Define an Argand diagram .

An Argand diagram is a way of representing complex numbers geometrically, in two dimensions. It gives every complex number its own position in the plane, so that algebra with complex numbers can be seen as a picture.

A complex number can be shown on an Argand diagram in two different ways. What are they?

As a point , marked at the coordinates given by its real and imaginary parts, or as a position vector , drawn as an arrow from the origin to that point. Both represent the same complex number.

How do you find the region satisfying two inequalities at once?

Shade each region lightly on the same diagram, using diagonal lines running in a different direction for each one. Where the two sets of lines cross is the intersection , the only part satisfying both, and that is what gets shaded boldly.

A LevelMathsCambridge (CIE)Pure 3FlashcardsComplex NumbersArgand Diagrams

Argand Diagrams (Cambridge (CIE) A Level Maths: Pure 3): Flashcards

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Define an Argand diagram.

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Define an Argand diagram.
An Argand diagram is a way of representing complex numbers geometrically, in two dimensions.
It gives every complex number its own position in the plane, so that algebra with complex numbers can be seen as a picture.
What do the two axes of an Argand diagram represent?
The horizontal axis is the real axis, labelled $Re$ , and the vertical axis is the imaginary axis, labelled $Im$ .
The complex number $x + y i$ is therefore placed at the point with coordinates $(x, y)$ .
True or False?
The number $2$ cannot be shown on an Argand diagram, because it has no imaginary part.
False.
A real number is a complex number whose imaginary part is $0$ , so $2$ is $2 + 0 i$ and is plotted at $(2, 0)$ .
Every real number has a place on an Argand diagram, on the horizontal axis.
A complex number can be shown on an Argand diagram in two different ways. What are they?
As a point, marked at the coordinates given by its real and imaginary parts, or as a position vector, drawn as an arrow from the origin to that point.
Both represent the same complex number.
Two complex numbers are drawn as position vectors and completed into a parallelogram. What does each diagonal show?
The diagonal starting at the origin gives their sum, $z_{1} + z_{2}$ .
The other diagonal, joining the two plotted points, gives their difference, $z_{1} - z_{2}$ .
A point on an Argand diagram has coordinates $(- 2, - 1)$ . Complete the complex number it represents.
$z =______+______i$
The completed complex number is:
$z = - 2 + (- 1) i = - 2 - i$
Writing the negative second coordinate inside a bracket first makes it clear that the term becomes $- i$ and not $+ i$ .
What is the geometric effect of adding a fixed complex number $w = a + b i$ to $z$ ?
It translates $z$ by the vector $[\begin{matrix} a \\ b \end{matrix}]$ .
Every point moves the same distance in the same direction, so nothing about the shape or size of a drawing changes.
Subtracting a fixed complex number $w = a + b i$ from $z$ also translates $z$ . Complete the translation vector.
$[\begin{matrix} ______ \\ ______ \end{matrix}]$
The completed vector is:
$[\begin{matrix} - a \\ - b \end{matrix}]$
Subtracting $w$ moves a point exactly as far as adding $w$ would, but in the opposite direction.
The point representing $3 + 2 i$ is plotted. Where does it move when $4 - 5 i$ is added?
To the point representing $7 - 3 i$ , at coordinates $(7, - 3)$ .
Adding $4 - 5 i$ moves every point $4$ to the right and $5$ down.
Why can complex numbers be added using column vectors?
Because $x + y i$ can be represented by the position vector $[\begin{matrix} x \\ y \end{matrix}]$ , and each of the two parts is combined on its own.
Adding two column vectors does precisely that, entry by entry.
What is the geometric effect of taking the complex conjugate of $z$ ?
It reflects $z$ in the real axis.
The real part is unchanged while the imaginary part changes sign, so the point crosses the horizontal axis to an equal distance on the other side.
True or False?
Taking the conjugate moves every point on an Argand diagram.
False.
A point on the real axis stays exactly where it is, because its imaginary part is already $0$ and changing the sign of $0$ changes nothing.
Those are the only points left in place.
Define the modulus of a complex number.
The modulus of $z$ , written $|z|$ , is its distance from the origin on an Argand diagram.
For $z = x + y i$ , Pythagoras gives $|z| = \sqrt{x^{2} + y^{2}}$ .
True or False?
For any two complex numbers, $|z_{1} + z_{2}| = |z_{1}| + |z_{2}|$ .
False.
Both $3 + 4 i$ and $- 3 + 4 i$ have a modulus of $5$ , but their sum is $8 i$ , whose modulus is $8$ rather than $10$ .
Define the argument of a complex number.
The argument of $z$ , written $\arg z$ , is the angle measured anticlockwise in radians from the positive real axis to the line joining the origin to $z$ .
The argument of $0$ is undefined, because no such line can be drawn.
Complete the range in which arguments are usually given.
$______< \arg z \leq______$
The completed range is:
$- π < \arg z \leq π$
Occasionally the range $0 \leq \arg z < 2 π$ is used instead.
True or False?
A complex number in the third or fourth quadrant has a negative argument.
True.
In the usual range the angle is measured from the positive real axis, and for a point below that axis the shorter turn is clockwise, which counts as negative.
Measuring anticlockwise instead would give an angle greater than $π$ , outside the range.
How do you find the argument of $- 1 - \sqrt{3} i$ ?
Sketch the point first to see which quadrant it lies in, then use $\tan$ on the two side lengths to get the acute angle to the real axis, here $\tan^{- 1} \sqrt{3} = \frac{π}{3}$ .
