Introduction
In the study of complex numbers, the triangle inequality is a fundamental property that relates the magnitude (or modulus) of a sum of numbers to the sum of their individual magnitudes. It states that for any two complex numbers $z$ and $w$:
$|z + w| \le |z| + |w|$
This inequality is geometrically intuitive: in the complex plane, it represents the fact that the length of one side of a triangle must be less than or equal to the sum of the lengths of the other two sides.
The Proof
To prove this algebraically, we use the property that $|z|^2 = z \bar{z}$, where $\bar{z}$ is the complex conjugate of $z$.
Step 1: Expand the square of the modulus
Let's start by calculating $|z + w|^2$: $|z + w|^2 = (z + w)(\overline{z + w})$
Using the property that the conjugate of a sum is the sum of the conjugates, we have: $|z + w|^2 = (z + w)(\bar{z} + \bar{w})$
Step 2: Distribute and simplify
Expanding the product: $|z + w|^2 = z\bar{z} + z\bar{w} + w\bar{z} + w\bar{w}$
Since $z\bar{z} = |z|^2$ and $w\bar{w} = |w|^2$, we get: $|z + w|^2 = |z|^2 + (z\bar{w} + \bar{z\bar{w}}) + |w|^2$
Note that $w\bar{z} = \overline{z\bar{w}}$. For any complex number $u$, $u + \bar{u} = 2 \text{Re}(u)$. Therefore: $|z + w|^2 = |z|^2 + 2 \text{Re}(z\bar{w}) + |w|^2$
Step 3: Apply the inequality property
We know that for any complex number $u$, $\text{Re}(u) \le |u|$. Since $|z\bar{w}| = |z||\bar{w}| = |z||w|$, we can write: $\text{Re}(z\bar{w}) \le |z\bar{w}| = |z||w|$
Substituting this into our equation: $|z + w|^2 \le |z|^2 + 2|z||w| + |w|^2$
Step 4: Conclusion
The right side is a perfect square: $|z + w|^2 \le (|z| + |w|)^2$
Taking the square root of both sides (since magnitudes are non-negative), we arrive at the final result: $|z + w| \le |z| + |w|$
Geometric Intuition
Imagine $z$ and $w$ as vectors in the complex plane originating from the origin. The sum $z+w$ creates a parallelogram. The vector $z+w$ represents the diagonal. The inequality essentially states that the shortest distance between two points is a straight line, confirming that the magnitude of the diagonal is no greater than the sum of the sides of the triangle.