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Imaginary Number ○꠹｜Definition｜1st｜20251119205401-00-⌔
Imaginary number
An imaginary number is the product of a real number and the imaginary unit i,1 which is defined by its property i = −1.23 The square of an imaginary number bi is − b. For example, 5 i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary.4
Originally coined in the 17th century by René Descartes5 as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler in the 18th century, and Augustin-Louis Cauchy and Carl Friedrich Gauss in the early 19th century.
An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.6 Imaginary numbers are often called purely imaginary to distinguish them from complex numbers more generally; the set of all imaginary numbers is sometimes denoted , where denotes the set of real numbers.
Printed 2026-06-28.
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Link to original Footnotes
j is usually used in engineering contexts where i has other meanings (such as electrical current). ↩
Uno Ingard, K. (1988). “Chapter 2”. Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X. ↩
Weisstein, Eric W. “Imaginary Number”. mathworld.wolfram.com. Retrieved 2020-08-10. ↩
Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9. ↩
Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. p. 121. ISBN 978-0-8176-4337-9. Extract of page 121 ↩
Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 978-1-4390-4379-0. ↩
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