Complex Numbers : Argand Diagram

In summary, on an Argand diagram, the region R where l iz + 1 + 3i l is less than or equal to 3 can be drawn as a circle with radius 3 and center at (-3,1). The modulus of l (1-y) + (3 + x)i l is also equal to 3. To draw this loci, the equation can be manipulated to l z - ( -3 + i ) l less than or equal to 3.
  • #1
Delzac
389
0
On an Argand diagram, sketch the region R where the following inequalities are satisfied:

l iz + 1 + 3i l less than or equal to 3

How do you draw this loci?
Do i manipulate the equation?

if so i got this :

l z - ( -3 + i ) l less than or equal to 3i

But how in the world do you draw this?

And is :

( l iz + 1 + 3i l less than or equal to 3 )= ( l z* + 1 + 3i l less than or equal to 3)

If so can is it possible to draw the z* loci and relate it to z's loci.

Any help will be greatly appreciated. Thanks.
 
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  • #2
z is some complex number of the form x+iy. What is the modulus of l iz + 1 + 3i l? (Hint: Simplify iz + 1 + 3i to the form A+iB and then find the modulus.)
 
  • #3
If i am going to let z = x + yi

Then i will get the following results :

l (1-y) + (3 + x)i l Less than or = 3

if so, do i draw a circle with radius 3, centre ( -1, -3) ?

So how this feels wrong.
 
  • #4
Delzac said:
If i am going to let z = x + yi

Then i will get the following results :

l (1-y) + (3 + x)i l Less than or = 3

if so, do i draw a circle with radius 3, centre ( -1, -3) ?

So how this feels wrong.

It should be a circle with radius 3, centre (-3,1)... what's the modulus of l (1-y) + (3 + x)i l ?
 
  • #5
Delzac said:
On an Argand diagram, sketch the region R where the following inequalities are satisfied:

l iz + 1 + 3i l less than or equal to 3

How do you draw this loci?
Do i manipulate the equation?

if so i got this :

l z - ( -3 + i ) l less than or equal to 3i

You should have l z - ( -3 + i ) l less than or equal to 3. so that's just a circle (and everything inside the circle) centered at -3+i.
 
  • #6
k i got it, thanks.
 

What are complex numbers and what is an Argand diagram?

Complex numbers are numbers that contain both a real part and an imaginary part. They are represented in the form a + bi, where a and b are real numbers and i is the imaginary unit. An Argand diagram is a graphical representation of complex numbers in which the real part is plotted on the x-axis and the imaginary part is plotted on the y-axis.

What is the purpose of an Argand diagram?

The purpose of an Argand diagram is to visualize complex numbers in the complex plane. It allows us to see the relationship between the real and imaginary parts of a complex number and to perform operations on them, such as addition, subtraction, multiplication, and division.

How do you plot a complex number on an Argand diagram?

To plot a complex number a + bi on an Argand diagram, you would first locate the point (a, b) on the complex plane. Then, draw a line from the origin to the point (a, b) to represent the complex number. The length of this line represents the magnitude of the complex number, and the angle it makes with the positive real axis represents the argument of the complex number.

What is the relationship between an Argand diagram and polar coordinates?

An Argand diagram is closely related to polar coordinates, as the distance from the origin to a point on the complex plane represents the magnitude of the complex number, and the angle it makes with the positive real axis represents the argument of the complex number. This allows us to easily convert between rectangular and polar forms of complex numbers.

How are operations performed on complex numbers using an Argand diagram?

In an Argand diagram, addition and subtraction of complex numbers can be performed by adding or subtracting the corresponding vectors. Multiplication of complex numbers can be visualized as a rotation and scaling of the complex plane, while division can be visualized as a rotation and scaling in the opposite direction. These operations can be easily performed by using the geometric properties of an Argand diagram.

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