Imaginary number

"Imaginary Numbers" redirects here. For the 2013 EP by The Maine, see Imaginary Numbers (EP).

The powers of i
are cyclic:
is a 4th
root of unity

An imaginary number is the product of a real number and the imaginary unit i,[note 1] which is defined by its property i2 = −1.[1][2] The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary.[3]

Originally coined in the 17th century by René Descartes[4] 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.[5]


Cite error: There are <ref group=note> tags on this page, but the references will not show without a {{reflist|group=note}} template (see the help page).

  1. ^ Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.
  2. ^ Weisstein, Eric W. "Imaginary Number". mathworld.wolfram.com. Retrieved 2020-08-10.
  3. ^ Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9.
  4. ^ 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
  5. ^ Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.). Cengage Learning. p. 66. ISBN 978-1-4390-4379-0.