Question 1

What are imaginary numbers?

Accepted Answer

Imaginary numbers are numbers that give a negative result when squared. The imaginary unit i is defined as √(-1). Complex numbers combine real and imaginary parts in the form a + bi, where a is the real part and b is the imaginary part.

Question 2

How do you add imaginary numbers?

Accepted Answer

To add complex numbers, add the real parts together and the imaginary parts together separately. For example: (2 + 3i) + (1 + 4i) = (2+1) + (3+4)i = 3 + 7i.

Question 3

How do you multiply imaginary numbers?

Accepted Answer

Use the FOIL method: (a+bi)(c+di) = ac + adi + bci + bdi². Since i² = -1, this becomes (ac-bd) + (ad+bc)i. For example: (2+3i)(1+4i) = 2 - 12 + (8+3)i = -10 + 11i.

Question 4

How do you divide complex numbers?

Accepted Answer

Multiply both numerator and denominator by the conjugate of the denominator. The conjugate of (a+bi) is (a-bi). This eliminates the imaginary part from the denominator, making division possible.

Question 5

What is the complex conjugate?

Accepted Answer

The complex conjugate of a + bi is a - bi. It's found by changing the sign of the imaginary part. Multiplying a complex number by its conjugate gives a real number: (a+bi)(a-bi) = a² + b².

Question 6

What is the magnitude of a complex number?

Accepted Answer

The magnitude (or modulus) of a complex number a + bi is |z| = √(a² + b²). It represents the distance from the origin to the point (a, b) in the complex plane.

Question 7

What is polar form of complex numbers?

Accepted Answer

Polar form expresses a complex number using magnitude r and angle θ: z = r(cos θ + i sin θ) or z = re^(iθ). Where r = √(a² + b²) and θ = arctan(b/a).

Question 8

What is i in mathematics?

Accepted Answer

i is the imaginary unit, defined as √(-1). By definition, i² = -1, i³ = -i, and i⁴ = 1. This pattern repeats for higher powers of i.

Question 9

Where are imaginary numbers used?

Accepted Answer

Complex numbers are essential in electrical engineering (AC circuits), quantum mechanics, signal processing, control theory, fluid dynamics, and many other fields. They simplify calculations involving oscillations and waves.

Question 10

Can you take the square root of a negative number?

Accepted Answer

Yes, using imaginary numbers. For example, √(-4) = 2i, √(-9) = 3i. The square root of any negative number can be expressed using i.

Imaginary Numbers Calculator - Complex Number Calculator & i Calculator

First Complex Number (z₁)

Second Complex Number (z₂)

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Polar Form

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Complex Number Calculator Features

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How the Imaginary Numbers Calculator Works

Complex Number Operations

Mathematical Foundation

Applications of Complex Numbers

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Complex Number Calculator Examples

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Division Example

Conjugate Example

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