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Pure · Complex numbers

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The Argand diagram, modulus and argument

Picturing a complex number z = a + bi as a point (or vector) on the Argand diagram, its modulus |z| = √(a²+b²) as the distance from the origin, and its argument arg z as the angle from the positive real axis — taken as the principal value in (−π, π], which means getting the quadrant right, not just arctan(b/a). Geometric effects: the conjugate reflects in the real axis, addition is vector addition, and |z₁ − z₂| is the distance between the two points.

Pure · Complex numbers

The Argand diagram, modulus and argument

Picturing a complex number z = a + bi as a point (or vector) on the Argand diagram, its modulus |z| = √(a²+b²) as the distance from the origin, and its argument arg z as the angle from the positive real axis — taken as the principal value in (−π, π], which means getting the quadrant right, not just arctan(b/a). Geometric effects: the conjugate reflects in the real axis, addition is vector addition, and |z₁ − z₂| is the distance between the two points.

Why it works

A real number lives on a number line. A complex number z=a+biz = a + bi needs two coordinates — the real part aa and the imaginary part bb — so it lives on a plane. The Argand diagram is exactly that plane: the horizontal axis is the real axis and the vertical axis is the imaginary axis. The number z=a+biz = a + bi is plotted at the point (a,b)(a, b), and you can equally think of it as the vector arrow from the origin OO to that point. So 3+4i3 + 4i sits 33 across and 44 up; −2−i-2 - i sits in the bottom-left.

Modulus — how far from the origin. The modulus ∣z∣|z| is the length of that vector: by Pythagoras on the right-angled triangle with legs ∣a∣|a| and ∣b∣|b|, ∣z∣=a2+b2.|z| = \sqrt{a^2 + b^2}. It is always ≥0\ge 0 and only depends on the sizes of the parts, so signs do not matter: ∣−1+i∣|{-1} + i| and ∣1+i∣|1 + i| are both 2\sqrt{2}. Two traps to avoid: it is not a+ba + b and not a2+b2a^2 + b^2 — you must square, add, then take the square root.

Argument — the direction. The argument arg⁡z\arg z is the angle the vector makes with the positive real axis, measured anticlockwise as positive and clockwise as negative. An angle is only fixed up to full turns of 2π2\pi, so we pin down one value: the principal argument in the range −π<arg⁡z≤π.-\pi < \arg z \le \pi. Points above the real axis get a positive argument (up to π\pi); points below get a negative one (down to just above −π-\pi).

The key skill is the quadrant, not the button. The acute angle inside the triangle is the reference angle α=arctan⁡∣b∣∣a∣,\alpha = \arctan\frac{|b|}{|a|}, but arg⁡z\arg z is not simply arctan⁡(b/a)\arctan(b/a) — that only works in the first quadrant. Sketch the point first, then read off the argument from where it lands:
  • 1st quadrant (a>0, b>0a>0,\ b>0):  arg⁡z=α\ \arg z = \alpha.
  • 2nd quadrant (a<0, b>0a<0,\ b>0):  arg⁡z=π−α\ \arg z = \pi - \alpha.
  • 3rd quadrant (a<0, b<0a<0,\ b<0):  arg⁡z=−(π−α)\ \arg z = -(\pi - \alpha).
  • 4th quadrant (a>0, b<0a>0,\ b<0):  arg⁡z=−α\ \arg z = -\alpha.
For example z=−1−iz = -1 - i has reference angle arctan⁡11=π4\arctan\frac{1}{1} = \frac{\pi}{4}, but it is in the third quadrant, so arg⁡z=−(π−π4)=−3π4\arg z = -\left(\pi - \frac{\pi}{4}\right) = -\frac{3\pi}{4} — not π4\frac{\pi}{4}, even though arctan⁡−1−1=arctan⁡1=π4\arctan\frac{-1}{-1} = \arctan 1 = \frac{\pi}{4}. The two minus signs cancel inside the fraction and throw the angle into the wrong quadrant; the sketch saves you.

Geometry for free. Because zz is a point on a plane, the algebra of complex numbers has pictures:
  • Conjugate = reflection in the real axis. z∗=a−biz^* = a - bi is the mirror image of
z=a+biz = a + bi across the real axis. Reflecting keeps the distance from OO, so ∣z∗∣=∣z∣|z^*| = |z|, and it flips the angle above/below the axis, so arg⁡z∗=−arg⁡z\arg z^* = -\arg z.
  • Addition = vector addition. z1+z2z_1 + z_2 is found by the parallelogram (tip-to-tail)
rule on the two arrows, exactly like adding vectors.
  • ∣z1−z2∣|z_1 - z_2| = the distance between the points. The vector from the point for z2z_2
to the point for z1z_1 is z1−z2z_1 - z_2, so its length ∣z1−z2∣|z_1 - z_2| is the straight-line distance between them on the diagram — the complex-number version of the distance formula.