Introduction to the Modulus of a Complex Number
In mathematics, a complex number $z$ is expressed in the form $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$).
The modulus (or absolute value) of a complex number, denoted by $|z|$, represents the distance of the point $(a, b)$ from the origin $(0, 0)$ in the complex plane. It is defined as:
$|z| = \sqrt{a^2 + b^2}$
A useful property of the modulus is that the square of the modulus is equal to the product of the complex number and its conjugate ($z$ and $\bar{z}$):
$|z|^2 = z \cdot \bar{z}$
The Identity Proof
We are tasked to prove the following identity for two complex numbers $z$ and $w$:
$|z + w|^2 = |z|^2 + |w|^2 + 2\text{Re}(z\bar{w})$
Step-by-Step Proof
Step 1: Apply the property $|u|^2 = u \cdot \bar{u}$ Using the fundamental property of the modulus, we can express the left side as:
$|z + w|^2 = (z + w) \cdot \overline{(z + w)}$
Step 2: Use the property of conjugates The conjugate of a sum is the sum of the conjugates: $\overline{z + w} = \bar{z} + \bar{w}$. Substituting this back into the equation:
$|z + w|^2 = (z + w)(\bar{z} + \bar{w})$
Step 3: Expand the product Using the distributive property (FOIL method):
$|z + w|^2 = z\bar{z} + z\bar{w} + w\bar{z} + w\bar{w}$
Step 4: Substitute modulus definitions We know that $z\bar{z} = |z|^2$ and $w\bar{w} = |w|^2$:
$|z + w|^2 = |z|^2 + |w|^2 + z\bar{w} + w\bar{z}$
Step 5: Utilize properties of conjugates Note that $w\bar{z}$ is the conjugate of $z\bar{w}$ because $\overline{z\bar{w}} = \bar{z} \cdot \bar{\bar{w}} = \bar{z}w = w\bar{z}$. In complex algebra, for any complex number $u$, $u + \bar{u} = 2\text{Re}(u)$. Letting $u = z\bar{w}$:
$z\bar{w} + \overline{z\bar{w}} = 2\text{Re}(z\bar{w})$
Conclusion: Substituting this back into our equation from Step 4, we arrive at the final result:
$|z + w|^2 = |z|^2 + |w|^2 + 2\text{Re}(z\bar{w})$
This completes the proof. This identity is extremely useful in physics and engineering, particularly in signal processing and quantum mechanics, where it relates the norm of a vector sum to the inner product of the vectors.