Complex Number Plane
Key Questions
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The complex plane is a Cartesian-like plane where each complex number is a point, the x coordinate being the real part of the complex number and the y coordinate the imaginary part. In other words,
#z=a+bi# in the complex plane is the point#(a,b)# in the Cartesian plane.
Dean R.
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1
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Jun 21 2018
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In simple terms the modulus of a complex number is its size.
If you picture a complex number as a point on the complex plane, it is the distance of that point from the origin.
If a complex number is expressed in polar coordinates (i.e. as
#r(cos theta + i sin theta)# ), then it's just the radius (#r# ).If a complex number is expressed in rectangular coordinates - i.e. in the form
#a+ib# - then it's the length of the hypotenuse of a right angled triangle whose other sides are#a# and#b# .From Pythagoras theorem we get:
#|a+ib|=sqrt(a^2+b^2)# .
George C.
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May 11 2015
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Answer:
#1 = (1, 0)# and#i = (0, 1)# Explanation:
The complex number plane is usually considered as a two dimensional vector space over the reals. The two coordinates represent the real and imaginary parts of the complex numbers.
As such, the standard orthonormal basis consists of the number
#1# and#i# ,#1# being the real unit and#i# the imaginary unit.We can consider these as vectors
#(1, 0)# and#(0, 1)# in#RR^2# .In fact, if you start from a knowledge of the real numbers
#RR# and want to describe the complex numbers#CC# , then you can define them in terms of pairs of real numbers with arithmetic operations:#(a, b) + (c, d) = (a+c, b+d)" "# (this is just addition of vectors)#(a, b) * (c, d) = (ac-bd, ad+bc)# The mapping
#a -> (a, 0)# embeds the real numbers in the complex numbers, allowing us to consider real numbers as just complex numbers with a zero imaginary part.Note that:
#(a, 0) * (c, d) = (ac, ad)# which is effectively scalar multiplication.
George C.
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Jul 21 2017