The imaginary unit, usually denoted by i, is a mathematical constant that is a solution to the quadratic equation x² = −1, which is not solved by any real number. Any real-number multiple of the imaginary unit is called an imaginary number.

Combining the real numbers with the imaginary unit using addition and multiplication generates a new number system called the complex numbers, which consists of all numbers of the form a + bi with real numbers a and b.

There are two complex square roots of −1: the imaginary unit i and its additive inverse −i. More generally, every complex number has two complex-valued square roots which are additive inverses of each other, except for zero, which has zero as its (double) square root.

Historically the imaginary unit was denoted by ⁠${\displaystyle {\sqrt {-1}}}$⁠, though this is now rare. In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead. For example, in electrical engineering the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current.

Square roots of negative numbers are called imaginary because in early-modern mathematics, only what are now called real numbers, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so the square root of a negative number was previously considered undefined or nonsensical. The name imaginary is generally credited to René Descartes, and Isaac Newton used the term as early as 1670. The i notation was introduced by Leonhard Euler.

A unit is an undivided whole, and unity or the unit number is the number one (1).

The imaginary unit i is defined solely by the property that its square is −1: $${\displaystyle i^{2}=-1.}$$

With i defined this way, it follows directly from algebra that i and −i are both square roots of −1.