A complex number has two parts: a real part and an imaginary part. The imaginary unit i satisfies i² = -1. Every real number is a complex number with b = 0. Complex numbers fill a 2D plane rather than a 1D line, giving every polynomial equation exactly as many roots as its degree.
Multiplying by i is a 90-degree counterclockwise rotation. Multiplying by i twice (i.e. by i²) is a 180-degree rotation, which turns 1 into -1. So i² = -1 is not an algebraic trick; it is a rotation.
Over the real numbers, x²+1=0 has no solution. Over the complex numbers it has two: i and -i. The Fundamental Theorem of Algebra says: extend to complex numbers and every polynomial of degree n has exactly n roots.
Complex numbers extend the real line to a 2D plane by introducing i, where i squared equals -1. Every complex number z = a + bi has a real part a, imaginary part b, modulus |z| = sqrt(a squared + b squared), and argument arg(z) = atan(b/a). Multiplication by e^(i*theta) rotates by theta radians. The Fundamental Theorem of Algebra states every polynomial of degree n has exactly n complex roots counting multiplicity. Complex numbers are the foundation of quantum mechanics, signal processing, and Euler's identity.