My Favorite One-Liners: Part 46

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them. Today’s one-liner is something I’ll use after completing some monumental calculation. For example, if $z, w \in \mathbb{C}$, the proof of the triangle inequality is no joke, as it requires the following as lemmas:

• $\overline{z + w} = \overline{z} + \overline{w}$
• $\overline{zw} = \overline{z} \cdot \overline{w}$
• $z + \overline{z} = 2 \hbox{Re}(z)$
• $|\hbox{Re}(z)| \le |z|$
• $|z|^2 = z \cdot \overline{z}$
• $\overline{~\overline{z}~} = z$
• $|\overline{z}| = |z|$
• $|z \cdot w| = |z| \cdot |w|$

With all that as prelude, we have

$|z+w|^2 = (z + w) \cdot \overline{z+w}$

$= (z+w) (\overline{z} + \overline{w})$

$= z \cdot \overline{z} + z \cdot \overline{w} + \overline{z} \cdot w + w \cdot \overline{w}$

$= |z|^2 + z \cdot \overline{w} + \overline{z} \cdot w + |w|^2$

$= |z|^2 + z \cdot \overline{w} + \overline{z} \cdot \overline{~\overline{w}~} + |w|^2$

$= |z|^2 + z \cdot \overline{w} + \overline{z \cdot \overline{w}} + |w|^2$

$= |z|^2 + 2 \hbox{Re}(z \cdot \overline{w}) + |w|^2$

$\le |z|^2 + 2 |z \cdot \overline{w}| + |w|^2$

$= |z|^2 + 2 |z| \cdot |\overline{w}| + |w|^2$

$= |z|^2 + 2 |z| \cdot |w| + |w|^2$

$= (|z| + |w|)^2$

In other words,

$|z+w|^2 \le (|z| + |w|)^2$.

Since $|z+w|$ and $|z| + |w|$ are both positive, we can conclude that

$|z+w| \le |z| + |w|$.

QED

In my experience, that’s a lot for students to absorb all at once when seeing it for the first time. So I try to celebrate this accomplishment:

Anybody ever watch “Home Improvement”? This is a Binford 6100 “more power” mathematical proof. Grunt with me: RUH-RUH-RUH-RUH!!!

Leave a comment

2 Comments

This site uses Akismet to reduce spam. Learn how your comment data is processed.