Triangle Inequality/Complex Numbers/Proof 2

Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\cmod z$ denote the modulus of $z$.

Then:

$\cmod {z_1 + z_2} \le \cmod {z_1} + \cmod {z_2}$

Proof

Let $z_1 = a_1 + i a_2, z_2 = b_1 + i b_2$.

\(\ds \cmod {z_1 + z_2}\)

\(\le\)

\(\ds \cmod {z_1} + \cmod {z_2}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {\paren {a_1 + b_1}^2 + \paren {a_2 + b_2}^2}^{\frac 1 2}\)

\(\le\)

\(\ds \paren { {a_1}^2 + {a_2}^2}^{\frac 1 2} + \paren { {b_1}^2 + {b_2}^2}^{\frac 1 2}\)

Definition of Complex Modulus

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {a_1 + b_1}^2 + \paren {a_2 + b_2}^2\)

\(\le\)

\(\ds {a_1}^2 + {a_2}^2 + {b_1}^2 + {b_2}^2 + 2 \paren { {a_1}^2 + {a_2}^2}^{\frac 1 2} \paren { {b_1}^2 + {b_2}^2}^{\frac 1 2}\)

squaring both sides

\(\ds \leadstoandfrom \ \ \)

\(\ds {a_1}^2 + 2 a_1 b_1 + {b_1}^2 + {a_2}^2 + 2 a_2 b_2 + {b_2}^2\)

\(\le\)

\(\ds {a_1}^2 + {a_2}^2 + {b_1}^2 + {b_2}^2 + 2 \paren { {a_1}^2 + {a_2}^2}^{\frac 1 2} \paren { {b_1}^2 + {b_2}^2}^{\frac 1 2}\)

multiplying out

\(\ds \leadstoandfrom \ \ \)

\(\ds a_1 b_1 + a_2 b_2\)

\(\le\)

\(\ds \paren { {a_1}^2 + {a_2}^2}^{\frac 1 2} \paren { {b_1}^2 + {b_2}^2}^{\frac 1 2}\)

simplifying

\(\ds \leadstoandfrom \ \ \)

\(\ds \paren {a_1 b_1 + a_2 b_2}^2\)

\(\le\)

\(\ds \paren { {a_1}^2 + {a_2}^2} \paren { {b_1}^2 + {b_2}^2}\)

Cauchy's Inequality

\(\ds \leadstoandfrom \ \ \)

\(\ds {a_1}^2 {b_1}^2 + 2 a_1 b_1 a_2 b_2 + {a_2}^2 {b_2}^2\)

\(\le\)

\(\ds {a_1}^2 {b_1}^2 + {a_2}^2 {b_2}^2 + {a_1}^2 {b_2}^2 + {a_2}^2 {b_1}^2\)

\(\ds \leadstoandfrom \ \ \)

\(\ds 2 a_1 b_1 a_2 b_2\)

\(\le\)

\(\ds {a_1}^2 {b_2}^2 + {a_2}^2 {b_1}^2\)

\(\ds \leadstoandfrom \ \ \)

\(\ds 0\)

\(\le\)

\(\ds \paren {a_1 b_2}^2 - 2 \paren {a_1 b_2} \paren {a_2 b_1} + \paren {a_2 b_1}^2\)

\(\ds \leadstoandfrom \ \ \)

\(\ds 0\)

\(\le\)

\(\ds \paren {a_1 b_2 - a_2 b_1}^2\)

The final statement is a tautology, and all implications are reversible.

Hence the result.

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Graphical Representations of Complex Numbers. Vectors: $7 \ \text{(a)}$