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Mathematics · Ch 4 — Complex Numbers and Quadratic Equations

Algebra of Complex Numbers

4.3

Algebra of Complex Numbers

Addition of Complex Numbers

Complex numbers are added by adding their real parts and their imaginary parts separately. If z1=a+ibz_1 = a + ib and z2=c+idz_2 = c + id, then

z1+z2=(a+c)+i(b+d)z_1 + z_2 = (a + c) + i(b + d)

For example, (3+2i)+(−1+5i)=(3−1)+i(2+5)=2+7i(3 + 2i) + (-1 + 5i) = (3 - 1) + i(2 + 5) = 2 + 7i.

Note

Addition of complex numbers is just like adding two vectors in the plane — you add the horizontal components (real parts) and the vertical components (imaginary parts) independently.

Subtraction of Complex Numbers

Subtraction follows the same pattern. For z1=a+ibz_1 = a + ib and z2=c+idz_2 = c + id,

z1−z2=(a−c)+i(b−d)z_1 - z_2 = (a - c) + i(b - d)

So (5−3i)−(2+4i)=(5−2)+i(−3−4)=3−7i(5 - 3i) - (2 + 4i) = (5 - 2) + i(-3 - 4) = 3 - 7i.

Multiplication of Complex Numbers

Multiplication uses the distributive law and the fact that i2=−1i^2 = -1. For z1=a+ibz_1 = a + ib and z2=c+idz_2 = c + id,

z1z2=(a+ib)(c+id)=a(c+id)+ib(c+id)=ac+iad+ibc+i2bd=ac+iad+ibc−bd=(ac−bd)+i(ad+bc)\begin{aligned} z_1 z_2 &= (a + ib)(c + id) \\ &= a(c + id) + ib(c + id) \\ &= ac + iad + ibc + i^2 bd \\ &= ac + iad + ibc - bd \\ &= (ac - bd) + i(ad + bc) \end{aligned}

Thus the product formula is:

(a+ib)(c+id)=(ac−bd)+i(ad+bc)(a + ib)(c + id) = (ac - bd) + i(ad + bc)

For instance, (2+3i)(1−4i)=2(1)+2(−4i)+3i(1)+3i(−4i)=2−8i+3i−12i2=2−5i+12=14−5i(2 + 3i)(1 - 4i) = 2(1) + 2(-4i) + 3i(1) + 3i(-4i) = 2 - 8i + 3i - 12i^2 = 2 - 5i + 12 = 14 - 5i.

Watch out

A common mistake is forgetting that i2=−1i^2 = -1 turns the last term into −bd-bd, not +bd+bd. Always write i2i^2 explicitly before simplifying.

Division of Complex Numbers

To divide one complex number by another, we multiply numerator and denominator by the conjugate of the denominator. The conjugate of c+idc + id is c−idc - id. For z1=a+ibz_1 = a + ib and z2=c+idz_2 = c + id (with z2≠0z_2 \neq 0),

z1z2=a+ibc+id=(a+ib)(c−id)(c+id)(c−id)\frac{z_1}{z_2} = \frac{a + ib}{c + id} = \frac{(a + ib)(c - id)}{(c + id)(c - id)}

The denominator becomes a real number:

(c+id)(c−id)=c2−(id)2=c2−i2d2=c2+d2(c + id)(c - id) = c^2 - (id)^2 = c^2 - i^2 d^2 = c^2 + d^2

So the division formula is:

a+ibc+id=ac+bdc2+d2+ibc−adc2+d2\frac{a + ib}{c + id} = \frac{ac + bd}{c^2 + d^2} + i\frac{bc - ad}{c^2 + d^2}

For example, 1+2i3−4i=(1+2i)(3+4i)(3−4i)(3+4i)=3+4i+6i+8i29+16=3+10i−825=−5+10i25=−15+25i\frac{1 + 2i}{3 - 4i} = \frac{(1 + 2i)(3 + 4i)}{(3 - 4i)(3 + 4i)} = \frac{3 + 4i + 6i + 8i^2}{9 + 16} = \frac{3 + 10i - 8}{25} = \frac{-5 + 10i}{25} = -\frac{1}{5} + \frac{2}{5}i.

Tip

When dividing, always check that the denominator's conjugate is used. The product of a complex number and its conjugate is always a positive real number: zzˉ=a2+b2z \bar{z} = a^2 + b^2.

