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.

Tags

CCSS.HSN.CN.A.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.

Tags

CCSS.HSN.CN.A.1

Front

How do you add complex numbers?

Back

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

Tags

CCSS.HSN.CN.A.2

Front

What is the conjugate of a complex number?

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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CCSS.HSN.CN.A.3

Front

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 a² + b², which is a real number.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL method). For example, (a + bi)(c + di) = ac + adi + bci + bdi², and remember that i² = -1.

Tags

CCSS.HSN.CN.A.2

Front

What is the modulus of a complex number?

Back

The modulus of a complex number a + bi is given by √(a² + b²). It represents the distance from the origin to the point (a, b) in the complex plane.

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 real axis, calculated using θ = arctan(b/a).

Front

What is the square of a complex number (a + bi)?

Back

The square of a complex number (a + bi) is (a + bi)² = a² + 2abi + (bi)² = a² - b² + 2abi.

Front

What is the value of i^4?

Back

The value of i^4 is 1, since i^4 = (i²)² = (-1)² = 1.

Tags

CCSS.HSN.CN.A.2