The Modulus Argument That Split in Two
A student must solve z² = −5 + 12i for complex z and mark both solutions on an Argand diagram. He converts −5 + 12i to modulus–argument form, finds the modulus is 13, and computes the argument as arctan(12/(−5)) ≈ −67.4°, which he writes as a negative angle. He then takes the square root of the modulus and halves the argument, obtaining one root, and is unsure how to produce the second. He also wants to verify his answers by squaring them back, and he is worried because his calculator's arctan gave a fourth-quadrant angle even though −5 + 12i clearly lies in the second quadrant.
1. Find the correct modulus and principal argument of −5 + 12i, explain the calculator discrepancy, and hence find both square roots.
Read the worked answer
Modulus: |−5 + 12i| = √(25 + 144) = √169 = 13. Correct.
Argument: the point (−5, 12) is in the second quadrant (x < 0, y > 0). The reference angle is arctan(12/5) ≈ 67.4°, so the principal argument is 180° − 67.4° = 112.6° (≈ 1.965 rad). The calculator returned −67.4° because arctan(12/(−5)) only knows the ratio and cannot see the quadrant; the student must adjust by adding 180°: −67.4° + 180° = 112.6°. This is the source of his worry.
Square roots: modulus √13, arguments 112.6°/2 = 56.3° and 56.3° + 180° = 236.3°.
Root 1: √13(cos 56.3° + i sin 56.3°) ≈ 3.606(0.5547 + 0.8321i) ≈ 2.00 + 3.00i. Root 2: the negative, ≈ −2.00 − 3.00i.
2. Verify the roots by squaring and explain the general rule for the number and placement of nth roots.
Read the worked answer
Check (2 + 3i)² = 4 + 12i + 9i² = 4 + 12i − 9 = −5 + 12i. Correct. The other root is its negative, (−2 − 3i)² = (2 + 3i)² = −5 + 12i, also correct.
General rule: the equation zⁿ = w has exactly n distinct roots, equally spaced in argument by 360°/n around a circle of radius |w|^{1/n}. For n = 2 the two roots differ by 180°, which is why the second root is simply the negative of the first. The student's single root was correct but incomplete; halving one argument gives one root, and adding 180° gives the other. A reliable procedure: compute modulus and quadrant-correct argument first, then split the argument, then verify by raising back to the power.