30. Let z = a + bi and w = c + di. Use geometry to show...

30. Let $z = a + bi$ and $w = c + di$. Use geometry to show that (a) $|z + w| \le |z| + |w|$ (b) $|z + w| \ge |z| - |w|$ Under what conditions are these inequalities actual equalities? The results of this problem are sometimes called the \"triangle inequalities.\" Why?

Transcript

00:01 Okay, so here we're given two complex numbers, a plus i -b, w is c plus id, and we're asked to prove a certain statement.

00:11 So the first one is that the conjugate of the sum is equal to the sum of the conjugates.

00:19 So let's just write this down.

00:21 So of course, let's start.

00:23 Z plus w is the number a plus c plus i times b plus d.

00:30 Which implies that the conjugate of this is given by a plus c minus i, b plus d.

00:40 So we can indeed simplify this as being a minus ib plus c minus id, but then this is exactly z bar plus w bar.

00:53 Okay, so this proves the first statement.

00:56 Second statement that the real part of w is less than or equal than the norm of w.

01:05 So let's recall now what the norm of w is, right? so by definition, the norm of w is the square root of the real part of w squared plus the imaginary part of w squared.

01:18 I'm not even going to use c and d in here.

01:21 There's no need.

01:22 Right? and now, of course, the square of a number is always.

01:27 Positive.

01:28 So this amount in here is greater than or equal to zero, which implies that then the whole square root of the real part of w squared plus that positive amount will be greater than equal than the square root simply of the real part.

01:43 Okay.

01:44 And here this is simply the modulus of real right.

01:47 So the square root cancel out with the square.

01:50 So this is a conclusion...

Question

30. Let $z = a + bi$ and $w = c + di$. Use geometry to show that (a) $|z + w| \le |z| + |w|$ (b) $|z + w| \ge |z| - |w|$ Under what conditions are these inequalities actual equalities? The results of this problem are sometimes called the \"triangle inequalities.\" Why?

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