Edexcel A Level Further Maths: Core Pure:复习笔记1.1.6 Regions in Argand Diagrams

How do I sketch inequalities such asor  on an Argand diagram?

• The inequality is satisfied by all complex numbers, , with a real part less than 2
  • On an Argand diagram, this would be the region to the left of the vertical line
  • The vertical line itself, , should be drawn dotted to show that points on this line are not permitted due to the strict inequality (<)
• In general, dotted lines are used with strict inequalities ( < or > ) and solid lines are used with inequalities that can be equal ( ≤ or ≥ )
• Here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the region to the left of the dotted vertical line (or solid vertical line for ≤)
  • : shade the region to the right of the dotted vertical line (or solid vertical line for ≥)
• Similarly, here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the region below the dotted horizontal line (or solid horizontal line for ≤)
  • : shade the region above the dotted horizontal line (or solid horizontal line for ≥)

Sketching the inequalities and

How do I sketch inequalities such as  on an Argand diagram?

• For a given complex number , the inequality represents all complex numbers, , that lie inside the circle of radius , centred at
• Here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the region inside the circle of radius , centred at
  • : shade the (unbounded) region outside circle of radius , centred at
  • Again use a dotted line for strict inequalities (< and >) and a solid line for weak inequalities ( ≤ or ≥ )

Sketching the inequalities and

How do I sketch inequalities such as  on an Argand diagram?

• For two given complex numbers and , the inequality  represents all complex numbers, , that lie closer to  than to
• Here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the region on the side of the perpendicular bisector of  and that includes the point
  • : shade the region on the side of the dotted perpendicular bisector of  and that includes the poin
• A good way to remember which side to shade is that the inequality sign points (like an arrow) to the side to be shaded

Sketching the inequalities  and

How do I sketch inequalities such as on an Argand diagram?

• For a given complex number, , the inequality represents all complex numbers, , that have an argument between α and β, as measured from the pointα
• Here is how to represent the following inequalities as regions on an Argand diagram…
  • : shade the wedge-shaped region between the half-line and the half-line , with an open circle at to show the exclusion of this point (to avoid the undefined value of )

Sketching the inequalities

How do I draw multiple inequalities on the same Argand diagram?

• To sketch a region that satisfies multiple inequalities, we need to find the intersection of all the regions (where all regions overlap)
  • E.g. the region satisfied by  can be found as follows…
    • Using diagonal lines in the same direction, lightly shade the region (the outside of a circle of radius 2 about the origin)
    • Using diagonal lines in a different direction, lightly shade the region  (the inside of a circle of radius 5 about the origin)
    • Where the diagonal lines cross each other highlights the region satisfying both inequalities; this should be shaded boldly
  • E.g. the region satisfied by and  can be found as follows…
    • Using diagonal lines in the same direction, lightly shade the region (the side including  from the perpendicular bisector of the points i and 1)
    • Using diagonal lines in a different direction, lightly shade the region (the wedge-shaped region from 0 to  radians from the origin)
    • Where the diagonal lines cross each other highlights the region satisfying both inequalities; this should be shaded boldly

The inequalities   on the left, the inequalities and  on the right

How do I find the greatest (or least) values of or  in a region?

• Every complex number, , that lies within a region on an Argand diagram will have its own modulus, , that is measured from the origin
• For any shaped region…
  • The least value of  is the distance from the origin to the nearest point in the region
  • The greatest value of  is the distance from the origin to the farthest point in the region
  • Sometimes the least value is zero (if the origin is in the region) and/or the greatest value is infinite (if the region is unbounded)
• For circular regions that do not contain the origin…
  • Find the length from the origin to the centre of the circle…
  • then add a radius for the greatest value of or subtract a radius for the least value of
• Every complex number, , that lies within a region on an Argand diagram will have its own argument, , as measured from the origin
• For any shaped region…
  • The least value of is  the smallest angle a line through the origin can make to a point in the region
  • The greatest value of  is the largest angle a line through the origin can make to a point in the region
• For circular regions that do not contain the origin…
  • Find θ, the argument of the centre…
  • Find α, the angle between a tangent to the circle through the origin and the line from the origin to the centre…
    • This can be done using trigonometry, as a radius meets a tangent at right angles
  • The greatest value of is  and the least value of is

Find greatest and least values of and