CIE A Level Maths: Pure 3

Revision Notes

8.2.5 Inequalities & Regions in Argand Diagrams

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Inequalities & Regions in Argand Diagrams

How do I sketch inequalities such as bold Re blank bold italic z less than bold italic k  or bold Im blank bold italic z less than bold italic k on an Argand diagram?

  • The inequality Re blank z less than 2 is satisfied by all complex numbers, z equals x plus straight i y, with a real part less than 2
    • On an Argand diagram, this would be the region to the left of the vertical line x equals 2
    • The vertical line itself, x equals 2, 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…
    • Re blank z less than k: shade the region to the left of the dotted vertical line (or solid vertical line for ≤)
    • Re blank z greater than k: 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…
    • Im blank z less than k: shade the region below the dotted horizontal line (or solid horizontal line for ≤)
    • Im blank z greater than k: shade the region above the dotted horizontal line (or solid horizontal line for ≥)

8-2-5_notes_fig1

Sketching the inequalities bold Re bold space bold italic z bold greater than bold 3 bold space bold italic a bold italic n bold italic d bold space bold Im bold space bold italic z bold less or equal than bold 4

How do I sketch inequalities such as open vertical bar bold italic z minus bold italic a close vertical bar less than bold italic k on an Argand diagram?

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

8-2-5_notes_fig2

Sketching the inequalities  and 

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

  • For two given complex numbers a spaceand b, the inequality open vertical bar z minus a close vertical bar less than vertical line z minus b vertical line represents all complex numbers, z, that lie closer to a than to b
  • Here is how to represent the following inequalities as regions on an Argand diagram…
    • stretchy vertical line z minus a stretchy vertical line bold less than bold vertical line bold italic z bold minus bold italic b bold vertical line: shade the region on the side of the perpendicular bisector of a and b that includes the point bold italic a
    • stretchy vertical line z minus a stretchy vertical line bold greater than bold vertical line bold italic z bold minus bold italic b bold vertical line: shade the region on the side of the dotted perpendicular bisector of a and b that includes the point bold italic b
  • A good way to remember which side to shade is that the inequality sign points (like an arrow) to the side to be shaded

 8-2-5_notes_fig3

Sketching the inequalities stretchy vertical line z minus 2 stretchy vertical line bold less than stretchy vertical line z minus 2 i stretchy vertical line and stretchy vertical line z minus i stretchy vertical line bold greater or equal than stretchy vertical line z minus open parentheses 3 plus 4 i close parentheses stretchy vertical line

How do I sketch inequalities such as bold italic alpha bold less than bold arg bold invisible function application bold left parenthesis bold z bold minus bold a bold right parenthesis bold less than bold italic beta on an Argand diagram?

  • For a given complex number, a, the inequality alpha less than arg invisible function application left parenthesis z minus a right parenthesis less than beta represents all complex numbers, z, that have an argument between alpha and beta, as measured from the point alpha
  • Here is how to represent the following inequalities as regions on an Argand diagram…
    • alpha less than arg left parenthesis z minus a right parenthesis less than beta: shade the wedge-shaped region between the half-line arg left parenthesis z minus a right parenthesis equals alpha and the half-line arg invisible function application left parenthesis z minus a right parenthesis equals beta, with an open circle at z space equals space a to show the exclusion of this point (to avoid the undefined value of arg space 0)

8-2-5_notes_fig4

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 2 less than open vertical bar z close vertical bar less or equal than 5 can be found as follows…
      • Using diagonal lines in the same direction, lightly shade the region vertical line z vertical line greater than 2 space(the outside of a circle of radius 2 about the origin)
      • Using diagonal lines in a different direction, lightly shade the region open vertical bar z close vertical bar less or equal than 5 (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 open vertical bar z minus straight i close vertical bar less than vertical line z minus 1 vertical line and 0 less or equal than arg invisible function application z less or equal than fraction numerator 3 pi over denominator 4 end fraction can be found as follows…
      • Using diagonal lines in the same direction, lightly shade the region open vertical bar z minus straight i close vertical bar less than vertical line z minus 1 vertical line (the side including z equals straight i from the perpendicular bisector of the points straight i and 1)
      • Using diagonal lines in a different direction, lightly shade the region 0 less or equal than arg invisible function application z less or equal than fraction numerator 3 pi over denominator 4 end fraction (the wedge-shaped region from 0 to fraction numerator 3 straight pi over denominator 4 end fraction radians from the origin)
      • Where the diagonal lines cross each other highlights the region satisfying both inequalities; this should be shaded boldly

8-2-5_notes_fig5

The inequalities bold 2 bold less than stretchy vertical line z stretchy vertical line bold less or equal than bold 5  on the left, the inequalities stretchy vertical line z minus i stretchy vertical line bold less than stretchy vertical line z minus 1 stretchy vertical line  and  bold 0 bold less or equal than bold arg bold space bold italic z bold less than fraction numerator bold 3 bold pi over denominator bold 4 end fractionon the right

How do I find the greatest (or least) values of stretchy vertical line z stretchy vertical line or begin mathsize 22px style bold arg bold italic space bold italic z end style in a region?

  • Every complex number, z, that lies within a region on an Argand diagram will have its own modulus, open vertical bar z close vertical bar, that is measured from the origin
  • For any shaped region…
    • The least value of vertical line z vertical line is the distance from the origin to the nearest point in the region
    • The greatest value of vertical line z vertical line 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 vertical line z vertical line or subtract a radius for the least value of vertical line z vertical line
  • Every complex number, z, that lies within a region on an Argand diagram will have its own argument, arg space z, as measured from the origin
  • For any shaped region…
    • The least value of is arg space z the smallest angle a line through the origin can make to a point in the region
    • The greatest value of arg space z 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 theta, the argument of the centre…
    • Find alpha, 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 arg space z is theta space plus space alpha and the least value of arg space z is theta space minus space alpha

8-2-5_notes_fig6

Find greatest and least values of  open vertical bar z close vertical bar and arg space z

Worked example

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3T~_4SUb_8-2-5_example_fig1-part-2

oa5Q2N4J_8-2-5_example_fig1-part-3

Exam Tip

  • When shading an unbounded region in the exam, make sure you extend your shading outwards, crossing any axes where necessary.
  • When shading multiple inequalities to find a common region, lots of methods are accepted; these include lightly shading individual regions to see where they overlap, or starting to shade along the boundaries of individual regions to find the common region, or marking individual regions to find the region with multiple marks, etc.

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Mark

Author: Mark

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.