Properties of Addition and Multiplication

Complex numbers satisfy the same algebraic properties as real numbers. Let z1,z2,z3z_1, z_2, z_3 be complex numbers.

Closure: The sum z1+z2z_1 + z_2 and product z1z2z_1 z_2 are both complex numbers.

Commutativity: z1+z2=z2+z1z_1 + z_2 = z_2 + z_1 and z1z2=z2z1z_1 z_2 = z_2 z_1.

Associativity: (z1+z2)+z3=z1+(z2+z3)(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) and (z1z2)z3=z1(z2z3)(z_1 z_2) z_3 = z_1 (z_2 z_3).

Identity: There exist unique complex numbers 0=0+0i0 = 0 + 0i (additive identity) and 1=1+0i1 = 1 + 0i (multiplicative identity) such that z+0=zz + 0 = z and z⋅1=zz \cdot 1 = z for every complex number zz.

Inverse: Every complex number z=a+ibz = a + ib has an additive inverse −z=−a−ib-z = -a - ib satisfying z+(−z)=0z + (-z) = 0. Every non-zero complex number zz has a multiplicative inverse z−1z^{-1} satisfying z⋅z−1=1z \cdot z^{-1} = 1, given by:

z−1=1z=zˉ∣z∣2=a−iba2+b2z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2} = \frac{a - ib}{a^2 + b^2}

Distributivity: z1(z2+z3)=z1z2+z1z3z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3.

Important

The set of complex numbers with these operations forms a field — it satisfies all the same algebraic laws as the real numbers. This means you can manipulate complex expressions exactly as you do real expressions, with the single extra rule i2=−1i^2 = -1.

The Conjugate of a Complex Number

The conjugate of z=a+ibz = a + ib is zˉ=a−ib\bar{z} = a - ib. Geometrically, it reflects zz across the real axis.

Properties of Conjugates:

  1. z1+z2‾=z1ˉ+z2ˉ\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}
  2. z1−z2‾=z1ˉ−z2ˉ\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}
  3. z1z2‾=z1ˉ⋅z2ˉ\overline{z_1 z_2} = \bar{z_1} \cdot \bar{z_2}
  4. (z1z2)‾=z1ˉz2ˉ\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}, provided z2≠0z_2 \neq 0
  5. zˉ‾=z\overline{\bar{z}} = z
  6. z+zˉ=2Re(z)z + \bar{z} = 2 \text{Re}(z) and z−zˉ=2iIm(z)z - \bar{z} = 2i \text{Im}(z)
  7. zzˉ=a2+b2=∣z∣2z \bar{z} = a^2 + b^2 = |z|^2, a non-negative real number
›Proof

Proof of property 1: Let z1=a+ibz_1 = a + ib, z2=c+idz_2 = c + id. Then z1+z2=(a+c)+i(b+d)z_1 + z_2 = (a + c) + i(b + d). Its conjugate is (a+c)−i(b+d)=(a−ib)+(c−id)=z1ˉ+z2ˉ(a + c) - i(b + d) = (a - ib) + (c - id) = \bar{z_1} + \bar{z_2}.

Proof of property 3: z1z2=(ac−bd)+i(ad+bc)z_1 z_2 = (ac - bd) + i(ad + bc). Its conjugate is (ac−bd)−i(ad+bc)(ac - bd) - i(ad + bc). Meanwhile, z1ˉ⋅z2ˉ=(a−ib)(c−id)=ac−iad−ibc+i2bd=(ac−bd)−i(ad+bc)\bar{z_1} \cdot \bar{z_2} = (a - ib)(c - id) = ac - iad - ibc + i^2 bd = (ac - bd) - i(ad + bc). The two are equal.

Proof of property 4: Using property 3, z1‾=z2⋅z1z2‾=z2ˉ⋅(z1z2)‾\overline{z_1} = \overline{z_2 \cdot \frac{z_1}{z_2}} = \bar{z_2} \cdot \overline{\left(\frac{z_1}{z_2}\right)}. Since z2≠0z_2 \neq 0, z2ˉ≠0\bar{z_2} \neq 0, so (z1z2)‾=z1ˉz2ˉ\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}.

Modulus of a Complex Number

The modulus (or absolute value) of z=a+ibz = a + ib is ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}, a non-negative real number representing the distance from the origin to the point (a,b)(a, b) in the complex plane.

Properties of Modulus:

  1. ∣z∣≥0|z| \geq 0, with ∣z∣=0|z| = 0 if and only if z=0z = 0
  2. ∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1| |z_2| …
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