Enter Rectangular Components
Use x as the real component and y as the imaginary coefficient.
Formula Used
z = x + yi
|z| = √(x² + y²)
θ = atan2(y, x)
z̄ = x − yi
z = |z|[cos(θ) + i sin(θ)]
z = |z|e^(iθ)
Here, x is the real component, y is the imaginary coefficient, |z| is the modulus, and θ is the argument.
How to Use This Calculator
- Enter the horizontal value as the real component.
- Enter the vertical value as the imaginary component.
- Select the required angle unit and angle range.
- Choose precision, formatting, symbol, and zero tolerance options.
- Press the conversion button to display every available result.
- Use the copy controls when moving values into another calculation.
Rectangular and Complex Number Guide
Understanding Rectangular and Complex Forms
A rectangular pair describes a point with horizontal and vertical values. In complex notation, the horizontal value becomes the real component. The vertical value becomes the imaginary coefficient. The imaginary unit is written as i, where i squared equals negative one. Therefore, the point x, y becomes z equals x plus yi. This notation connects coordinate geometry with algebra. It supports electrical analysis, signal processing, control systems, and mathematics. A conversion preserves signs, decimal accuracy, and the relationship between components.
Why the Conversion Matters
Complex numbers store two related quantities inside one expression. That compact structure makes many calculations easier. Rotations, oscillations, wave behavior, and alternating current often use complex values. Rectangular form is especially useful for addition and subtraction. Polar form is often better for multiplication, division, powers, and roots. This calculator presents both views together. You can inspect the complex expression, modulus, argument, conjugate, normalized value, and related notations. Seeing every result helps you choose the best form for the next calculation.
Reading the Real and Imaginary Parts
The real component lies on the x-axis. Positive values move right; negative values move left. The imaginary component lies on the y-axis. Positive values move upward; negative values move downward. Signs must remain unchanged during conversion. For example, the pair three, negative four becomes three minus four i. A zero imaginary value creates a purely real number. A zero real value creates a purely imaginary number. When both values are zero, the modulus is zero and the argument is undefined.
Precision and Rounding Choices
Input values may contain integers, decimals, or scientific notation. The selected precision controls displayed results, not the conversion logic. More decimal places can help with engineering work and repeated calculations. Fewer places make results easier to read. Rounding may slightly change a displayed modulus or angle. Keep precision when results feed another formula. The calculator can remove trailing zeros for cleaner notation. It can also preserve fixed decimal places when consistent formatting is required for reports, worksheets, or comparison tables.
Using Magnitude and Argument
The modulus measures distance from the origin to the point. It is always zero or positive. The argument measures the direction of the point from the positive real axis. The atan2 function determines the correct quadrant. Angles can be shown in radians or degrees. A negative angle and its coterminal angle describe the same direction. Polar notation combines modulus and argument. Exponential notation uses Euler's number with an imaginary exponent. These forms reveal rotation and scaling better than rectangular notation.
Checking and Applying Results
After conversion, compare the real and imaginary parts with your inputs. Confirm that the sign between them is correct. Check the quadrant suggested by component signs. The conjugate should reverse the imaginary sign. Multiplying a complex number by its conjugate produces the modulus squared. That identity provides a verification. You can copy the result into equations, circuit calculations, transforms, or geometry work. Save decimal precision for later steps. Clear the form before starting an unrelated calculation to avoid accidental reuse.
Frequently Asked Questions
1. What is rectangular form?
Rectangular form describes a complex number as x + yi. The value x is real, while y is the coefficient of the imaginary unit.
2. How does a coordinate pair become complex notation?
Use the horizontal coordinate as the real part. Use the vertical coordinate as the imaginary coefficient. The pair (x, y) becomes x + yi.
3. What happens when the imaginary component is negative?
The plus sign changes to a minus sign. For example, x = 5 and y = -2 produce 5 - 2i.
4. Can the calculator use j instead of i?
Yes. Select j as the imaginary symbol. Engineers often use j because i commonly represents electric current.
5. What is the modulus of a complex number?
The modulus is the distance from the origin. It equals the square root of the real component squared plus the imaginary component squared.
6. Why does the calculator use atan2?
The atan2 function uses both component signs. It identifies the correct quadrant and avoids ambiguity found in a basic inverse tangent calculation.
7. What is the conjugate?
The conjugate keeps the real component and reverses the imaginary sign. The conjugate of 3 + 4i is 3 - 4i.
8. Why is the argument undefined at zero?
The origin has no direction from itself. Therefore, the zero complex number has modulus zero but no unique argument.
9. What does zero tolerance do?
Zero tolerance treats very small component values as zero. This can remove floating-point noise and create cleaner displayed expressions.
10. When should trailing zeros remain visible?
Keep trailing zeros when a report requires consistent decimal places or when displayed precision communicates measurement accuracy.
11. Can rectangular form be converted back from polar form?
Yes. Multiply the modulus by cosine of the angle for the real part. Multiply the modulus by sine for the imaginary part.