Blogging today from Last Drop Cafè in Claremont, CA. I think I found a new favorite drink: mint mocha made with soy milk.
Here are some basics of the conjugate and modulus of complex numbers.
Let z = x+i*y, with i=√1
Conjugate
Usually labeled "zbar" (z with a line over it), the conjugate is also labeled conj(z) and z*.
conj(z) = x  i*y
Modulus or Absolute Value
Not surprisingly, the modulus, also called the absolute value of the complex number z is defined as:
z = √(x^2 + y^2)
Properties
Let's explore some properties of the conjugate and modulus.
z + conj(z) = (x + i*y) + (x  i*y) = 2*x
z  conj(z) = (x + i*y)  (x  i*y) = 2*i*y
(conj(z))^2 = (x  i*y)^2 = x^2  2*i*x*y + (i*y)^2 = x^2  y^2  2*i*x*y
conj(z)^2 + z^2 = (x  i*y)^2 + (x + i*y)^2 = x^2  2*i*x*y  y^2 + x^2 + 2*i*x*y  y^2
= 2*(x^2  y^2)
conj(z) * z = (x  i*y) * (x + i*y) = x^2 + i*x*y  i*x*y  i^2*y^2 = x^2 + y^2 = z^2
which easily leads to: z = √(z * conj(z)) and
z1 * z2 = √(z1 * conj(z1) * z2 * conj(z2)) = √(z1 * conj(z1)) * √(z2 * conj(z2)) = z1 * z2
and:
z1/z2 = √((z1 * conj(z1))/(z2 * conj(z2))) = √((z1 * conj(z1))/√((z2 * conj(z2)) = z1/z2
Source: Wuncsh, David A. Complex Numbers with Applications. 2nd Edition. AddisonWesley Publishing Company. Reading, MA. 1994
A blog is that is all about mathematics and calculators, two of my passions in life.
Saturday, November 22, 2014
Complex Analysis: The Conjugate, the Modulus, and its Properties
Subscribe to:
Post Comments (Atom)
HP Prime: Area by Quadratic Splines
HP Prime: Area by Quadratic Splines Introduction The program QUADSUM calculates the area under the curve described by the set o...

Casio fx991EX Classwiz Review Casio FX991EX The next incarnation of the fx991 line of Casio calculators is the fx991 EX. ...

TI36X Pro Review This is a review of the TI36X Pro Calculator by Texas Instruments. History Originally, this was the TI30X Pro that w...

One of the missing features of the TI82/83/84 family is the ability to convert between bases. Here are two programs in TIBasic to help...
No comments:
Post a Comment