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