Linear Algebra A Level Complex Number Magnitude Inequality

Question

Any ideas how to prove this inequality of any two complex numbers?

$$ ||z| - |w||\leq|z + w |$$

My intuition told me to square both sides:

$$ (|z| - |w|) ^ 2 \leq |z + w|^2 \\ |z|^2 - 2|z||w| + |w|^2 \leq |(z + w)(z+w)| \\z \bar z - 2|z||w| + w \bar w \leq |zz+2wz + ww| $$ In polar complex notation $ z = a + ib, w = c + id$ where $ a,b,c,d \ \epsilon \ R: $ $$ a^2+b^2 - 2\sqrt {a^2+b^2} \sqrt{c^2 + d^2} + c^2 + d^2 \leq |a^2 + 2abi-b^2 + 2((ac-bd) + (bc + ad)i) + c^2 + 2cdi - d^2| $$ Simplify RHS in polar complex notation: $$ |a^2 + c^2 - (b^2 + d^2) + 2ac - 2bd + 2(bc + ad + cd + ab)i| $$

If I take the magnitude of the RHS, maybe the terms can cancel out the LHS and become something like: $$0\leq a^2 + b^2$$ Then it becomes obvious that the inequality is true. Is expanding a reasonable technique or is there a better way to prove this?

Answer 1

A shorter way is to note the difference form $|z+w| \ge |z| - |w|$ of the triangle inequality*. Similarly we have $|z+w|=|w+z| \ge |w|-|z|$. Together these imply $|z+w|\ge\max\{|z| - |w|,|w|-|z|\} = \big| |z|-|w| \big|$.

*It's worth remembering why $|z+w| \ge |z| - |w|$ is a version of the triangle inequality: it's because $|z| = |(z+w)-w| \le |z+w|+|-w| = |z+w|+|w|$.

Answer 2

By the triangle inequality, $|z| = |(z + w) - w| \leq |z + w| + |w|$, so $|z|-|w|\leq |z+w|$.

Interchanging $z$ and $w$ then gives $-(|z|-|w|)\leq |z+w|$.

This means that both $|z|-|w|$ and its opposite are no more than $|z+w|$. In particular, $-\Big||z|-|w|\Big| \leq \underbrace{\Big||z|-|w|\Big| \leq |z + w|}_\textrm{what you want to show}$.

Answer 3

If $w=0$ or $z=0$ the result is trivial. Without loss of generality assume $0<|z|\le |w|$ and take $Z=z/w,$ then $0<|Z|\le 1$ and we need to show that $$1\le|Z|+|Z+1|.$$ Draw the triangle whose vertices are $0, Z$ and $Z+1$ and use the fact that "the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the remaining side."

Answer 4

Consider the three segments connecting origin, $z$, and $-w$. The difference in length between any two of these cannot exceed the length of the third segment.

Equality occurs when the origin, $z$, and $-w$ are collinear with the origin not in between