The point is in the third quadrant, so the argument is $- (π - \frac{π}{3}) = - \frac{2 π}{3}$ .
What is the modulus-argument (polar) form of a complex number?
$z = r (\cos θ + i \sin θ)$ , where $r$ is the modulus and $θ$ is the argument.
It follows from substituting $x = r \cos θ$ and $y = r \sin θ$ into $z = x + y i$ .
A complex number is written as $3 (\cos \frac{π}{5} - i \sin \frac{π}{5})$ . What is its argument?
$- \frac{π}{5}$ , not $\frac{π}{5}$ .
The minus sign means this is not yet in polar form. Using $\cos (- θ) \equiv \cos θ$ and $\sin (- θ) \equiv - \sin θ$ rewrites it with a plus sign and reveals the true argument.
When two complex numbers are multiplied, complete what happens to their moduli and their arguments.
$|z_{1} z_{2}| =______, \arg (z_{1} z_{2}) =______$
The completed results are:
$|z_{1} z_{2}| = |z_{1}| |z_{2}|, \arg (z_{1} z_{2}) = \arg z_{1} + \arg z_{2}$
Dividing works the same way in reverse: the moduli are divided and the arguments subtracted.
Two complex numbers are multiplied and the argument comes out as $\frac{7 π}{6}$ . What now?
Subtract $2 π$ to bring it back into the usual range, giving $- \frac{5 π}{6}$ .
Adding or subtracting $2 π$ lands on the same direction from the origin, so the complex number itself is unchanged.
Define a locus on an Argand diagram.
A locus is the set of all points representing complex numbers $z$ that satisfy a given equation.
On the diagram it appears as a line or a curve, made up of every point that fits.
What loci do $Re (z) = k$ and $Im (z) = k$ give?
$Re (z) = k$ is the vertical line $x = k$ , and $Im (z) = k$ is the horizontal line $y = k$ .
Every point on the first has real part $k$ , and every point on the second has imaginary part $k$ .
What locus does $|z - a| = k$ give, and why?
A circle of radius $k$ centred at the point $a$ .
This is because $|z - a|$ is the distance between the points representing $z$ and $a$ , so the equation says every such $z$ sits a fixed distance $k$ away from $a$ .
The locus $|z - (3 + 4 i)| = 10$ is a circle. Complete its Cartesian equation.
${(x -______)}^{2} + {(y -______)}^{2} =______$
The completed equation is:
${(x - 3)}^{2} + {(y - 4)}^{2} = 100$
The centre $3 + 4 i$ becomes the coordinates $(3, 4)$ , and the radius appears squared on the right-hand side.
What is the centre of the circle $|z - 8 i + 3| = 12$ ?
$- 3 + 8 i$ .
Rewrite the equation as $|z - (- 3 + 8 i)| = 12$ first, because the centre can only be read off once the expression has been put into the form $|z - a|$ .
True or False?
The origin lies inside the circle $|z - (3 + 4 i)| = 10$ .
True.
The distance from the centre $3 + 4 i$ to the origin is $\sqrt{3^{2} + 4^{2}} = 5$ , which is less than the radius of $10$ .
Checking this before sketching tells you whether the circle crosses the axes.
What locus does $|z - a| = |z - b|$ give?
The perpendicular bisector of the line joining the points $a$ and $b$ .
The equation says $z$ is equally far from $a$ and from $b$ , and that is exactly the set of points making up the perpendicular bisector.
What locus does $\arg (z - a) = α$ give?
A half-line starting at the point $a$ , making an angle $α$ with the positive real axis.
The point $a$ itself is marked with a small open circle, since taking $z = a$ would require $\arg 0$ .
When is a boundary drawn dotted on an Argand diagram, and when solid?
Dotted for a strict inequality, $<$ or $>$ , because points on the boundary itself are not included.
Solid for $\leq$ or $\geq$ , where the boundary points do satisfy the inequality.
Which region does $Re (z) > 3$ describe?
Everything to the right of the vertical line $x = 3$ , with that line drawn dotted.
The region is unbounded, so the shading should be extended outwards, crossing the axes where it needs to.
Which region does $|z - a| < k$ describe?
Everything inside the circle of radius $k$ centred at $a$ , with the circle itself dotted.
Reversing it to $|z - a| > k$ instead gives the unbounded region outside that circle.
Which side of the perpendicular bisector does $|z - a| < |z - b|$ shade?
The side containing the point $a$ , because the inequality says $z$ is closer to $a$ than to $b$ .
A quick check is that the inequality sign points, like an arrow, towards the side that gets shaded.
Which region does $α < \arg (z - a) < β$ describe?
A wedge-shaped region with its vertex at $a$ , lying between the two half-lines at angles $α$ and $β$ measured from that point.
Only the part between them satisfies both inequalities at once.
How do you find the region satisfying two inequalities at once?
Shade each region lightly on the same diagram, using diagonal lines running in a different direction for each one.
Where the two sets of lines cross is the intersection, the only part satisfying both, and that is what gets shaded boldly.
True or False?
Every region on an Argand diagram has a greatest value of $|z|$ .
False.
An unbounded region, such as the outside of a circle, stretches away without limit, so there is no greatest value.
At the other end, the least value is $0$ whenever the region contains the origin.
A region is the disc $|z - 5 i| \leq 2$ . How do you find the greatest value of $|z|$ ?
Find the distance from the origin to the centre, here $5$ , then add the radius, giving $7$ .
Subtracting the radius instead gives the least value, $3$ , and both work because the origin lies outside this circle.
A region is the disc $|z - 5 i| \leq 2$ . How do you find the least value of $\arg z$ ?
Draw the tangent from the origin to the circle on the side giving the smallest angle.
A tangent meets a radius at a right angle, so $\sin α = \frac{2}{5}$ gives the angle $α$ between that tangent and the line from the origin to the centre.
The least argument is then $\frac{π}{2} - α = 1.16$ radians.

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