complex number

A complex number bundles a real number and an imaginary number into a single object, written a + bi. Think of it as a two-part recipe: a is the “ordinary” real ingredient and bi is the imaginary ingredient. Together they form a number you can add, subtract, multiply, and divide just like ordinary numbers, as long as you remember the one rule i^2 = -1.

Every complex number has the form a + bi where a and b are real numbers; a is the real part and b is the imaginary part. Real numbers are simply the special case b = 0, and imaginary numbers are the special case a = 0. The whole collection is written C, the complex numbers, and it contains the real numbers R inside it.

Complex numbers are not exotic curiosities — they are the natural home for solving polynomial equations. Inside C every quadratic, cubic, or higher-degree equation has all of its solutions, which is the deep reason the field of complex numbers was worth inventing.

One honest caveat: unlike real numbers, complex numbers cannot be ordered in a way that respects arithmetic. There is no consistent notion of one complex number being “greater than” another, so inequalities like a + bi > c + di simply do not make sense.