Imaginary Number Calculator Symbolab

Imaginary Number Operations:

\[ i^2 = -1 \] \[ (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i \] \[ (a + bi) \times (c + di) = (ac - bd) + (ad + bc)i \] \[ \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \]

Operation:

First number (a):

First number (b):

Second number (c):

Second number (d):

Result:

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1. What is the Imaginary Unit?

The imaginary unit (denoted by i) is defined by the property that i² = -1. Complex numbers are numbers of the form a + bi, where a and b are real numbers.

2. How Does the Calculator Work?

The calculator performs operations on complex numbers using these formulas:

\[ (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i \] \[ (a + bi) \times (c + di) = (ac - bd) + (ad + bc)i \] \[ \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \]

Where:

a, c — Real parts
b, d — Imaginary coefficients
i — Imaginary unit (√-1)

3. Importance of Imaginary Numbers

Details: Complex numbers are fundamental in many areas of mathematics, physics, and engineering, including electrical engineering, quantum mechanics, and signal processing.

4. Using the Calculator

Tips: Enter the real and imaginary parts of both complex numbers, select the operation, and click Calculate. Division by zero is undefined.

5. Frequently Asked Questions (FAQ)

Q1: What is i²?
A: i² = -1 by definition of the imaginary unit.

Q2: Can you take the square root of a negative number?
A: Yes, √-x = i√x for x > 0.

Q3: What are complex numbers used for?
A: They're used in AC circuit analysis, quantum physics, control theory, and many other fields.

Q4: What's the complex conjugate?
A: The conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate gives a real number.

Q5: How do you represent complex numbers graphically?
A: On the complex plane, with the real part on the x-axis and imaginary part on the y-axis.