Enter real and imaginary parts. See quotient, graph, steps, and downloads. Practice precise complex division with clean outputs and examples.
The graph places the dividend, divisor, and quotient on the complex plane.
| Dividend | Divisor | Quotient | Real Part | Imaginary Part |
|---|---|---|---|---|
| 4 + 6i | 1 + 2i | 3 - 0i | 3 | 0 |
| 7 + 5i | 2 - 1i | 1.8 + 3.4i | 1.8 | 3.4 |
| 3 - 8i | 4 + 2i | -0.2 - 1.9i | -0.2 | -1.9 |
| 9 + 3i | 3 + 3i | 2 - 1i | 2 | -1 |
For \((a + bi) \div (c + di)\), multiply both parts by the conjugate of the divisor.
Conjugate: \(c - di\)
Expanded form: \((a + bi)(c - di) \div ((c + di)(c - di))\)
Real part: \((ac + bd) \div (c^2 + d^2)\)
Imaginary part: \((bc - ad) \div (c^2 + d^2)\)
Final quotient: \(\frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i\)
Imaginary number division is a standard part of complex arithmetic. A complex number has a real part and an imaginary part. The imaginary unit is written as i, where i² equals -1. Division looks harder than addition or subtraction, but the method is very systematic.
The main idea is to remove the imaginary term from the denominator. You do that by multiplying the numerator and denominator by the conjugate of the divisor. The conjugate changes the sign between the real and imaginary terms. This creates a real denominator because \((c + di)(c - di)\) becomes \(c^2 + d^2\).
After expansion, the quotient separates into a real part and an imaginary part. That makes the answer easier to read and compare. Students often use this form in algebra, trigonometry, electrical work, and signal analysis. Engineers also use complex division in circuit calculations, control systems, and wave models.
This calculator helps you avoid manual errors. It shows the quotient, denominator value, conjugate, magnitudes, and angles. The graph places each complex value on the Argand plane, so you can see direction and distance from the origin. That visual check is helpful during learning and review.
The export tools are useful for assignments, notes, and reports. The example table gives quick practice references. The step section explains the exact process, so the tool works as both a solver and a study guide. Use it whenever you need a fast and clean complex number division result.
It divides one complex number by another complex number. You enter real and imaginary parts for both numbers, then the calculator returns the quotient in standard a + bi form.
The conjugate removes the imaginary term from the denominator. That makes the denominator a real number, which lets the quotient be written clearly as separate real and imaginary parts.
Division by zero is undefined. If both divisor parts are zero, the calculator shows an error because no valid quotient exists in that case.
Yes. It displays the original division, the conjugate step, and the formulas used for the real and imaginary parts. This helps with checking classwork and homework.
The graph plots the dividend, divisor, and quotient on the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.
Yes. The calculator accepts positive numbers, negative numbers, and decimals. This makes it suitable for simple examples and more advanced complex arithmetic problems.
They describe each complex number in polar style. Magnitude shows distance from the origin, and angle shows direction measured from the positive real axis.
Yes. Use the CSV button for spreadsheet style export, or the PDF button for a simple document copy of the current result and key values.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.