How to do complex numbers (modulus, argument, polar form) on the Casio fx-CG50
Arithmetic with i, the modulus and argument, and converting between Cartesian and polar/exponential form. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
AA HL and AI HL: arithmetic, modulus–argument form, and (AI HL) complex numbers for sinusoidal models and AC circuits. In AA HL most complex-number work is on the non-calculator paper, so use the GDC to check.
Practise it: HL AA complex numbers · HL AI complex numbers.
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Complex numbers on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
- SHIFT MENU (SET UP): Complex Mode = a+bi, Angle = Rad. EXIT.
- MENU 1 (RUN-MATRIX). Type (3 + 4i) ÷ (1 − 2i) where i is SHIFT 0. EXE gives −1 + 2i.
- OPTN F3 (COMPLEX): F2 (Abs) for |z|, F3 (Arg) for arg z, F4 (Conjg) for z*. Use Ans for the last answer.
- Polar form: F6 (▷) → ▶r∠θ after the number; ▶a+bi converts back.
- In Real mode (the default) a complex answer gives Non-Real ERROR: switch to a+bi.
Complex numbers on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
- mode → a+bi, RADIAN.
- Type (3 + 4i) / (1 − 2i); i is 2nd ..
- math → CMPLX: abs(, angle(, conj(, real(, imag(, ►Polar, ►Rect.
Complex numbers on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS; the CX II CAS is not allowed in IB, A Level or IGCSE exams.
- Settings (home/on → Settings → Document Settings): Real or Complex = Rectangular, Angle = Radian.
- Type (3+4i)/(1−2i) with i from the π▸ key. enter.
- abs(ans), angle(ans), conj(ans) (menu → Number → Complex Number Tools). ▶Polar converts to polar form.
Worked example
z = (3 + 4i) ÷ (1 − 2i). Find z in the form a + bi, |z| and arg z.
Answers (checked with Python)
z = −1 + 2i, |z| = √5 = 2.23607, arg z = 2.03444 radians (116.565°).
Common calculator mistakes
- Argument in degrees when the question wants radians (or the other way round).
- The argument is in (−π, π]: a question asking for 0 ≤ θ < 2π needs 2π added to negative values.
- Missing brackets: 3 + 4i ÷ 1 − 2i is not (3 + 4i) ÷ (1 − 2i).
- Real mode: Non-Real ERROR or Math ERROR instead of a complex answer.
Do it in Desmos
Not in the exam. Desmos is not an approved calculator for IB exams, so never rely on it for Paper 2. It is excellent for learning, checking a GDC answer and building graphs for your Internal Assessment (IA).
- Type z = (3 + 4i)/(1 − 2i). Desmos switches to complex mode and plots z as a point on the Argand diagram.
- |z| and arg(z) give the modulus and argument (in radians).
The interactive graph loads here when you scroll to it.
Or open desmos.com/calculator and type:
z=frac{3+4i}{1-2i}|z|arg(z)
More calculator skills
Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the OS versions shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. Worked-example answers were recomputed independently in Python.