# imaginary unit

## English

English Wikipedia has an article on:
Wikipedia the complex or Cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis.

### Etymology

So named because it takes on the role of unit for the imaginary part of a complex number.

### Noun

imaginary unit (plural imaginary units)

1. (number theory, complex analysis, quaternion theory) An imaginary number (in the case of complex numbers, usually denoted $i$ ) that is defined as a solution to the equation $x^{2}=-1$ .
• 2013, J. A. Sparenberg, Hydrodynamic Propulsion and Its Optimization, Springer, page 183,
In Section 4.3 we introduced the imaginary unit $i$ with $(i)^{2}=-1$ , in the complex space domain. In section 4.4 we introduced the complex unit $j$ with $(j)^{2}=-1$ , in connection with the time dependency of the shape (4.4.16) of the swimming profile. These different imaginary units have no interaction, we leave unaltered the product $ij$ .
• 2014, Dennis G. Zill, Warren S. Wright, Advanced Engineering Mathematics, Ascend Learning (Jones & Bartlett Learning), 5th Edition, page 793,
We now simply say that $i$ is the imaginary unit and define it by the property $i^{2}=-1$ . Using the imaginary unit, we build a general complex number out of two real numbers.
• 2021, Bruce Hunt, Locally Mixed Symmetric Spaces, Springer, page 123,
Consider now in addition to the given $\mathbb {H} \subset \mathbb {O}$ , one of the totally imaginary units not contained in $\mathbb {H}$ , and note that ${SU}(2)$ acts transitively on the set of totally imaginary units of a quaternion algebra; as we are assuming $\alpha \notin \mathbb {H}$ , we may in fact assume that $\mathbb {O} =\mathbb {H} \oplus \mathbb {H} '$ with $\alpha \in \mathbb {H} '$ .

#### Usage notes

• The imaginary unit of complex analysis is usually denoted $i$ . In some fields (for instance, electrical engineering), however, it is customarily denoted $j$ , to avoid confusion with the symbol for electric current.
• The complex numbers are generated by assuming a single imaginary unit, $i$ , and constructing the numbers $a+bi$ , where $a$ and $b$ are real numbers.
• The quaternions (regardable as an extension of the complex numbers) are similarly generated by assuming three distinct imaginary units, $i,j,k$ , and constructing the numbers $a+bi+cj+dk$ .