3.9 - Rotations and square roots

3.9 - Rotations and square roots

Multiplying by ii turns a point through a right angle: can a calculation really perform a rotation? You will multiply and divide using moduli and arguments, interpret complex operations geometrically, and find both square roots of a complex number with exact Cartesian working.

Multiplying and dividing in polar form

In Cartesian form, i(1+i)=−1+ii(1+i)=-1+i. The distance from the origin stays 2\sqrt2, but the direction changes from π/4\pi/4 to 3π/43\pi/4. The factor ii, whose argument is π/2\pi/2, has added its own angle.

To see the general pattern, expand two unit-modulus factors:

(cos⁡α+isin⁡α)(cos⁡β+isin⁡β)=(cos⁡αcos⁡β−sin⁡αsin⁡β)+i(sin⁡αcos⁡β+cos⁡αsin⁡β)=cos⁡(α+β)+isin⁡(α+β).\begin{aligned} (\cos\alpha+i\sin\alpha)(\cos\beta+i\sin\beta) &=(\cos\alpha\cos\beta-\sin\alpha\sin\beta)\\ &\quad+i(\sin\alpha\cos\beta+\cos\alpha\sin\beta)\\ &=\cos(\alpha+\beta)+i\sin(\alpha+\beta). \end{aligned}

The last step is the angle-addition identity. Multipliers rr and ss contribute the ordinary real product rsrs. Division reverses multiplication, so it divides those moduli and subtracts the angles.

Polar multiplication and division

For z1=reiαz_1=re^{i\alpha} and z2=seiβz_2=se^{i\beta}, with r,s>0r,s>0,

z1z2=rs ei(α+β),z1z2=rs ei(α−β).z_1z_2=rs\,e^{i(\alpha+\beta)},\qquad \frac{z_1}{z_2}=\frac rs\,e^{i(\alpha-\beta)}.

Hence ∣z1z2∣=∣z1∣∣z2∣|z_1z_2|=|z_1||z_2| and ∣z1/z2∣=∣z1∣/∣z2∣|z_1/z_2|=|z_1|/|z_2|. Arguments add or subtract up to a whole number of turns. These complex rules are not listed in MF19.

If a sum of angles is outside the requested argument interval, add or subtract 2π2\pi. A modulus is a positive distance, so divide moduli normally; do not subtract them.

Worked example 1: a product crossing the argument boundary

Question. Let u=3e2πi/3u=3e^{2\pi i/3} and v=2eπi/2v=2e^{\pi i/2}. Express uvuv as reiθre^{i\theta}, where r>0r>0 and −π<θ≤π-\pi<\theta\le\pi.

Plan. Multiply moduli and add angles, then bring the angle into the specified interval.

uv=6ei(2π/3+π/2)=6e7πi/6=6e−5πi/6.uv=6e^{i(2\pi/3+\pi/2)}=6e^{7\pi i/6} =6e^{-5\pi i/6}.

Answer. 6e−5πi/66e^{-5\pi i/6}.

Check. u=−3/2+(33/2)iu=-3/2+(3\sqrt3/2)i and v=2iv=2i. Their Cartesian product is −33−3i-3\sqrt3-3i, which has modulus 66 and direction −5π/6-5\pi/6.

Keep the two numbers and change multiplication to division. The moduli divide and the angle operation reverses.

Worked example 2: a quotient

Question. For the same u=3e2πi/3u=3e^{2\pi i/3} and v=2eπi/2v=2e^{\pi i/2}, express u/vu/v in the same required form.

Plan. Divide the moduli and subtract the denominator's angle.

uv=32ei(2π/3−π/2)=32eiπ/6.\frac uv=\frac32e^{i(2\pi/3-\pi/2)} =\frac32e^{i\pi/6}.

Answer. (3/2)eiπ/6(3/2)e^{i\pi/6}.

Check. The quotient is 33/4+3i/43\sqrt3/4+3i/4. Multiplying this by 2i2i gives −3/2+(33/2)i=u-3/2+(3\sqrt3/2)i=u.

The compact argument equation needs care: principal arguments need not add without adjustment. For instance, two arguments 3π/43\pi/4 add to 3π/23\pi/2, but the product's argument in (−π,π](-\pi,\pi] is −π/2-\pi/2. The directions agree modulo 2π2\pi.

Watch out

For u/vu/v, the angle is arg⁡u−arg⁡v\arg u-\arg v, in that order. Subtracting a negative denominator angle means adding its magnitude. Division by zero remains undefined.

Use a negative denominator angle here.

Geometric operations

A fixed addition moves every point by the same displacement. For example, adding 3−i3-i moves 1+2i1+2i to 4+i4+i: three units right and one unit down. Subtracting 3−i3-i reverses that translation. It does not generally preserve distance from the origin.

Conjugation changes x+iyx+iy to x−iyx-iy, reflecting the point in the real axis. For example, 1+2i1+2i and 1−2i1-2i lie equally far above and below that axis. Real points are unchanged.

Multiplication as a transformation

For a fixed nonzero c=seiβc=se^{i\beta}, the map z↦czz\mapsto cz multiplies distances from the origin by s=∣c∣s=|c| and rotates points anticlockwise by β\beta about the origin. A negative β\beta describes a clockwise turn.

The inverse map z↦z/cz\mapsto z/c has scale factor 1/s1/s and rotation angle −β-\beta. These geometric descriptions are not listed in MF19.

The origin stays fixed. This follows from the polar rules for every other point, while c⋅0=0c\cdot0=0. If c=0c=0, multiplication sends every point to the origin and there is no rotation angle.

Worked example 3: enlarging and rotating

Question. Describe fully the transformation w=2izw=2iz. Find the image of z=1+iz=1+i.

Plan. Read the modulus and argument of the multiplier, then check the image in Cartesian form.

2i=2eiπ/2,w=2i(1+i)=2i+2i2=−2+2i.2i=2e^{i\pi/2},\qquad w=2i(1+i)=2i+2i^2=-2+2i.

Answer. An enlargement with scale factor 22, centre the origin, combined with an anticlockwise rotation of π/2\pi/2 about the origin. The image is −2+2i-2+2i.

Check. The distance changes from 2\sqrt2 to 8=22\sqrt8=2\sqrt2; the direction changes from π/4\pi/4 to 3π/43\pi/4, an increase of π/2\pi/2.

Replace multiplication by division using the same constant. Both geometric effects reverse.

Worked example 4: the inverse geometric effects

Question. Describe fully the transformation w=z/(2i)w=z/(2i). Find the image of z=1+iz=1+i.

Plan. Use reciprocal scale factor and opposite rotation, then simplify with 1/i=−i1/i=-i.

w=−i2(1+i)=12−12i.w=-\frac i2(1+i)=\frac12-\frac12i.

Answer. An enlargement with scale factor 1/21/2, centre the origin, combined with a clockwise rotation of π/2\pi/2 about the origin. The image is 1/2−i/21/2-i/2.

Check. The new distance is 2/2\sqrt2/2 and the new angle is −π/4-\pi/4. Multiplication by 2i2i recovers 1+i1+i.

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