How to do complex numbers (modulus, argument, polar form) on the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II
On the fx-CG50: MENU → 1 (RUN-MATRIX) → OPTN → F3 (COMPLEX).On the TI-84 Plus CE: mode → a+bi.
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.
Which IB course and paper
| Course | GDC papers | What you use it for |
|---|---|---|
| AA SL | Not in the syllabus | — |
| AA HL | Papers 2 and 3 (Paper 1 is calculator-free) | Most complex-number questions are on Paper 1 (no calculator); use the GDC on Papers 2 and 3 and to check your algebra. |
| AI SL | Not in the syllabus | — |
| AI HL | Papers 1, 2 and 3 | Arithmetic, modulus–argument form and complex numbers in sinusoidal models. |
A GDC is allowed on every AI paper and on AA Papers 2 and 3; AA Paper 1 is calculator-free, so learn the method by hand as well.
Topic notes: Complex numbers and Argand diagrams (AI HL) · Complex numbers (AA HL).
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).
Watch the key presses on the fx-CG5016 presses, one at a time
The walkthrough needs JavaScript. The same key presses are in the numbered steps below.
- 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.
Watch the key presses on the TI-84 Plus CE6 presses, one at a time
The walkthrough needs JavaScript. The same key presses are in the numbered steps below.
- 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 a different model: check the exam rules before using one).
- 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.
For teachers: Show on the board Print a keystroke card Teaching GDC skills
Quick answers
How do I type i (imaginary numbers) on the fx-CG50?
MENU 1 (RUN-MATRIX). Type (3 + 4i) ÷ (1 − 2i) where i is SHIFT 0. EXE gives −1 + 2i.
How do I type i (imaginary numbers) on the TI-84 Plus CE?
Type (3 + 4i) / (1 − 2i); i is 2nd ..
How do I type i (imaginary numbers) on the TI-Nspire CX II?
Type (3+4i)/(1−2i) with i from the π▸ key. enter.
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°).
Now try it on an exam-style question
Use the keystrokes above on IB-style complex numbers questions. Free to try, no sign-up.
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.
- Exact answers: when a question asks for an exact modulus or argument (√5, π/4), a decimal from the calculator scores less.
- Angle mode: calculus with trig functions (and most AA trig) needs radians; bearings and triangle questions use degrees. Check the mode before the first trig question of the paper.
- Accuracy: unless the question says otherwise, IB markschemes want answers exact or to 3 significant figures. Keep the full calculator value for later parts and round only the answer you write down (x = 2.094551… is written 2.09, not 2.1 or 2.0946).
If you get an error
fx-CG50: Ma ERROR
Ma ERROR Casio's Math ERROR: the result is outside the allowable calculation range, or you are trying to do an illegal mathematical operation, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check your input values and change them so the result is within the allowable range.
TI-84 Plus CE: ERR:DOMAIN, ERR:SYNTAX, NONREAL ANSWERS, ERR:DATA TYPE
ERR:DOMAIN A value given to a function or instruction is outside the values it accepts. In a regression, TI says it means the data do not suit the model you chose. Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Exponential and power models need positive data. Fix: Change the value that is out of range, or choose a model that suits the data.
ERR:SYNTAX The command contains a syntax error: a misplaced function, argument, bracket or comma. Check: TI points out that subtract − and negative (−) are different keys, and using the wrong one causes this error. Then check the brackets and commas. Fix: Find the misplaced function, argument, bracket or comma and correct it.
NONREAL ANSWERS In Real mode, the calculation gave a complex result. Check: The square root of a negative number, or an equation with no real solution. Fix: If a complex answer is expected, switch the mode to a+bi; otherwise check the inputs.
ERR:DATA TYPE An argument is the wrong type, such as a complex number where a real number is needed, or a matrix stored to a list. Check: A complex result used in a real-only command, or a list and a matrix mixed up. Fix: Use the type the command expects.
TI-Nspire CX II: Domain error, Syntax error
Domain error An argument must be in a specified domain, and this one is not (TI's example: rand(0)). Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Fix: Change the value that is out of range.
Syntax error The syntax checker found a problem. Check: A missing bracket or comma. Fix: Correct the expression and try again.
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).
Open desmos.com/calculator (free, in any browser) and type:
z=frac{3+4i}{1-2i}|z|arg(z)
Questions
Can I use a calculator for complex numbers in the exam?
In IB Maths you can use one approved GDC on every AI paper and on AA Papers 2 and 3, in exam mode. AA Paper 1 is calculator-free, so learn the method by hand as well.
What are the fx-CG50 keystrokes for complex numbers?
SHIFT MENU (SET UP): Complex Mode = a+bi, Angle = Rad. EXIT. Then MENU 1 (RUN-MATRIX). Type (3 + 4i) ÷ (1 − 2i) where i is SHIFT 0. EXE gives −1 + 2i. Then OPTN F3 (COMPLEX): F2 (Abs) for |z|, F3 (Arg) for arg z, F4 (Conjg) for z*. Use Ans for the last answer. The full steps and the other calculators are in the keystrokes section above.
Does this work on other calculators?
Yes. This page has keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II. The worked example has the same answers on every model, so use it to check.
What should I write as working?
Write the maths, not the key presses: say what you are finding in normal notation (the equation, the probability statement or the integral), then the calculator's answer exact or to 3 significant figures unless the question says otherwise. Keep full accuracy in later parts.
How do I type i (imaginary numbers) on the fx-CG50?
MENU 1 (RUN-MATRIX). Type (3 + 4i) ÷ (1 − 2i) where i is SHIFT 0. EXE gives −1 + 2i.
How do I type i (imaginary numbers) on the TI-84 Plus CE?
Type (3 + 4i) / (1 − 2i); i is 2nd ..
How do I type i (imaginary numbers) on the TI-Nspire CX II?
Type (3+4i)/(1−2i) with i from the π▸ key. enter.
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 models 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. A calculator with no tab here has no checked steps for this skill yet. Worked-example answers were recomputed independently in Python.