The inverse tangent loses the quadrant

3 + 4i

53.13°the argument

53.13°what arctan returns

agreesverdict

All four numbers have modulus 5. Only the argument tells them apart, and only two of the four have an argument arctan can find.

The figure needs JavaScript. It plots 3 + 4i, −3 + 4i, −3 − 4i and 3 − 4i on an Argand diagram, all at distance 5 from the origin, with arguments 53.13°, 126.87°, −126.87° and −53.13°. The inverse tangent of b/a returns 53.13, −53.13, 53.13 and −53.13, so it is wrong by 180° for the two numbers on the left. Everything below stands on its own.

The parts of a complex number

Start from i² = −1, and write

z = a + bi

Then, for z = 3 + 4i:

Real part, Re(z) = 3. A real number, not 3 + 0i.
Imaginary part, Im(z) = 4. Also a real number: it is the 4, not the 4i.
Conjugate, z* = 3 − 4i. Flip the sign of the imaginary part and nothing else.
Modulus, |z| = √(3² + 4²) = 5. The distance from the origin.
Argument, arg(z) = 53.13°, or 0.927 radians. The angle from the positive real axis.

Plot a + bi at the point (a, b) and the whole of this becomes geometry: the complex plane, or Argand diagram. The modulus and argument are then the polar coordinates of that point, which is the whole of 1.13.

The conjugate's one job is to make things real.

z z* = (3 + 4i)(3 − 4i) = 9 − 16i² = 9 + 16 = 25

which is |z|². That is why dividing complex numbers works at all: multiply top and bottom by the conjugate of the bottom and the bottom becomes a real number you can divide by.

Why arctan is not enough

The argument is the angle whose tangent is b/a, so arctan(b/a) looks like the answer. It is not, because arctan only ever returns a value between −90° and 90°, and that covers only the right-hand half of the plane. Take the four numbers with modulus 5:

3 + 4i, first quadrant: arctan gives 53.13°, and that is right.
−3 + 4i, second quadrant: arctan gives −53.13°, and the answer is 126.87°. Add 180.
−3 − 4i, third quadrant: both signs cancel, so arctan gives +53.13°, and the answer is −126.87°. Subtract 180.
3 − 4i, fourth quadrant: arctan gives −53.13°, and that is right.

The third case is the dangerous one. A number in the bottom left is reported in the top right, and the only thing that would have told you is sketching it.

So the method has two steps, always.

Sketch it. Mark a along the real axis and b up the imaginary one, and see which quadrant you are in. Ten seconds.
Find the acute angle to the real axis from arctan(|b| / |a|), then place it: θ in the first, 180 − θ in the second, θ − 180 in the third, −θ in the fourth.

The convention is that the argument lies in −180° < arg(z) ≤ 180°, so a third-quadrant number gets a negative argument rather than one over 180. Adding 360 to any answer does not change the number, so an argument of 233.13° is the same point; it is just not the principal value an examiner expects.

Arithmetic in Cartesian form

Adding is componentwise, exactly like vectors:

(3 + 4i) + (1 − 2i) = 4 + 2i

Multiplying is ordinary expansion, with i² = −1 at the end:

(3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 + 8 − 2i = 11 − 2i

and a check comes free: |11 − 2i| = 11.18, which is 5 × √5, the product of the two moduli. Moduli multiply, which is the fact 1.13 is built on.

Dividing uses the conjugate:

(3 + 4i)/(1 − 2i) = (3 + 4i)(1 + 2i)/5 = (−5 + 10i)/5 = −1 + 2i

and the check is to multiply back: (−1 + 2i)(1 − 2i) = −1 + 2i + 2i + 4 = 3 + 4i. One line, and it catches every sign slip, which on this sub-topic is most of them.

On the GDC: complex numbers

Both machines handle complex arithmetic, and both will give you a modulus and an argument without the quadrant problem, because they use a two-argument function internally. Knowing the quadrant rule still matters, because Paper 1 has no calculator.

When you may use it. Analysis Paper 1 is non-calculator, and complex-number questions there want exact answers: a modulus of 5 and an argument of π/3 or 3π/4, not a decimal. Paper 2 and Paper 3 allow it, and that is where 53.1° is acceptable.

TI-Nspire CX II

The imaginary unit is π i, the dedicated i key, not the letter i from the keypad
doc → Settings → Document Settings, set Real or Complex to Rectangular and Angle to Degree
abs(-3-4i) gives 5; angle(-3-4i) gives -126.87, the right answer
Compare tan⁻¹(-4/-3), which gives 53.13. Put the two on one screen

Casio fx-CG50

MENU → Run-Matrix, then SHIFT MENU SET UP and set Complex Mode to a+bi and Angle to Deg
i is SHIFT 0; OPTN → F3 COMPLEX holds Abs, Arg, Conjg, ReP and ImP
Arg(-3-4i) gives -126.87
SET UP is per-application on this machine, so set Complex Mode in Run-Matrix before you start, not in Graph

The mark people lose. Writing arctan(b/a) as the argument. It is right in two quadrants out of four and silently 180° out in the other two, and the modulus is no help because all four of these numbers have modulus 5. On Paper 1, where there is no angle( function to save you, the sketch IS the method. Draw the point before you touch the arctan.

Your turn

1. Find |3 + 4i|.

2. Find arg(3 + 4i) in degrees, to 2 decimal places.

3. Now find arg(−3 − 4i) in degrees, to 2 decimal places.

4. Find the real part of (3 + 4i)/(1 − 2i).

5. Why does arctan(b/a) fail for −3 − 4i?

Question 5. Why does the inverse tangent of b over a fail for minus 3 minus 4i? Because the modulus is negative. Because the two minus signs cancel in the fraction, so arctan cannot tell this number from 3 + 4i. Because the calculator rounds. Because tangent is undefined in the third quadrant.

Where the marks go

1 markThe modulus.

1 markThe acute angle from arctan.

1 markPlacing it in the right quadrant, with a sketch or a stated reason.

The third mark is a mark for the sketch. Examiners award it because the quadrant is the only part of the argument that cannot be read off a calculator on Paper 1, and a correct acute angle in the wrong quadrant scores one out of three.