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
Further Maths (Core Pure): arithmetic, modulus–argument form, loci and roots of polynomials. Use the calculator to check exact answers.
Practise it: Algebra and functions.
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.
Complex numbers on the Casio fx-991EX ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991EX ClassWiz: scientific (no graphs).
- MENU 2 (Complex).
- Type (3 + 4i) ÷ (1 − 2i): i is ENG. =.
- |z|: SHIFT hyp (Abs). Argument, conjugate and ▶r∠θ are in OPTN.
- Set the angle unit to radians in SETUP for the argument in radians.
Complex numbers on the Casio fx-991CW ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991CW ClassWiz: scientific (no graphs); the newer ClassWiz with a HOME menu of apps.
- HOME → Complex.
- Type the calculation with i (in the Complex app the i key is available on the keypad, see the User's Guide) and EXE.
- Abs, Argument, Conjugate and the polar conversion are in CATALOG → Complex.
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 allowed in A Level exams. Use it to learn, to check a calculator answer, and to see what a function or data set looks like.
- 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.
- Casio fx-991EX / fx-570EX Quick Start Guides : Casio fx-991EX ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-991CW / fx-570CW User's Guide : Casio fx-991CW ClassWiz, the fixed ROM (these models take no OS updates).
- 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.