Imaginary unit

Description

The imaginary unit is the complex number $i = 0 + 1 i$. It satisfies $i^2 = -1$, see i squared is minus one. It is a solution of the equation $x^2 + 1 = 0$. The other solution is $-i$. In the complex plane, $i$ is the point $(0, 1)$, at distance $1$ above $0$.

Remarks

The powers of $i$ repeat with period $4$: $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$ and $i^4 = 1$. Multiplication by $i$ turns the complex plane by a quarter turn about $0$, counterclockwise. In electrical engineering, people often write $j$ for $i$.

Explanations

L1 human

The imaginary unit is a new number whose square is $-1$. No real number has this property, so $i$ is not on the real number line. It sits one unit above $0$ in the plane of complex numbers.