * 2 / DENOMINATOR
"{" + piFraction(A_ANGLE, true) + "}"
"{" + polarForm(A_RADIUS, A_ANGLE, USE_EULER_FORM) + "}"
cos( A_ANGLE ) * A_RADIUS
sin( A_ANGLE ) * A_RADIUS
"\\red{" + B_RADIUS + "}"
randRange(1, DENOMINATOR - 1 )
B_ANGLE_NUMERATOR * PI * 2 / DENOMINATOR
"\\red{" + piFraction(B_ANGLE, true) + "}"
"\\red{" + polarForm(B_RADIUS, B_ANGLE, USE_EULER_FORM) + "}"
cos( B_ANGLE ) * B_RADIUS
sin( B_ANGLE ) * B_RADIUS
"\\blue{" + ANSWER_RADIUS + "}"
( A_ANGLE_NUMERATOR + B_ANGLE_NUMERATOR ) % DENOMINATOR
ANSWER_ANGLE_NUMERATOR * PI * 2 / DENOMINATOR
"\\blue{" + piFraction(ANSWER_ANGLE, true) + "}"
cos( ANSWER_ANGLE ) * ANSWER_RADIUS
sin( ANSWER_ANGLE ) * ANSWER_RADIUS
"{" + piFraction((A_ANGLE_NUMERATOR + B_ANGLE_NUMERATOR) * PI * 2 / DENOMINATOR, true) + "}"
Multiply the following complex numbers:
{\color{blue}z} = v \cdot {\color{red}w} = \left(A_REP\right) \cdot \left(B_REP\right)
graphInit({
range: [ [ -10, 10 ], [ -10 ,10 ] ],
scale: 20,
tickStep: 1,
axisArrows: "->"
});
drawComplexChart( 10, DENOMINATOR );
label( [A_REAL, A_IMAG], "\\color{black} v", "left" );
circle( [A_REAL, A_IMAG], 1 / 4, {
fill: KhanUtil.BLACK,
stroke: "none"
});
label( [B_REAL, B_IMAG], "\\color{red} w", "right" );
circle( [B_REAL, B_IMAG], 1 / 4, {
fill: KhanUtil.RED,
stroke: "none"
});
graph.currComplexPolar = new ComplexPolarForm( DENOMINATOR, 10 );
graph.currComplexPolar.color = BLUE;
redrawComplexPolarForm();
[
graph.currComplexPolar.getAngleNumerator(),
graph.currComplexPolar.getRadius()
]
var angle = guess[0];
var radius = guess[1];
if (angle === 0 && radius === 1) {
return "";
}
return angle === ANSWER_ANGLE_NUMERATOR &&
radius === ANSWER_RADIUS;
redrawComplexPolarForm(guess[0], guess[1]);
redrawComplexPolarForm(guess[0], guess[1]);
The multiplication of two polar forms is given by multiplying the absolute values und add the angles.
The first number, v = A_REP,
has angle A_ANGLE_REP
and absolute value A_RADIUS_REP.
The second number, {\red w} = B_REP,
has angle B_ANGLE_REP
and absolute angle B_RADIUS_REP.
The absolute value of the result is thus
A_RADIUS_REP \cdot B_RADIUS_REP = ANSWER_RADIUS_REP.
The sum of the angles is A_ANGLE_REP + B_ANGLE_REP = INTERMEDIATE_ANGLE_REP.
The angle INTERMEDIATE_ANGLE_REP ist grösser als 2 \pi.
Eine komplexe Zahl kehrt zum Ausgangspunkt zurück, wenn der Winkel um 2 \pi erhöht wird.
Thus, the angle here is INTERMEDIATE_ANGLE_REP - 2 \pi = ANSWER_ANGLE_REP to be set to.
The sum of the angles is A_ANGLE_REP + B_ANGLE_REP = ANSWER_ANGLE_REP.