Complex Number Calculator

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divide complex numbers calculator

Divide complex numbers step by step

On the Operations tab, enter z₁ and z₂ and choose ÷. The steps show the multiplication by the conjugate, the numerator, the real denominator and the result, then check it in polar form.

Why the conjugate solves division

Dividing z₁ = a + bi by z₂ = c + di (with c and d not both zero) requires removing the i from the denominator. The trick is to multiply numerator and denominator by the conjugate of z₂, c − di, because (c + di)(c − di) = c² + d², a positive real number. The numerator becomes (ac + bd) + (bc − ad)i, and the quotient is (ac + bd)/(c² + d²) + ((bc − ad)/(c² + d²))i. Division by zero does not exist, and the tool warns you.
In polar form division is even shorter: z₁/z₂ has modulus |z₁|/|z₂| and argument arg z₁ − arg z₂. That is why the check step appears next to the algebraic work. For rectangular inputs the result is exact: 1/(3 + 4i) = 3/25 − (4/25)i, in fractions, and the decimal approximation follows right below.
Useful cases: dividing by a real number just divides both parts (the tool short step); dividing by i is the same as multiplying by −i, because 1/i = −i; and the reciprocal of z is the conjugate divided by the squared modulus, 1/z = (a − bi)/(a² + b²). In circuits, the same arithmetic gives the current I = V/Z from the complex voltage and impedance.

Worked divisions

Classic division

Input: (3+4i)/(1-2i)
Expected output: −1 + 2i

Numerator −5 + 10i, denominator 5.

Purely imaginary result

Input: (1+i)/(1-i)
Expected output: i

Numerator (1 + i)² = 2i; denominator 2.

Reciprocal of a complex number

Input: 1/(3+4i)
Expected output: 3/25 − (4/25)i

Conjugate over |z|² = 25.

Fractional result

Input: (2+3i)/(4-i)
Expected output: 5/17 + (14/17)i

Numerator (2 + 3i)(4 + i) = 5 + 14i; denominator 17.

Division by a pure imaginary

Input: (5-2i)/(2i)
Expected output: −1 − (5/2)i

The conjugate of 2i is −2i, and 1/i = −i.

Division by a real number

Input: (10+5i)/5
Expected output: 2 + i

Each part is divided by 5.

Full tool FAQ

How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator, so the denominator becomes a real number. In (3 + 4i) ÷ (1 − 2i), the conjugate of 1 − 2i is 1 + 2i. The numerator is (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i, and the denominator is 1² + 2² = 5. The result is −1 + 2i.

How do you find the modulus and argument of a complex number?

How do you convert from polar to rectangular form?

What is De Moivre's formula and how do you compute powers?

How do you compute roots of complex numbers?

What is i to the power n, for large n?

How do you solve a quadratic equation with a negative discriminant?

What is the difference between i and j?

How do you compute the impedance of an RLC circuit?

Which input formats are accepted and how precise is it?

Frequently asked questions

Why do you multiply by the conjugate?

Because the product of a number and its conjugate is real: (c + di)(c − di) = c² + d². The denominator stops having an i and the division becomes a split into real and imaginary parts. Multiplying top and bottom by the same number does not change the value of the fraction.

Can I divide numbers given in polar form?

Yes. 6∠90° divided by 3∠30° gives 2∠60°: the moduli divide and the angles subtract. Type with ∠ or with the less-than sign, and the result appears in every form, with the Argand plane.

What happens if the divisor is zero?

Division by 0 + 0i does not exist. The tool shows a warning instead of a result, including when the zero shows up inside an expression, such as 1/(1-1).

Next steps

Why the conjugate solves division

Dividing z₁ = a + bi by z₂ = c + di (with c and d not both zero) requires removing the i from the denominator. The trick is to multiply numerator and denominator by the conjugate of z₂, c − di, because (c + di)(c − di) = c² + d², a positive real number. The numerator becomes (ac + bd) + (bc − ad)i, and the quotient is (ac + bd)/(c² + d²) + ((bc − ad)/(c² + d²))i. Division by zero does not exist, and the tool warns you.

In polar form division is even shorter: z₁/z₂ has modulus |z₁|/|z₂| and argument arg z₁ − arg z₂. That is why the check step appears next to the algebraic work. For rectangular inputs the result is exact: 1/(3 + 4i) = 3/25 − (4/25)i, in fractions, and the decimal approximation follows right below.

Useful cases: dividing by a real number just divides both parts (the tool short step); dividing by i is the same as multiplying by −i, because 1/i = −i; and the reciprocal of z is the conjugate divided by the squared modulus, 1/z = (a − bi)/(a² + b²). In circuits, the same arithmetic gives the current I = V/Z from the complex voltage and impedance.

Worked divisions

Classic division

Input: (3+4i)/(1-2i)
Expected output: −1 + 2i

Numerator −5 + 10i, denominator 5.

Purely imaginary result

Input: (1+i)/(1-i)
Expected output: i

Numerator (1 + i)² = 2i; denominator 2.

Reciprocal of a complex number

Input: 1/(3+4i)
Expected output: 3/25 − (4/25)i

Conjugate over |z|² = 25.

Fractional result

Input: (2+3i)/(4-i)
Expected output: 5/17 + (14/17)i

Numerator (2 + 3i)(4 + i) = 5 + 14i; denominator 17.

Division by a pure imaginary

Input: (5-2i)/(2i)
Expected output: −1 − (5/2)i

The conjugate of 2i is −2i, and 1/i = −i.

Division by a real number

Input: (10+5i)/5
Expected output: 2 + i

Each part is divided by 5.

Full tool FAQ

How do you divide complex numbers?

How do you find the modulus and argument of a complex number?

How do you convert from polar to rectangular form?

What is De Moivre's formula and how do you compute powers?

How do you compute roots of complex numbers?

What is i to the power n, for large n?

How do you solve a quadratic equation with a negative discriminant?

What is the difference between i and j?

How do you compute the impedance of an RLC circuit?

Which input formats are accepted and how precise is it?

Frequently asked questions

Why do you multiply by the conjugate?

Can I divide numbers given in polar form?

Yes. 6∠90° divided by 3∠30° gives 2∠60°: the moduli divide and the angles subtract. Type with ∠ or with the less-than sign, and the result appears in every form, with the Argand plane.

What happens if the divisor is zero?

Division by 0 + 0i does not exist. The tool shows a warning instead of a result, including when the zero shows up inside an expression, such as 1/(1-1).