Multiply 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. It is used to extend the real number system to include solutions to equations that do not have real solutions.

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 conjugate of a complex number a + bi?

Back

The conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part.

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What is the product of a complex number and its conjugate?

Back

The product of a complex number a + bi and its conjugate a - bi is a^2 + b^2.

Front

What is the geometric interpretation of complex numbers?

Back

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

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How do you express a complex number in polar form?

Back

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

Front

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

Back

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

Front

What is the argument of a complex number?

Back

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

Front

How do you add two complex numbers?

Back

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