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.

The square root of a complex number is found by taking the square root of its modulus and half the angle. In this case, the square root of -4i is -2i.

The argument (angle) of a complex number a + bi is arctan(b/a). In this case, arctan((-1)/1) = -π/4.

The reciprocal of a complex number a + bi is found by dividing 1 by the complex number and multiplying the result by the conjugate of the complex number.

The argument (angle) of a complex number a + bi is arctan(b/a). In this case, arctan(2/0) approaches π/2.

The modulus of a complex number is always non-negative. It represents the distance from the origin to the point in the complex plane.

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.

Expand and simplify the product of the complex numbers. The real part is found by multiplying the real parts and the imaginary parts separately and then adding them up.

Multiply the complex numbers using the distributive property. The product of (2 + 3i) and (1 - 2i) is 7 + 5i.

Expand and simplify the product of the complex numbers. The imaginary part is found by multiplying the real part of one complex number with the imaginary part of the other and vice versa, then adding them up.

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

Expand and simplify the product of the complex numbers. The real part is found by multiplying the real parts and the imaginary parts separately and then adding them up.