The complex number 3i is plotted on the complex plane. You then multiply it by i. Which point represents the result?
A(0, 3i) — the imaginary part stays the same and the real part gains i
B(−3, 0) — the result is −3, which lies on the negative real axis
C(3, 3) — both real and imaginary parts shift by the multiplier
D(0, −3) — multiplication by i reflects across the real axis
Multiplying by i rotates a complex number 90° counterclockwise. 3i has modulus 3 and argument 90° (π/2). Multiplying by i adds another 90° to the argument: 90° + 90° = 180°. A complex number at distance 3 from the origin at angle 180° is −3 + 0i, the point (−3, 0) on the negative real axis. Option A is a common confusion between the algebraic computation (i × 3i = 3i² = −3) and incorrect geometric intuition.
Question 2 Multiple Choice
When you multiply two complex numbers z₁ and z₂, what happens to their moduli and arguments?
AThe moduli add and the arguments multiply — analogous to how exponents work
BThe real parts multiply and the imaginary parts multiply separately
CThe moduli multiply and the arguments add
DThe moduli add and the arguments add — multiplication is like vector addition with angles
Complex multiplication combines scaling and rotation: |z₁z₂| = |z₁||z₂| (moduli multiply) and arg(z₁z₂) = arg(z₁) + arg(z₂) (arguments add). Option B is the most common misconception — treating complex multiplication as if each component multiplied independently like real multiplication. This fails: (a + bi)(c + di) ≠ ac + bdi. Option D confuses multiplication with addition, where the real and imaginary parts do add separately.
Question 3 True / False
Adding two complex numbers is geometrically equivalent to vector addition: the real parts and imaginary parts each add independently.
TTrue
FFalse
Answer: True
Correct. Adding z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ gives (x₁+x₂) + i(y₁+y₂) — exactly what you get by placing the two arrows head-to-tail in the plane. The real and imaginary parts behave as independent components, making the complex plane isomorphic to ℝ² as a vector space under addition.
Question 4 True / False
The rule i² = −1 is an arbitrary algebraic convention chosen to extend the real numbers, with no geometric meaning in the complex plane.
TTrue
FFalse
Answer: False
i² = −1 is a geometric fact, not an arbitrary convention. Multiplying by i rotates a complex number 90° counterclockwise (i has modulus 1 and argument π/2). Applying this rotation twice — i.e., computing i² — gives a 180° total rotation, which maps any point z to −z. So i² acting on any complex number gives its negation, which is exactly multiplication by −1. The 'arbitrary' rule is actually an expression of what happens when you rotate 90° twice.
Question 5 Short Answer
Explain what happens geometrically when you multiply a complex number by i, and use this to explain why i² = −1.
Think about your answer, then reveal below.
Model answer: Multiplying by i rotates the complex number 90° counterclockwise around the origin, because i has modulus 1 and argument π/2 = 90°. When multiplying, moduli multiply (1 × anything = anything) and arguments add (90° added to any angle). Applying this twice — i² — means two successive 90° rotations, which is a 180° rotation. A 180° rotation maps any complex number z to −z, so i² applied to 1 gives −1. The algebraic rule i² = −1 is a restatement of what two quarter-turns do to the plane.
This is one of the most illuminating facts in complex analysis: the 'mysterious' definition of i becomes completely natural when you see complex numbers geometrically. The imaginary unit is simply a 90° rotation operator, and its square is a 180° flip — which is exactly multiplication by −1.