How complex-number arithmetic works
A complex number has a real part and an imaginary part (i, where i² = −1). Addition and subtraction combine matching parts; multiplication expands via the distributive property and substitutes i² = −1. Division multiplies numerator and denominator by the denominator's conjugate to make the denominator a real number.
- a, b — real and imaginary parts of the first number z₁
- c, d — real and imaginary parts of the second number z₂
- i — imaginary unit, i² = −1
More detail
Why division is different
Complex division multiplies numerator and denominator by the denominator's conjugate (c−di), turning the denominator into the real number c²+d². When c²+d² is 0 — meaning z₂ is 0 — division is undefined and the result shows as “—”.
Check the sign too. A negative imaginary part appears above as “−…i”. Multiplication and division set the sign of the real and imaginary parts independently, so don't infer one from the other.
Frequently asked questions
What is (3+4i) + (1+2i)?
Add matching parts: (3+4i)+(1+2i) = (3+1)+(4+2)i = 4+6i. Enter z1 real = 3, imaginary = 4, operation = Add, z2 real = 1, imaginary = 2 above to see it computed live.
Why is complex multiplication different from real multiplication?
Because i²=−1. Expanding (a+bi)×(c+di) with the distributive property gives ac+adi+bci+bdi²; substituting i²=−1 simplifies it to (ac−bd)+(ad+bc)i. For example (1+2i)×(3+4i) = (1×3−2×4)+(1×4+2×3)i = −5+10i.
Why does the calculator show “—” for division?
The division formula's denominator is c²+d². If z2's real part (c) and imaginary part (d) are both 0, the denominator is 0 and division is undefined, so the calculator shows “—” instead of a number.
What does the imaginary unit i mean?
i is the imaginary unit satisfying i²=−1, introduced because real numbers alone can't express the square root of a negative number. A complex number a+bi consists of a real part a and an imaginary part b (the coefficient of i).
How is complex division calculated?
Multiply numerator and denominator by the denominator's conjugate (c−di) to turn the denominator into the real number c²+d²: (a+bi)÷(c+di) = [(ac+bd)+(bc−ad)i]/(c²+d²). For example (1+2i)÷(3+4i) = [(1×3+2×4)+(2×3−1×4)i]/(3²+4²) = (11+2i)/25 = 0.44+0.08i.