How this instrument works
Dividing complex numbers cannot be done the way real numbers are divided, because c + di is not a single scalar you can invert directly. The standard fix multiplies both numerator and denominator by the denominator's conjugate, c − di. Since (c + di)(c − di) = c² + d², that product is guaranteed real, and the awkward complex divisor is gone — the identical conjugate trick used to rationalize a denominator like 1 ⁄ (2 + √3).
Expanding the numerator after that multiplication, (a + bi)(c − di) = (ac + bd) + (bc − ad)i. The two coefficients look almost symmetric but are not: the real part adds ac and bd, while the imaginary part subtracts ad from bc. That one sign flip comes straight from i² = −1 turning a subtraction in the expansion into an addition for the real term, and it is the detail that trips up a division worked by hand.
There is a second way to see the same operation: in polar form, dividing one complex number by another divides their distances from the origin and subtracts their angles. Two views, one answer. A useful edge case sits at d = 0, where the denominator is purely real and the formulas above collapse to ordinary fraction division, a ⁄ c + (b ⁄ c)i — and at c = d = 0 the denominator is 0 + 0i, where no quotient exists at all.
- Enter the numerator's two parts into Numerator real, a and Numerator imaginary, b — together they describe a + bi.
- Enter the denominator's two parts into Denominator real, c and Denominator imaginary, d, describing c + di.
- Read Result: real part for the real component of the quotient.
- Read Result: imaginary part for the coefficient of i in the quotient.
- Watch for c and d both at 0 — that denominator is 0 + 0i, and division by it is undefined.
Worked example — (1 + 2i) divided by (3 + 4i)
Set Numerator real, a to 1 and Numerator imaginary, b to 2, describing 1 + 2i; set Denominator real, c to 3 and Denominator imaginary, d to 4, describing 3 + 4i. Multiplying by the conjugate 3 − 4i turns the denominator into c² + d² = 9 + 16 = 25, and the numerator expands to (1)(3) + (2)(4) = 11 for the real part and (2)(3) − (1)(4) = 2 for the imaginary part, so Result: real part reads 11 ⁄ 25 = 0.44 and Result: imaginary part reads 2 ⁄ 25 = 0.08.
Check it by multiplying back: (0.44 + 0.08i)(3 + 4i) = 1.32 + 1.76i + 0.24i + 0.32i² = 1.32 + 2i − 0.32 = 1 + 2i, exactly the numerator this sheet started from. That round trip — divide, then multiply the quotient by the original denominator — is the fastest hand check for any complex division, and it lands on the nose here.
Questions
What is the formula for dividing complex numbers?
For (a + bi) ⁄ (c + di), multiply top and bottom by the denominator's conjugate, c − di, then read off Re = (ac + bd) ⁄ (c² + d²) and Im = (bc − ad) ⁄ (c² + d²). Both share the same real denominator, c² + d², which is exactly what the conjugate multiplication was for.
Why does the real part formula add while the imaginary part formula subtracts?
Expanding (a + bi)(c − di) gives (ac + bd) + (bc − ad)i. The −di term multiplies bi to give −bdi², and since i² = −1 that becomes +bd, an addition. The same −di term multiplies a to give −adi, which stays subtracted in the imaginary term. One sign survives the i² flip and the other does not.
What happens if the denominator's imaginary part is 0?
The formulas reduce to ordinary real division: Re becomes a ⁄ c and Im becomes b ⁄ c, since d = 0 clears every bd and ad term that touches it, leaving only c² underneath. Dividing 4 + 2i by the real number 2, for instance, gives 2 + 1i — each part divided separately.
Can a complex number be divided by 0 + 0i?
No. The shared denominator c² + d² is 0 only when both c and d are 0, and dividing by zero is exactly as undefined here as it is for real numbers. This sheet flags that input rather than returning a result.
How does complex division relate to division in polar form?
In polar form, dividing two complex numbers divides their moduli, the distances from the origin, and subtracts their angles. It is the same quotient reached a different way — the rectangular method used here works directly from the real and imaginary parts without ever finding an angle.
What is the most common mistake when dividing complex numbers by hand?
Multiplying the denominator by its conjugate but forgetting to multiply the numerator by that same factor, which changes the value of the fraction instead of just rewriting it. Both the top and the bottom need the identical conjugate for the quotient to stay equal to the original.