Core Pure 1 Chapter 2: Argand Diagrams

20 flashcards · Mathematics · by TheoTyson

1. What does the xxx -axis represent on an argand diagram?

   Real axis

2. What does the yyy -axis represent on an argand diagram?

   Imaginary axis

3. What are the coordinates of z=x +z=x\,+z=x+ i yyy ?

   P(x, y)P(x,\,y)P(x,y)

4. What is the vector of z=x +z=x\,+z=x+ i yyy ?

   xy\begin{matrix} x \\ y \end{matrix}xy​

5. What is the modulus of a complex number?

   Distance from the origin to that number on an Argand diagram. (Magnitude of the vector)

6. What is the modulus of z=x +z=x\,+z=x+ i yyy ?

   ∣z∣=x2+y2\lvert z \rvert = \sqrt{x^2+y^2}∣z∣=x2+y2 ​

7. What is the argument of a complex number?

   The angle between the positive real axis and the line joining the imaginary number to the origin, measured in an anticlockwise direction.

8. What is the argument of z=x +z=x\,+z=x+ i yyy ?

   tan θ=yxtan \: \theta = \frac{y}{x}tanθ=xy​

9. What is the principal argument?

   The argument of any complex number is in the range −π&lt;θ≤π-\pi &lt; \theta \leq \pi−π<θ≤π

10. What is the modulus argument form when ∣z∣=r\lvert z \rvert = r∣z∣=r and arg =θarg \: = \thetaarg=θ ?

    z=r(cos&MediumSpace;θ&MediumSpace;+z = r(cos \: \theta \: +z=r(cosθ+ i sin&MediumSpace;θ)sin \: \theta)sinθ)

11. How do you multiply two complex numbers in modulus argument form?

    z1∗z2=∣z1∣∣z2∣(cos⁡(θ1+θ2)+isin⁡(θ1+θ2))z_1 * z_2 = |z_1||z_2| (\cos{(\theta_1 + \theta_2)} + i\sin{(\theta_1 + \theta_2)})z1​∗z2​=∣z1​∣∣z2​∣(cos(θ1​+θ2​)+isin(θ1​+θ2​))

12. How do you divide two complex numbers in modulus argument form?

    z1z2=∣z1∣∣z2∣(cos⁡(θ1−θ2)+isin⁡(θ1−θ2))\frac{z_1}{z_2} = \frac{|z_1|}{|z_2|} (\cos{(\theta_1 - \theta_2)} + i\sin{(\theta_1 - \theta_2)})z2​z1​​=∣z2​∣∣z1​∣​(cos(θ1​−θ2​)+isin(θ1​−θ2​))

13. What is the distance between two complex numbers?

    z1=x1&ThinSpace;+z_1 = x_1 \, + z1​=x1​+ i y1y_1y1​ and z2=x2&ThinSpace;+z_2 = x_2 \, +z2​=x2​+ i y2y_2y2​ Distance = ∣z2−z1∣\lvert z_2 - z_1 \rvert∣z2​−z1​∣

14. What is the equation for a circle on an argand diagram, and what are the centre and radius?

    ∣z−z1∣=r\lvert z - z_1 \rvert = r∣z−z1​∣=r ∣z−(x1&ThinSpace;+\lvert z - (x_1 \, +∣z−(x1​+ i y1)∣=ry_1) \rvert = ry1​)∣=r Centre = (x1,y1)(x_1, y_1)(x1​,y1​) Radius = rrr

15. What is the argand diagram form to cartesian form of a circle equation?

    ∣z−z1∣=r\lvert z - z_1 \rvert = r∣z−z1​∣=r ∣z−(x1&ThinSpace;+\lvert z - (x_1 \, +∣z−(x1​+ i y1)∣=ry_1) \rvert = ry1​)∣=r (x−x1)2&ThinSpace;+(y−y1)2=r2(x -x_1)^2 \, + (y - y_1)^2 = r^2(x−x1​)2+(y−y1​)2=r2

16. What is the equation for a perpendicular bisector on an argand diagram, and what are the points?

    ∣z−z1∣=∣z−z2∣\lvert z - z_1 \rvert = \lvert z - z_2 \rvert∣z−z1​∣=∣z−z2​∣ ∣z−(x1&ThinSpace;\lvert z - (x_1 \,∣z−(x1​ +++ i y1)∣=∣z−(x2&ThinSpace;+y_1) \rvert = \lvert z - (x_2 \, +y1​)∣=∣z−(x2​+ i y2)∣y_2) \rverty2​)∣ Perpendicular bisector of the the line joining the coordinates (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​) .

17. What is the equation for a half line on an argand diagram, and what are the point and angle?

    arg(z−z1)=θarg(z - z_1)= \thetaarg(z−z1​)=θ arg(z−(x1&ThinSpace;+arg(z - (x_1 \, +arg(z−(x1​+ i y)=θy) = \thetay)=θ Half line from the point (x1,y1)(x_1, y_1)(x1​,y1​) making an angle θ\thetaθ with a line from (x1,y1)(x_1, y_1)(x1​,y1​) parallel to the xxx -axis.

18. What is the argand diagram form to cartesian form of a half line equation?

    arg(z−z1)=θarg(z - z_1) = \thetaarg(z−z1​)=θ arg(z−(x1&ThinSpace;+arg(z - (x_1 \, +arg(z−(x1​+ i y1)=θy_1) = \thetay1​)=θ y−y1=tan&ThinSpace;θ(x−x1)y - y_1 = tan \, \theta(x - x_1)y−y1​=tanθ(x−x1​)

19. What do dashed lines represent for regions?

    "> &lt;or&gt;&lt; or &gt;<or> (Not inclusive)

20. What do solid lines represent for regions?

    ≤or≥\leq or \geq≤or≥ (Inclusive)