Regions in Argand Diagrams (Edexcel A Level Further Maths: Core Pure): Revision Note

Inequalities & regions in Argand diagrams

How do I sketch inequalities such as $\mathbf{Re}\mathit{ }\mathit{z}<\mathit{k}$  or $\mathbf{Im}\mathit{ }\mathit{z}<\mathit{k}$ on an Argand diagram?

• The inequality $\mathrm{Re} z<2$ is satisfied by all complex numbers, $z=x+\mathrm{i}y$, with a real part less than 2

  • On an Argand diagram, this would be the region to the left of the vertical line $x=2$

  • The vertical line itself, $x=2$, 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…

  • $\mathrm{Re} z<k$: shade the region to the left of the dotted vertical line (or solid vertical line for ≤)

  • $\mathrm{Re} z>k$: 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…

  • $\mathrm{Im} z<k$: shade the region below the dotted horizontal line (or solid horizontal line for ≤)

  • $\mathrm{Im} z>k$: shade the region above the dotted horizontal line (or solid horizontal line for ≥)

Sketching the inequalities $\mathbf{Re}\mathit{z}\mathbf{>}\mathbf{3}\mathit{a}\mathit{n}\mathit{d}\mathbf{Im}\mathit{z}\mathbf{\le }\mathbf{4}$

How do I sketch inequalities such as $\left|\mathit{z}-\mathit{a}\right|<\mathit{k}$ on an Argand diagram?

• For a given complex number $a$, the inequality $\left|z-a\right|<k$ 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…

  • $\left|z-a\right|<k$: shade the region inside the circle of radius $k$, centred at $a$

  • $\left|z-a\right|>k$: 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 ≥ )

Sketching the inequalities $\mathbf{\left|}\mathit{z}\mathbf{\right|}\mathbf{\le }\mathbf{5}$  and $\mathbf{|}\mathit{z}\mathbf{-}\mathbf{(}\mathbf{3}\mathbf{+}\mathbf{4}\mathbf{i}\mathbf{)}\mathbf{|}\mathbf{>}\mathbf{2}$

How do I sketch inequalities such as $\mathbf{|}\mathit{z}\mathit{-}\mathit{a}\mathbf{|}\mathbf{<}\mathbf{|}\mathit{z}\mathbf{-}\mathit{b}\mathbf{|}$ on an Argand diagram?

• For two given complex numbers $a$and $b$, the inequality $\left|z-a\right|<|z-b|$ 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…

  • $\left|z-a\right|\mathbf{<}\mathbf{|}\mathit{z}\mathbf{-}\mathit{b}\mathbf{|}$: shade the region on the side of the perpendicular bisector of $a$ and $b$ that includes the point $\mathit{a}$

  • $\left|z-a\right|\mathbf{>}\mathbf{|}\mathit{z}\mathbf{-}\mathit{b}\mathbf{|}$: shade the region on the side of the dotted perpendicular bisector of $a$ and $b$ that includes the point $\mathit{b}$

• A good way to remember which side to shade is that the inequality sign points (like an arrow) to the side to be shaded

Sketching the inequalities $\left|z-2\right|\mathbf{<}\left|z-2i\right|$ and $\left|z-i\right|\mathbf{\ge }\left|z-\left(3+4i\right)\right|$

How do I sketch inequalities such as $\mathit{\alpha }\mathbf{<}\mathbf{arg}\mathbf{}\mathbf{(}\mathbf{z}\mathbf{-}\mathbf{a}\mathbf{)}\mathbf{<}\mathit{\beta }$ on an Argand diagram?

• For a given complex number, $a$, the inequality $\alpha <\arg (z-a)<\beta$ 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…

  • $\alpha <\arg (z-a)<\beta$: shade the wedge-shaped region between the half-line $\arg (z-a)=\alpha$ and the half-line $\arg (z-a)=\beta$, with an open circle at $z=a$ to show the exclusion of this point (to avoid the undefined value of $\arg 0$)

Sketching the inequalities $\mathbf{0}\mathbf{<}\mathbf{arg}\mathbf{(}\mathbf{z}\mathbf{-}\left(1+2i\right)\mathbf{)}\mathbf{<}\frac{\mathbf{\pi }}{\mathbf{4}}$

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 $2<\left|z\right|\le 5$ can be found as follows…

    • Using diagonal lines in the same direction, lightly shade the region $\left|z\right|>2$(the outside of a circle of radius 2 about the origin)

    • Using diagonal lines in a different direction, lightly shade the region $\left|z\right|\le 5$ (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 $\left|z-\mathrm{i}\right|<|z-1|$ and $0\le \arg z\le \frac{3\pi }{4}$ can be found as follows…

    • Using diagonal lines in the same direction, lightly shade the region $\left|z-\mathrm{i}\right|<|z-1|$ (the side including $z=\mathrm{i}$ from the perpendicular bisector of the points $\mathrm{i}$ and 1)

    • Using diagonal lines in a different direction, lightly shade the region $0\le \arg z\le \frac{3\pi }{4}$ (the wedge-shaped region from 0 to $\frac{3\mathrm{\pi }}{4}$ radians from the origin)

    • Where the diagonal lines cross each other highlights the region satisfying both inequalities; this should be shaded boldly

The inequalities $\mathbf{2}\mathbf{<}\left|z\right|\mathbf{\le }\mathbf{5}$  on the left, the inequalities $\left|z-i\right|\mathbf{<}\left|z-1\right|$  and  $\mathbf{0}\mathbf{\le }\mathbf{arg}\mathit{z}\mathbf{<}\frac{\mathbf{3}\mathbf{\pi }}{\mathbf{4}}$on the right

How do I find the greatest (or least) values of $\left|z\right|$ or $\mathbf{arg}\mathit{z}$ in a region?

• Every complex number, $z$, that lies within a region on an Argand diagram will have its own modulus, $\left|z\right|$, that is measured from the origin

• For any shaped region…

  • The least value of $\left|z\right|$ is the distance from the origin to the nearest point in the region

  • The greatest value of $\left|z\right|$ 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 $\left|z\right|$ or subtract a radius for the least value of $\left|z\right|$

• Every complex number, $z$, that lies within a region on an Argand diagram will have its own argument, $\arg z$, as measured from the origin

• For any shaped region…

  • The least value of is $\arg z$ the smallest angle a line through the origin can make to a point in the region

  • The greatest value of $\arg z$ 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 $\arg z$ is $\theta +\alpha$ and the least value of $\arg z$ is $\theta -\alpha$

Find greatest and least values of  $\left|z\right|$ and $\arg z$