Imaginary and Complex Numbers

Front

What is an imaginary number?

Back

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit 'i', where i is defined as the square root of -1.

Tags

CCSS.HSN.CN.A.1

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.

Tags

CCSS.HSN.CN.A.1

Front

How do you multiply 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.

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 used to simplify division of complex numbers.

Tags

CCSS.HSN.CN.A.3

Front

How do you add complex numbers?

Back

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

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 |a + bi| = √(a² + b²). It represents the distance from the origin 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 x-axis in the complex plane, calculated as θ = arctan(b/a).

Front

How do you simplify the square root of a negative number?

Back

To simplify the square root of a negative number, factor out -1: √(-x) = i√x.

Front

What is the result of multiplying (4 - 3i) and (-7 - 2i)?

Back

The result is -34 + 13i.

Tags

CCSS.HSN.CN.A.2

Front

How do you simplify 3√-81?

Back

3√-81 = 3 * 9i = 27i.