Imaginary numbers are as valid as the negatives once were
A complex number has form a + bi, where a and b are real and i satisfies i² = −1. Sixteenth-century mathematicians treated square roots of negatives as symbolic tricks for cubic equations. They extend the reals into a field where every non-constant polynomial has a root.
A complex number combines a real part a and an imaginary part b in the expression a + bi, where i is the imaginary unit with i² = −1. The set forms a field extending real arithmetic with the same addition, subtraction, multiplication, and division laws. Despite the name, imaginary numbers are mathematically as valid as reals.
Historically, negative numbers' square roots seemed impossible because any nonzero real square is positive. Sixteenth-century solutions to cubic equations made such roots useful, though they were first dismissed as contrivances. The 17th century named them imaginary. The fundamental theorem of algebra guarantees every non-constant polynomial with real or complex coefficients has a complex root.
Each complex number maps to a point (a, b) in the Argand diagram: real part on the horizontal axis, imaginary on the vertical. Addition translates the plane; multiplication scales and rotates. Real numbers are complex numbers with zero imaginary part; purely imaginary numbers have zero real part.
Addition combines parts separately: (x + yi) + (u + vi) = (x + u) + (y + v)i. Multiplication expands via i² = −1, as in (2 − i)(3 + 4i) = 10 + 5i. The complex conjugate reflects across the real axis. The modulus |z| = √(x² + y²) gives distance from the origin. Electromagnetism often writes a + bj instead of a + bi because i denotes electric current. Geometric problems can be solved by translating shapes into complex arithmetic. The set of all complex numbers is denoted C or the blackboard-bold C in standard notation.
Source: Complex number
