Question

If z = x + iy, x, y real, prove that :
| x | + lZ।=V2।Z।​

Answer 1

If z = x + iy

|z| = √(x² + y²)

|z|² = x² + y² ………(1)

we know that,

( |x| + |y| )² = |x|² + |y|² + 2|x||y| ………...(2)

and ( |x| - |y| )² = |x|² + |y|² - 2|x||y|

we know that, whole square is always greater than zero.

( |x| - |y| )² ≥ 0

|x|² + |y|² - 2|x||y| ≥ 0

|x|² + |y|² ≥ 2|x||y|

from equation (2)

( |x| + |y| )² ≤ |x|² + |y|² + |x|² + |y|²

( |x| + |y| )² ≤ 2 (|x|² + |y|²)

( |x| + |y| )² ≤ 2 |z|²

both side square root

|x| + |y| ≤ √2 |z|

Hence proved

Step-by-step explanation:

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