An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.Originally coined in the 17th century as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler and Carl Friedrich Gauss. An imaginary number bi can be added to a real number a to form a complex number of the form...

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f(x) = x. == Definition == Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies f(x) = x for all elements x in M.In other words, the function value f(x) in M (that is, the codomain) is always the same input element x of M (now considered as the domain). The identity function on M is clearly...

In mathematics, the restriction of a function f is a new function f | A {\displaystyle f\vert _{A}} obtained by choosing a smaller domain A for the original function f {\displaystyle f} . The notation f ↾ A {\displaystyle f{\upharpoonright _{A}}} is also used. == Formal definition == Let f ...