Complex number

In mathematics, a complex number is a number of the form ⁠ a + b i {\displaystyle a+bi} ⁠, where a and b are real numbers, and i is the imaginary unit, a number that satisfies the equation ⁠ i 2 = − 1 {\displaystyle \textstyle i^{2}=-1} ⁠; the number a is called the real part of ⁠ a + b i {\displaystyle a+bi} ⁠, and b is called the imaginary part. The complex numbers form a number system which extends the real numbers and supports the same basic arithmetic operations of addition, subtraction, multiplication, and division, satisfying the same associative, commutative, and distributive laws. This makes the complex numbers a field. The set of complex numbers is denoted by either of the symbols C {\displaystyle \mathbb {C} } or C. Because any nonzero real number, whether positive or negative, squares to a positive number, taking the square root of a negative number was historically considered impossible. When, in the 16th century, such square roots were found helpful for determining the real solutions of cubic equations, they were originally considered a mere symbolic contrivance, and in the 17th century were given the name imaginary numbers. Despite the historical nomenclature, "imaginary" numbers are not fictitious: they are no less valid mathematically than real numbers, and they are essential to the scientific description of the physical world. Complex numbers allow solutions to all non-constant polynomial equations with real or complex coefficients, even those that have no solutions in real numbers, a result known as the fundamental theorem of algebra. The complex numbers also form a real vector space of dimension two, with { 1 , i } {\displaystyle \{1,i\}} as a standard basis. Thus each complex number z = a + i b {\displaystyle z=a+ib} can be uniquely associated to a point...