Imaginary number

The powers of $i$ are cyclic:
$\ \vdots$
$\ i^{-2}=-1{\phantom {i}}$
$\ i^{-1}=-i{\phantom {1}}$
$\ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}$
$\ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}$
$\ \ i^{2}\ =-1{\phantom {i}}$
$\ \ i^{3}\ =-i{\phantom {1}}$
$\ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}$
$\ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}$
$\ \vdots$
$i$ 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 $i 2 = -1$ .^[1]^[2] The square of an imaginary number $bi$ is $- b 2$ . For example, $5 i$ 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).

^ 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.

[2] Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.

[3] Weisstein, Eric W. "Imaginary Number". mathworld.wolfram.com. Retrieved 2020-08-10.

[4] Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9.

[5] 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

[6] 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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