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Greatest Integer Function

If Z 1, Z 2∈ ...

Question

If $Z_{1}, Z_{2}$ ∈ C, which of the following is true?

A

|| ≤ ||-||

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B

|| ≥ ||+||

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C

||| ± ||| ≤ |+| ≤ || + ||

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D

| - | ≤ || + ||

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Solution

The correct options are
B
|| ≥ ||+||

C
||| ± ||| ≤ |+| ≤ || + ||

D
| - | ≤ || + ||

Triangle inequality :-

|| $Z_{1} | - | Z_{2}$ || ≤ || $Z_{1} | - | Z_{2}$ ||

Other Relations

|\(Z_1 ± Z_2\)| ≤ | $Z_{1}$ |+| $Z_{2}$ |

|\(Z_1 ± Z_2\)| ≤ || $Z_{1}$ |-| $Z_{2}$ ||

From the given relations, b,c,d are right options.

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Similar questions

Which of the following is correct for any two complex number $z_{1}$ and $z_{2}$ ?

z_{1}

z_{2}

are two complex numbers, then prove that

| z_{1} | + | z_{2} | = ∣ ∣ ∣ \frac{z_{1} + z_{2}}{2} + \sqrt{z_{1} z_{2}} ∣ ∣ ∣ + ∣ ∣ ∣ \frac{z_{1} + z_{2}}{2} - \sqrt{z_{1} z_{2}} ∣ ∣ ∣

z_{1}

z_{2}

are any two complex numbers, then

∣ ∣ ∣ z_{1} + \sqrt{z_{1}^{2} - z_{2}^{2}} ∣ ∣ ∣ + ∣ ∣ ∣ z_{1} - \sqrt{z_{1}^{2} - z_{2}^{2}} ∣ ∣ ∣

Q. If for a complex number

z_{1}

z_{2}

arg (z_{1}) - arg (z_{2}) = 0

| z_{1} - z_{2} |

If arg ( $Z_{1}$ ) = $α$ , arg ( $Z_{2}$ ) = $β$ , the value of $| Z_{1} + Z_{2} |^{2}$ is

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