Dividing Complex Numbers

Front

What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Front

What is the complex conjugate of a complex number a + bi?

Back

The complex conjugate of a complex number a + bi is a - bi.

Front

How do you multiply two complex numbers (a + bi)(c + di)?

Back

To multiply two complex numbers, use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2. Since i^2 = -1, this simplifies to (ac - bd) + (ad + bc)i.

Front

What is the formula for dividing complex numbers?

Back

To divide complex numbers a + bi by c + di, multiply the numerator and denominator by the complex conjugate of the denominator: (a + bi)/(c + di) = [(a + bi)(c - di)]/[(c + di)(c - di)].

Front

What is the denominator after multiplying and before simplifying in the division of (2 + 3i)/(1 - 2i)?

Back

The denominator is 5.

Front

What is the numerator after multiplying and before simplifying in the division of (2 + 3i)/(1 - 2i)?

Back

The numerator is 8 + i.

Front

What is the result of dividing (4 + 2i) by (1 + i)?

Back

The result is 3 - i.

Front

What is the modulus of a complex number a + bi?

Back

The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²).

Front

How do you express a complex number in polar form?

Back

A complex number a + bi can be expressed in polar form as r(cos θ + i sin θ), where r is the modulus and θ is the argument (angle) of the complex number.

Front

What is the argument of a complex number a + bi?

Back

The argument of a complex number a + bi is the angle θ formed with the positive x-axis, calculated as θ = arctan(b/a).