What is the reflection of z in the real axis?erisedk said:Homework Statement
Reflection of the line ##\bar{a}z + a\bar{z} = 0## in the real axis is
Homework Equations
The Attempt at a Solution
I know that a line in the complex plane is represented as ##\bar{a}z + a\bar{z} + b= 0## and that its slope ##μ = \dfrac{-a}{\bar{a}}##. I'm not sure how to do this problem. I'm also not very good with complex geometry so please help.
Right. So if you have two points ##z## and ##\bar w##, how would you write their reflections notationally? What is the general rule you see here?erisedk said:Its conjugate.
You need to conjugate those complex numbers z which are on that line instead of conjugating the equation.erisedk said:However, the funny thing is if I take the conjugate of only ##z##, I get the desired answer, i.e. ##\bar{a}\bar{z} + az = 0##. I can't really explain that though.
If you have an equation for z that specifies a point, z=a say, then how do you write the equation for the reflection of that point? You would write ##\bar z=a## or ##z=\bar a##, not ##\bar z=\bar a##.erisedk said:As ##\bar{z}## and ##w##? That I need to take the conjugate of the equation of the line? But that gives me back the original line. However, the funny thing is if I take the conjugate of only ##z##, I get the desired answer, i.e. ##\bar{a}\bar{z} + az = 0##. I can't really explain that though.
Complex numbers are numbers that contain both a real part and an imaginary part. They are typically written in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary part is a multiple of the imaginary unit i, which is defined as the square root of -1.
To add or subtract complex numbers, you simply combine the real parts and the imaginary parts separately. For example, (2 + 3i) + (5 + 2i) = (2 + 5) + (3 + 2)i = 7 + 5i. Similarly, (2 + 3i) - (5 + 2i) = (2 - 5) + (3 - 2)i = -3 + i.
The conjugate of a complex number a + bi is a - bi. In other words, the conjugate of a complex number is the same number with the sign of the imaginary part changed. For example, the conjugate of 2 + 3i is 2 - 3i.
To multiply complex numbers, you use the FOIL method, just like with binomials. For example, (2 + 3i)(5 + 2i) = 10 + 4i + 15i + 6i^2 = 10 + 19i - 6 = 4 + 19i. To divide complex numbers, you use the fact that the product of a complex number and its conjugate is a real number. For example, (2 + 3i)/(5 + 2i) = (2 + 3i)(5 - 2i)/(5 + 2i)(5 - 2i) = (10 - 6i + 15i - 6i^2)/(25 - 10i + 10i - 4i^2) = (16 + 9i)/29.
In reflection, complex numbers are used as coordinates to represent points in the complex plane. The reflection of a point can be found by multiplying its complex coordinates by a reflection matrix. This process can be used to reflect any shape or object in the complex plane. Additionally, the reflection of a complex number about the real or imaginary axis can be found by taking the conjugate of the number.