A negative number can't have a real square root, since squaring any real number gives a non-negative result. To handle this, we define the imaginary unit i=−1, so that:
−25=−1⋅25=5i
A complex number has the form x+iy, where x and y are real numbers: x is the real part, y is the imaginary part.
Modulus, argument, and conjugate
Every complex number can be written in polar form, x+iy=r(cosθ+isinθ), where:
r=x2+y2,θ=tan−1(xy)
r is called the modulus of x+iy, written ∣x+iy∣, and θ is its argument. The conjugate of x+iy is x−iy, and multiplying a complex number by its own conjugate always gives a real number:
(x+iy)(x−iy)=x2+y2
Arithmetic on complex numbers
Add or subtract complex numbers by combining their real and imaginary parts separately: (1+2i)+(2+i)=3+3i.
Multiplying two complex numbers z1=x1+iy1 and z2=x2+iy2 uses i2=−1:
z1z2=(x1x2−y1y2)+i(x1y2+x2y1)
Continue learning
Complex numbers are exactly what shows up when a quadratic equation's discriminant is negative. The two roots become a conjugate pair like the ones above.