# imaginary unit

## English

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### 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 ${\displaystyle i}$) that is defined as a solution to the equation ${\displaystyle x^{2}=-1}$.
• 2013, J. A. Sparenberg, Hydrodynamic Propulsion and Its Optimization, Springer, page 183,
In Section 4.3 we introduced the imaginary unit ${\displaystyle i}$ with ${\displaystyle (i)^{2}=-1}$, in the complex space domain. In section 4.4 we introduced the complex unit ${\displaystyle j}$ with ${\displaystyle (j)^{2}=-1}$, 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 ${\displaystyle ij}$.
• 2014, Dennis G. Zill, Warren S. Wright, Advanced Engineering Mathematics, Ascend Learning (Jones & Bartlett Learning), 5th Edition, page 793,
We now simply say that ${\displaystyle i}$ is the imaginary unit and define it by the property ${\displaystyle i^{2}=-1}$. 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 ${\displaystyle \mathbb {H} \subset \mathbb {O} }$, one of the totally imaginary units not contained in ${\displaystyle \mathbb {H} }$, and note that ${\displaystyle {SU}(2)}$ acts transitively on the set of totally imaginary units of a quaternion algebra; as we are assuming ${\displaystyle \alpha \notin \mathbb {H} }$, we may in fact assume that ${\displaystyle \mathbb {O} =\mathbb {H} \oplus \mathbb {H} '}$ with ${\displaystyle \alpha \in \mathbb {H} '}$.

#### Usage notes

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