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 imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, which means i^2 = -1.

Front

How do you add two complex numbers?

Back

To add two complex numbers, you 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 two complex numbers?

Back

To subtract two complex numbers, you 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

To multiply two complex numbers (a + bi) and (c + di), use the distributive property: (a + bi)(c + di) = ac + adi + bci + bdi^2. Remember that i^2 = -1.

Front

What is the conjugate of a complex number?

Back

The conjugate of a complex number a + bi is a - bi. It is used to simplify the division of complex numbers.

Front

How do you multiply a complex number by a real number?

Back

To multiply a complex number by a real number, you multiply the real part and the imaginary part by that real number. For example, k(a + bi) = ka + kbi.

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is the distance from the origin to the point (a, b) in the complex plane, calculated as |a + bi| = √(a^2 + b^2).

Front

How do you divide two complex numbers?

Back

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

Front

What is the geometric representation of complex numbers?

Back

Complex numbers can be represented as points or vectors in a two-dimensional plane called the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.