Free Multiply Complex Numbers Calculator

Multiply Complex Numbers Calculator: Free Online Tool

The Multiply Complex Numbers Calculator is a free online tool that quickly computes the product of any two complex numbers, whether they are expressed in rectangular (a+ib) form or polar (magnitude and angle) form. This makes it essential for students, engineers, and anyone dealing with complex number multiplication. In addition to the final result, the calculator conveniently displays the product in both formats, allowing you to choose the representation that suits your current need.

Multiplying in Rectangular Form

When both numbers are given in the standard $a+ib$ notation, the product follows the binomial expansion:

$$(a+ib)(c+id) = (ac - bd) + i(ad + bc)$$

Here:

• The real part of the result is $ac - bd$
• The imaginary part is $ad + bc$

For example, multiplying $3+2i$ by $1+4i$ yields:

$$(3+2i)(1+4i) = (3\cdot1 - 2\cdot4) + i(3\cdot4 + 2\cdot1) = (3-8) + i(12+2) = -5 + 14i$$

Multiplying in Polar Form

When the complex numbers are provided in polar form—i.e., as $r e^{i\varphi}$ and $s e^{i\psi}$—the multiplication simplifies to a combination of magnitude multiplication and angle addition:

$$(r e^{i\varphi})(s e^{i\psi}) = (r s) e^{i(\varphi+\psi)}$$

Thus:

• The magnitude of the product is the product of the original magnitudes: $|z_1 \cdot z_2| = r \cdot s$
• The phase (argument) is the sum of the original phases: $\arg(z_1 \cdot z_2) = \varphi + \psi$

Polar multiplication is often more intuitive when visualizing rotations and scaling in the complex plane.

How to Use the Calculator

Using this complex number calculator is straightforward:

1. Input the first number. Choose whether to enter it in rectangular form (real and imaginary parts) or polar form (magnitude and phase angle). The interface lets you toggle between the two.
2. Input the second number. You may use either format independently; mixing rectangular and polar inputs is perfectly fine.
3. Compute. The tool instantly performs the complex number multiplication, applying the appropriate formulas internally.
4. Review the result. The output shows both the rectangular and polar representations, so you can copy the one you need without performing any conversion.

This flexibility makes the calculator ideal for checking homework, verifying manual calculations, or exploring the product of complex numbers in different forms.

Why Both Output Forms Matter

Rectangular form (a+ib) is natural for addition and subtraction, but multiplication often becomes cumbersome. In contrast, polar form transforms multiplication into simple operations on magnitudes and angles, revealing the geometric nature of complex numbers. By providing the result in both forms, this complex number calculator bridges the gap between algebraic computation and geometric intuition.

Sample Calculation Walkthrough

1. Multiply $2+3i$ and $4-5i$ in rectangular:

$$(2+3i)(4-5i) = (2\cdot4 - 3\cdot(-5)) + i(2\cdot(-5) + 3\cdot4) = (8 +15) + i(-10+12) = 23 + 2i$$

The polar representation of the result is approximately $\sqrt{23^{2}+2^{2}} e^{i\arctan(2/23)} \approx 23.087 e^{i0.0868}$, which the calculator also displays.

1. Multiply directly in polar: $5 e^{i\pi/3}$ and $2 e^{i\pi/6}$ give magnitude $5 \times 2 = 10$ and angle $\pi/3 + \pi/6 = \pi/2$, so the product is $10 e^{i\pi/2}$. The calculator shows both formats automatically.

Key Terminology

• Rectangular (Cartesian) form: $a+ib$, where $a$ is the real part and $b$ is the imaginary part.
• Polar form: $r e^{i\varphi}$ or $r(\cos\varphi + i\sin\varphi)$, where $r$ is the modulus (magnitude) and $\varphi$ is the argument (angle).
• Modulus of the product: Always equals the product of the moduli when using polar multiplication.

By automating the steps and showing both forms, this calculator helps you grasp the concept of multiplying imaginary numbers and see the connection between algebraic and geometric interpretations.