Imaginary and Complex Numbers

Front

What is the definition of 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.

Front

How do you simplify the square root of a negative number, such as √-45?

Back

To simplify √-45, factor out the negative: √-45 = √(45) * √(-1) = √(45) * i = 3i√5.

Front

What is the result of subtracting two complex numbers, such as (10 + 15i) - (48 - 30i)?

Back

To subtract complex numbers, subtract the real parts and the imaginary parts separately: (10 - 48) + (15i + 30i) = -38 + 45i.

Front

How do you find the sum of two complex numbers, for example, (5 - 2i) + (-7 + 8i)?

Back

To find the sum, add the real parts and the imaginary parts: (5 - 7) + (-2i + 8i) = -2 + 6i.

Front

What is the formula for multiplying two complex numbers, such as (4 – 3i)(-7 – 2i)?

Back

Use the distributive property (FOIL): (4)(-7) + (4)(-2i) + (-3i)(-7) + (-3i)(-2i) = -28 - 8i + 21i + 6 = -34 + 13i.

Front

What is the imaginary unit 'i' and what is its significance in complex numbers?

Back

The imaginary unit 'i' is defined as √-1. It allows for the extension of real numbers to complex numbers, which are of the form a + bi, where a and b are real 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 is the real part and b is the imaginary part.

Front

How do you convert a complex number from rectangular form to polar form?

Back

To convert a complex number a + bi to polar form, calculate the modulus r = √(a² + b²) and the argument θ = arctan(b/a), then express it as r(cos θ + i sin θ) or re^(iθ).

Front

What is the conjugate of a complex number, and how is it used?

Back

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

Front

What is the modulus of a complex number, and how is it calculated?

Back

The modulus of a complex number a + bi is calculated as |a + bi| = √(a² + b²). It represents the distance from the origin in the complex plane.