1.6.8.9 the square root of complex numbers
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NE 112 Linear algebra for nanotechnology engineering
Douglas Wilhelm Harder, LEL, [email protected]
1.6.8.9 The square rootof complex numbers
Introduction
• In this topic, we will
– Review the square root of real numbers
– Find the square roots of a complex number
• Define one to be the principal square root
– Look at a geometric interpretation
– Observe a property
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Review
• Note that if x > 0, then
• Similarly,
• Question: What is ?
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( ) ( )2 2
x x x= − =
( ) ( )2 2
x j x j x= − = −
Square root of a complex number
• Solution:
– If , then if w2 = z
– Suppose :
• If ,
then and
– Therefore,
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w z=z z =
w w =
( ) ( )2 22 2w w w z z = = = =
2w z= 2 =
2w z
=
3
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Square root of a complex number
• We will define to be
– That root with the greatest real component, and
– If real components are equal,the root with the greatest imaginary component
• The easiest is in the polar representation:
– Require that , in which case
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z
−
def
2z z
=
Square root of a complex number
• Note that our definition still works for real numbers:
– If z is a positive real, then ,
thus
The other root is
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0 0z j = + =
00
2z = = =
( )0 j − + = −
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Square root of a complex number
• Also, it works for negative numbers
– If z is a negative real, then ,
thus
The other root is
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0z j = − + =
2z j
= =
( )0 j j − + = −
Square root of a complex number
• Examples:
– If ,
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1 24
z j
= + =
1 2 24
z j
= + =
4 28
=
4 42 2 2 2 2 2
2 2j
+ −= +
1.0987 0.4551 j +
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Geometric interpretation
• The geometric interpretation of the square root is
– The square root of the magnitude
– Half the angle if
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Properties
Theorem: if and only if
– The square root of a complex number on the unit circleremains on the unit circle
Proof: if and only if ,
which is true if and only if or ,
but if and only if z = 0, so it must be that .
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1z =
z z=
QED
1z =
2z z=
0z = 1z =
0z = 1z =
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Summary
• In this topic, we’ve introduced the square root of a complex number
– Given any non-zero complex number z,
• There are two numbers that when squared equal z
– We defined one to be principal square root
• The magnitude is the square root of the magnitude
• The angle is half the angle if
– The square roots of complex numbers on the unit circle remain on that unit circle
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−
References
[1] https://en.wikipedia.org/wiki/Complex_number#Square_root
[2] https://en.wikipedia.org/wiki/Square_root#
Square_roots_of_negative_and_complex_numbers
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Acknowledgments
None so far.
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Colophon
These slides were prepared using the Cambria typeface. Mathematical equations use Times New Roman, and source code is presented using Consolas.
The photographs of flowers and a monarch butter appearing on the title slide and accenting the top of each other slide were taken at the Royal Botanical Gardens in October of 2017 by Douglas Wilhelm Harder. Please see
https://www.rbg.ca/
for more information.
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Disclaimer
These slides are provided for the NE 112 Linear algebra fornanotechnology engineering course taught at the University ofWaterloo. The material in it reflects the authors’ best judgment inlight of the information available to them at the time of preparation.Any reliance on these course slides by any party for any otherpurpose are the responsibility of such parties. The authors acceptno responsibility for damages, if any, suffered by any party as aresult of decisions made or actions based on these course slides forany other purpose than that for which it was intended.
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