Operations and Properties of Complex Numbers Quiz

The modulus (or magnitude) of a complex number a + bi is given by sqrt(a^2 + b^2). In this case, sqrt(3^2 + 4^2) = 5.

The conjugate of a complex number a - bi is a + bi. In this case, the conjugate is 2 + 3i.

The modulus (absolute value) of a complex number a + bi is given by sqrt(a^2 + b^2). In this case, sqrt((-3)^2 + 4^2) = 5.

The modulus (absolute value) of a complex number a + bi is given by sqrt(a^2 + b^2). In this case, sqrt((-5)^2 + (-12)^2) = 13.

The modulus (absolute value) of a complex number a + bi is given by sqrt(a^2 + b^2). In this case, sqrt(1^2 + (-1)^2) = √2.

The product of two complex conjugates (a + bi)(a - bi) is always a real number. It simplifies to a^2 + b^2.

To square a complex number a + bi, multiply it by itself. In this case, (1 + i)^2 = 1 - i.

The argument (angle) of a complex number a + bi is arctan(b/a). In this case, arctan(√3/(-1)) = π/3.

The product of two complex conjugates is always a real number, not necessarily a complex number.

Multiply the complex numbers using the distributive property. The product of (2 + i) and (3 - 2i) is 3 + 7i.

Multiply the complex numbers using the distributive property. The product of (2 + 3i) and (4 - 2i) is 14 + 4i.

Divide the complex numbers using the formula (a + bi)/(c + di) = (ac + bd)/(c^2 + d^2) + (bc - ad)/(c^2 + d^2)i.

To find the cube of a complex number a + bi, multiply it by itself twice. In this case, (1 - i)^3 = 1 + 3i.

The modulus (absolute value) of the difference of two complex numbers |z - w| is the same as the difference of their moduli |z| - |w|. In this case, |5 - 3| = 4.

Subtract the complex numbers by subtracting their real and imaginary parts separately. The difference of (2 + 3i) and (1 - 2i) is 3 + i.