Calculator
Example Data Table
| Input | Method | Standard form result |
|---|---|---|
| 3 + 4i | Rectangular input | 3 + 4i |
| 5 cis 53.130102° | a = r cos θ, b = r sin θ | 3 + 4i |
| (2 + 3i)(4 - i) | Expand and use i² = -1 | 11 + 10i |
| (7 + i) / (2 - 3i) | Multiply by conjugate | 0.8462 + 1.7692i |
Formula Used
Standard form is z = a + bi. Here a is the real part. Here b is the imaginary coefficient.
Polar conversion uses a = r cos θ and b = r sin θ.
The modulus is |z| = sqrt(a² + b²). The argument is θ = atan2(b, a).
The conjugate is a - bi. Division uses (a + bi) / (c + di) and multiplies by c - di.
Power form uses zⁿ = rⁿ cis(nθ). Root form uses r^(1/n) cis((θ + 2πk) / n).
How to Use This Calculator
Select the input type first. Use rectangular form for a + bi. Use polar or exponential form when radius and angle are known.
Enter the required values. Choose the operation you need. Pick decimal places for rounding. Press Calculate.
Read the result above the form. Review the steps and check table. Then download the CSV or PDF report.
Complex Numbers in Standard Form
A complex number has two parts. The real part sits first. The imaginary part follows with i. Standard form writes every value as a + bi. This layout is simple. It also makes arithmetic easier.
Why Standard Form Matters
Standard form helps students compare values. It shows signs clearly. It separates real movement from imaginary movement. Many algebra, geometry, and engineering problems use this form. Polar or exponential input can look compact. Yet final answers often need rectangular form. This calculator converts those inputs and explains the path.
Useful Conversion Ideas
For rectangular input, the value already has a and b. The calculator cleans the signs and rounds carefully. For polar input, the radius and angle create both parts. The real part equals r cos theta. The imaginary part equals r sin theta. Exponential form uses the same idea through Euler notation.
Advanced Operation Support
Complex arithmetic follows special rules. The key rule is i squared equals negative one. Addition and subtraction combine matching parts. Multiplication expands both binomials. Division uses the conjugate of the denominator. Powers use repeated angle rotation. Roots divide the angle into equal branches.
Reading the Results
The main result gives the standard form. Extra rows show modulus, argument, conjugate, and polar form. These checks help confirm accuracy. A zero imaginary part means the answer is real. A zero real part means the answer is purely imaginary.
Exporting Your Work
Use the download buttons after calculation. The CSV file stores rows for spreadsheets. The PDF file stores a neat report. Both are useful for homework notes, class examples, and quick verification.
Best Practice
Enter angles with the correct unit. Choose degrees only when the angle is written in degrees. Choose radians for pi based work. Use more decimal places for technical answers. Use fewer decimal places for simple class answers. Review the formula section before submitting. It explains every conversion used by this page.
Common Learning Tips
Always keep the sign with b. Write 4 - 3i, not 4 + -3i. Check division carefully. A small denominator can create large values. Roots may have more than one answer. For a third root, expect three standard form values. For a fourth root, expect four values.
FAQs
What is standard form for a complex number?
Standard form is a + bi. The value a is real. The value b is the coefficient of i. This form keeps both parts clear.
Can this calculator convert polar form?
Yes. Enter radius r and angle θ. Select degrees or radians. The calculator uses cosine for the real part and sine for the imaginary part.
Does exponential form use the same conversion?
Yes. The form r e^(iθ) converts like polar form. The real part is r cos θ. The imaginary part is r sin θ.
How is the conjugate found?
The conjugate changes only the imaginary sign. For a + bi, the conjugate is a - bi. It is useful in division.
Why is i squared equal to negative one?
The imaginary unit i is defined so i² = -1. This rule makes complex multiplication and powers work consistently.
Can I divide two complex numbers?
Yes. Choose the divide operation. The calculator multiplies by the denominator conjugate. Division by zero is not allowed.
Why do roots give many answers?
Complex angles can rotate by full turns. nth roots split those rotations into n branches. So an nth root gives n values.
What do the export buttons save?
The CSV saves table rows for spreadsheets. The PDF saves a simple report with the main result and supporting values.