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!!!

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