Modulus, argument and polar form

Q: How do I find the modulus, argument and polar form of a complex number on a TI-84 Plus CE?

1. Switch the calculator into complex (a + bi) mode. 2. MODE → a+bi. Enter i with 2nd → . ; MATH → CPX gives abs( (modulus), angle( (argument) and ▶Polar. 3. Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

Q: How do I find the modulus, argument and polar form of a complex number on a Casio fx-CG50 and fx-CG100?

1. Switch the calculator into complex (a + bi) mode. 2. In Run-Matrix use SHIFT → 0 for i; OPTN → CPLX gives Abs, Arg and ▶r∠θ (polar form). 3. Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

Q: How do I find the modulus, argument and polar form of a complex number on a TI-Nspire CX?

1. Switch the calculator into complex (a + bi) mode. 2. Settings → set Complex (Rectangular/Polar). Use abs() for modulus, angle() for argument; type i with the dedicated key. 3. Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

Q: What should I watch out for when I find the modulus, argument and polar form of a complex number?

Use radian mode for arguments in AA HL unless the question asks for degrees.

Q: How are marks awarded when I find the modulus, argument and polar form of a complex number in an IB exam?

In the worked example on this page (3 marks, Paper 2), the marks are: M1: Convert to polar form; A1: |z| = 5; A1: arg(z) = 0.927 (rad).

Work with a + bi directly - get the modulus, argument and polar/Euler form in one step.

In short: The modulus and argument of a complex number give its length and its angle from the positive real axis, which together make up the polar form r(cos θ + i sin θ). The GDC can convert between Cartesian and polar form and return both values directly. Check the calculator is in the angle mode the question expects.

When you'd use this

Converting a complex number from a + bi form into modulus-argument (polar) or Euler form.
Finding |z| or arg(z) directly, without sketching an Argand diagram by hand.
Multiplying or dividing complex numbers in polar form, where moduli multiply and arguments add.
Checking a polar form conversion you did by hand, especially the quadrant of the argument.

TI-84 Plus CE Casio fx-CG50 and fx-CG100 TI-Nspire CX

At a glance: TI-84 Plus CE, Casio fx-CG50 and TI-Nspire CX compared

	TI-84 Plus CE	Casio fx-CG50 and fx-CG100	TI-Nspire CX
Key sequence	MODE → a+bi. Enter i with 2nd → . ; MATH → CPX gives abs( (modulus), angle( (argument) and ▶Polar.	In Run-Matrix use SHIFT → 0 for i; OPTN → CPLX gives Abs, Arg and ▶r∠θ (polar form).	Settings → set Complex (Rectangular/Polar). Use abs() for modulus, angle() for argument; type i with the dedicated key.

On a TI-84 Plus CE

Switch the calculator into complex (a + bi) mode.
MODE → a+bi. Enter i with 2nd → . ; MATH → CPX gives abs( (modulus), angle( (argument) and ▶Polar.
Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

Tip: Use radian mode for arguments in AA HL unless the question asks for degrees.

On a Casio fx-CG50 and fx-CG100

Switch the calculator into complex (a + bi) mode.
In Run-Matrix use SHIFT → 0 for i; OPTN → CPLX gives Abs, Arg and ▶r∠θ (polar form).
Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

Tip: Use radian mode for arguments in AA HL unless the question asks for degrees.

On a TI-Nspire CX

Switch the calculator into complex (a + bi) mode.
Settings → set Complex (Rectangular/Polar). Use abs() for modulus, angle() for argument; type i with the dedicated key.
Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

Tip: Use radian mode for arguments in AA HL unless the question asks for degrees.

Try it yourself

Here's a real IB-style question that uses exactly this technique.

Medium Calculator Paper 2 [3 marks]

For z = 3 + 4i, find |z| and arg(z) in radians, to 3 significant figures.

|z| = 5, arg(z) = 0.927

Mark it

Correct 3 / 3 marks

Worked solution & mark scheme:

M1 Convert to polar form

A1 |z| = 5

A1 arg(z) = 0.927 (rad)

Common questions

When would I need to find the modulus, argument and polar form of a complex number in IB Maths?

Work with a + bi directly - get the modulus, argument and polar/Euler form in one step. Converting a complex number from a + bi form into modulus-argument (polar) or Euler form. Finding |z| or arg(z) directly, without sketching an Argand diagram by hand.

How do I find the modulus, argument and polar form of a complex number on a TI-84 Plus CE?

1. Switch the calculator into complex (a + bi) mode. 2. MODE → a+bi. Enter i with 2nd → . ; MATH → CPX gives abs( (modulus), angle( (argument) and ▶Polar. 3. Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

How do I find the modulus, argument and polar form of a complex number on a Casio fx-CG50 and fx-CG100?

1. Switch the calculator into complex (a + bi) mode. 2. In Run-Matrix use SHIFT → 0 for i; OPTN → CPLX gives Abs, Arg and ▶r∠θ (polar form). 3. Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

How do I find the modulus, argument and polar form of a complex number on a TI-Nspire CX?

1. Switch the calculator into complex (a + bi) mode. 2. Settings → set Complex (Rectangular/Polar). Use abs() for modulus, angle() for argument; type i with the dedicated key. 3. Check the argument is in the range the question wants (−π < θ ≤ π, or 0 to 2π).

What should I watch out for when I find the modulus, argument and polar form of a complex number?

Use radian mode for arguments in AA HL unless the question asks for degrees.

How are marks awarded when I find the modulus, argument and polar form of a complex number in an IB exam?

In the worked example on this page (3 marks, Paper 2), the marks are: M1: Convert to polar form; A1: |z| = 5; A1: arg(z) = 0.927 (rad).

Practise with your calculator

Questions that need this technique link back to this guide. Try one in practice mode, or see every GDC guide.