Multiply three or more numbers

Learning Objectives Multiply three or more complex numbers in polar form by multiplying their moduli and adding their arguments. Apply the product rule for complex numbers to a sequence of three or more multiplications. Use De Moivre's Theorem to raise a complex number to an integer power, which involves repeated multiplication. Convert complex numbers from rectangular form (a + bi) to polar form (r(cos θ + i sin θ)) to facilitate multiplication. Interpret the geometric effect of multiplying three or more complex numbers as a sequence of rotations and dilations in the complex plane. Solve problems involving chained transformations (rotations and scaling) using complex number multiplication. How could you calculate the final position of a point that is rotated 45°, scale...

TermDefinitionExample Complex Number in Polar FormA way to represent a complex number using its distance from the origin (modulus, r) and its angle relative to the positive real axis (argument, θ). The standard notation is z = r(cos θ + i sin θ) or z = r cis θ.The complex number 1 + i in rectangular form is √2(cos 45° + i sin 45°) in polar form. Here, r = √2 and θ = 45°. Modulus (r)The magnitude, or absolute value, of a complex number z = a + bi. It is the distance from the origin to the point (a, b) in the complex plane. It is calculated as r = |z| = √(a² + b²).For z = 3 + 4i, the modulus is r = √(3² + 4²) = √25 = 5. Argument (θ)The angle formed by the line segment from the origin to the complex number and the positive real axis. It is typically measured in degrees or radians.For z = 1 +...

Product Rule for Three or More Complex Numbers Given z_k = r_k(\cos \theta_k + i \sin \theta_k) for k = 1, 2, ..., n. \newline The product is: \newline z_1 z_2 ... z_n = (r_1 r_2 ... r_n) [\cos(\theta_1 + \theta_2 + ... + \theta_n) + i \sin(\theta_1 + \theta_2 + ... + \theta_n)] To multiply three or more complex numbers in polar form, multiply all their moduli together to get the new modulus, and add all their arguments together to get the new argument. De Moivre's Theorem (for integer powers) [r(\cos \theta + i \sin \theta)]^n = r^n(\cos(n\theta) + i \sin(n\theta)) Use this rule to efficiently raise a complex number to an integer power 'n'. This is a special case of repeated multiplication where all the numbers are identical. The 'r^n' part is where...