Complex Number Plane
Key Questions
-
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.
·
1
·
Jun 21 2018
-
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 area andb .From Pythagoras theorem we get:
|a+ib|=sqrt(a^2+b^2) .
George C.
·
1
·
May 11 2015
-
Answer:
1 = (1, 0) andi = (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 andi ,1 being the real unit andi the imaginary unit.We can consider these as vectors
(1, 0) and(0, 1) inRR^2 .In fact, if you start from a knowledge of the real numbers
RR and want to describe the complex numbersCC , 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.
·
1
·
Jul 21 2017