Complex Number Calculator

What Is a Complex Number?

A complex number is a number written in the form $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit defined by $i^2 = -1$. The value $a$ is the real part and $b$ is the imaginary part. Complex numbers extend the real number system and are useful when equations involve the square root of a negative number.

Complex Number Operations

For two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, the main arithmetic operations are:

How to Calculate Complex Numbers?

1. Step 1 (Identify the Parts): Write each number as $a+bi$ and identify its real and imaginary parts.
2. Step 2 (Choose the Operation): Add, subtract, multiply, or divide the two complex numbers as required.
3. Step 3 (Simplify Powers of $i$): Use $i^2=-1$ whenever products contain $i^2$.
4. Step 4 (Combine Like Parts): Keep the real terms together and combine the coefficients of $i$ separately.
5. Step 5 (Write in Standard Form): Express the final answer as $a+bi$.

Complex Number Examples

Magnitude and Absolute Value of a Complex Number

The magnitude, also called the modulus or absolute value, measures the distance of a complex number from the origin on the complex plane. For $z=a+bi$, it is calculated using:

For example, the magnitude of $3+4i$ is $\sqrt{3^2+4^2}=5$. Unlike the absolute value of a real number, which measures distance from zero on a number line, the complex modulus measures distance from the origin in the complex plane.

Conjugate of a Complex Number

To find the conjugate of a complex number, change the sign of its imaginary part. If $z=a+bi$, then its conjugate is $\overline{z}=a-bi$. The conjugate is especially useful when dividing complex numbers because multiplying a number by its conjugate produces a real value:

Complex Number in Polar Form

A complex number can also be represented in polar form using its magnitude $r$ and angle $\theta$. For $z=a+bi$:

Polar form is particularly useful for multiplication, division, and other operations involving angles and magnitudes.

Applications of Complex Numbers

Common Mistakes to Avoid

• Forgetting $i^2=-1$: The product $(bi)(di)$ becomes $bd i^2=-bd$, not $bd$.
• Mixing Real and Imaginary Parts: Real terms should be combined with real terms, while coefficients of $i$ should be combined separately.
• Using the Wrong Conjugate: The conjugate of $a+bi$ is $a-bi$, so only the sign of the imaginary part changes.
• Confusing Magnitude With the Imaginary Part: The modulus $|a+bi|$ is $\sqrt{a^2+b^2}$, not simply $b$.