imaginary unit

English

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 $a+bi+cj+dk$.

Translations

imaginary number

• Finnish: imaginaariyksikkö (fi)
• French: unité imaginaire (fr) f
• German: imaginäre Einheit (de) f
• Hindi: अधिकल्पित एकक m (adhikalpit ekak)
• Italian: unità immaginaria (it) f
• Japanese: 虚数単位 (ja) (kyosū tan'i)
• Korean: 허수단위 (heosudanwi)
• Latin: unitas imaginaria f
• Polish: jednostka urojona (pl) f
• Russian: мни́мая едини́ца (ru) f (mnímaja jediníca)
• Thai: หน่วยจินตภาพ (nùai-jin-dtà-pâap)