Porisma

Worked solution

Find exact moduli and arguments in eight positions

Determine, in exact form, the modulus and argument of:

(a) 22; (b) −3-3; (c) 7i7i; (d) −5i-5i;

(e) 1+i31+i\sqrt3; (f) 1−i31-i\sqrt3; (g) −1+i-1+i; (h) −1−i-1-i.

CJC · Example 15

(a) Positive real axis

Answer

Modulus 22; principal argument 00.

∣2∣=22+02=2|2|=\sqrt{2^2+0^2}=2. The point P(2,0)P(2,0) is on the positive real axis, so arg⁡(2)=0\arg(2)=0.

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Argand diagram for source case (a), point P(2,0). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (a), point P(2,0). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (a), point P(2,0). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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(b) Negative real axis

Answer

Modulus 33; principal argument π\pi.

∣−3∣=(−3)2+02=3|-3|=\sqrt{(-3)^2+0^2}=3. The negative real axis has principal argument π\pi in (−π,π](-\pi,\pi].

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Argand diagram for source case (b), point P(−3,0). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (b), point P(−3,0). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (b), point P(−3,0). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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(c) Positive imaginary axis

Answer

Modulus 77; principal argument π/2\pi/2.

∣7i∣=02+72=7|7i|=\sqrt{0^2+7^2}=7. The positive imaginary axis gives arg⁡(7i)=π/2\arg(7i)=\pi/2.

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Argand diagram for source case (c), point P(0,7). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (c), point P(0,7). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (c), point P(0,7). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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(d) Negative imaginary axis

Answer

Modulus 55; principal argument −π/2-\pi/2.

∣−5i∣=02+(−5)2=5|-5i|=\sqrt{0^2+(-5)^2}=5. The negative imaginary axis gives arg⁡(−5i)=−π/2\arg(-5i)=-\pi/2.

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Argand diagram for source case (d), point P(0,−5). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (d), point P(0,−5). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (d), point P(0,−5). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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(e) First quadrant

Answer

Modulus 22; principal argument π/3\pi/3.

∣1+i3∣=12+(3)2=2|1+i\sqrt3|=\sqrt{1^2+(\sqrt3)^2}=2. The point is in quadrant I, so arg⁡(1+i3)=tan⁡−1∣3/1∣=π/3\arg(1+i\sqrt3)=\tan^{-1}\lvert\sqrt3/1\rvert=\pi/3.

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Argand diagram for source case (e), point P(1,√3). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (e), point P(1,√3). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (e), point P(1,√3). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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(f) Fourth quadrant

Answer

Modulus 22; principal argument −π/3-\pi/3.

∣1−i3∣=12+(−3)2=2|1-i\sqrt3|=\sqrt{1^2+(-\sqrt3)^2}=2. In quadrant IV, arg⁡(1−i3)=−tan⁡−1∣−3/1∣=−π/3\arg(1-i\sqrt3)=-\tan^{-1}\lvert-\sqrt3/1\rvert=-\pi/3.

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Argand diagram for source case (f), point P(1,−√3). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (f), point P(1,−√3). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (f), point P(1,−√3). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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(g) Second quadrant

Answer

Modulus 2\sqrt2; principal argument 3π/43\pi/4.

∣−1+i∣=12+12=2|-1+i|=\sqrt{1^2+1^2}=\sqrt2. In quadrant II, α=tan⁡−1∣1/(−1)∣=π/4\alpha=\tan^{-1}\lvert1/(-1)\rvert=\pi/4, so arg⁡(−1+i)=π−α=3π/4\arg(-1+i)=\pi-\alpha=3\pi/4.

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Argand diagram for source case (g), point P(−1,1). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (g), point P(−1,1). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (g), point P(−1,1). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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(h) Third quadrant

Answer

Modulus 2\sqrt2; principal argument −3π/4-3\pi/4.

∣−1−i∣=(−1)2+(−1)2=2|-1-i|=\sqrt{(-1)^2+(-1)^2}=\sqrt2. In quadrant III, α=tan⁡−1∣(−1)/(−1)∣=π/4\alpha=\tan^{-1}\lvert(-1)/(-1)\rvert=\pi/4, so arg⁡(−1−i)=−π+α=−3π/4\arg(-1-i)=-\pi+\alpha=-3\pi/4.

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Argand diagram for source case (h), point P(−1,−1). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (h), point P(−1,−1). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
Argand diagram for source case (h), point P(−1,−1). Horizontal Re and vertical Im use the same unit scale. The signed angle from the positive real axis selects the principal argument stated in the working.
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Solution notes

Use the source interval (−π,π](-\pi,\pi] for principal arguments. For (h), calculator modulus procedure: MATH → CMPLX → 5:abs(; enter −1−i and press ENTER. The screen gives 1.414213562, checking 2\sqrt2. For argument: MATH → CMPLX → 4:angle (; enter −1−i and press ENTER in radian mode. The screen gives −2.35619449, checking −3π/4-3\pi/4.

The source points to Appendix C on page 42 for further modulus/argument rules and labels them non-examinable in its H2 syllabus. This retains the source guidance, without asserting a current syllabus classification.