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 i = √-1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, i.e., i = √-1.

Front

What is the conjugate of a complex number?

Back

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

Front

How do you add complex numbers?

Back

To add complex numbers, add their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

How do you subtract complex numbers?

Back

To subtract complex numbers, subtract their real parts and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

Front

What is the product of two complex numbers?

Back

The product of two complex numbers (a + bi)(c + di) is given by ac + adi + bci + bdi^2, which simplifies to (ac - bd) + (ad + bc)i.

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by |a + bi| = √(a^2 + b^2).

Front

How do you divide complex numbers?

Back

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, to divide (a + bi) by (c + di), multiply by (c - di)/(c - di).

Front

What is the square of the imaginary unit 'i'?

Back

The square of the imaginary unit 'i' is i^2 = -1.

Front

What is the polar form of a complex number?

Back

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