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IrishBoy123

  • one year ago

complex numbers

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  1. IrishBoy123
    • one year ago
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    |z| = 2 prove \[|z^2 - 7 | \ge 3\]

  2. idku
    • one year ago
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    why can't you just plug it in?

  3. idku
    • one year ago
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    |2| raised to the second power is 4. Or is this something different than alg?

  4. freckles
    • one year ago
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    z is a complex number a+bi doesn't have to be 2

  5. idku
    • one year ago
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    oh, ok.

  6. freckles
    • one year ago
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    just the magnitude of a+bi has to be 2

  7. freckles
    • one year ago
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    \[|z^2-7|=|7-z^2|\ge|7|-|z^2|=|7|-|z|^2=7-4=3\]

  8. IrishBoy123
    • one year ago
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    et voila! https://gyazo.com/f606fea88651c5a8242fb8ea5f2e9db9 what about the LHS of that?

  9. freckles
    • one year ago
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    \[|z^2-7| =|z^2+(-7)| \le |z^2|+|(-7)|=|z|^2+7=4+7=11\]

  10. freckles
    • one year ago
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    \[3 \le |z^2-7| \le 11\]

  11. freckles
    • one year ago
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    wait did you mean right hand side of that?

  12. freckles
    • one year ago
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    because the first thing was the left hand side of that one thingy

  13. IrishBoy123
    • one year ago
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    https://gyazo.com/7c7699ae9dad7798832fed473b22ba9d

  14. freckles
    • one year ago
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    \[3=7-4=7-2^2=7-|z|^2=7-|z^2|=|7|-|z^2| \le |7-z^2|=|z^2-7| \\ \le |z^2|+|-7|=|z|^2+7=2^2+7=4+7=11\] this is both left and right hand sides

  15. Loser66
    • one year ago
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    \[|z^2|=|z^2 -7 +7|\leq |z^2 -7|+|7|\] Hence \(|z^2 -7|\geq|z^2|-|7| \) Therefore \(||z^2 -7||\geq|z^2|-|7||=|4-7|=3\)

  16. freckles
    • one year ago
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    \[|7|-|z^2| \le |7-z^2| \le |7|+|-z^2| \\ 7-|z|^2 \le |7-z^2| \le 7+|z|^2 \\ 7-2^2 \le |7-z^2| \le 7+2^2 \\ 3 \le |7-z^2| \le 11 \\ 3 \le |z^2-7| \le 11 \] just in case that one thing was hard to read

  17. Loser66
    • one year ago
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    are you sleeping? @IrishBoy123 hahaha

  18. freckles
    • one year ago
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    no that is his cat with his bum on the keyboard

  19. IrishBoy123
    • one year ago
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    that was even more epic! i'm gonna another one in a new thread. i can do them now but then i can post a final question that will connect them all up.

  20. IrishBoy123
    • one year ago
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    you two are just mad. sleeping with a bum on my keyboard or something!

  21. freckles
    • one year ago
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    i have no cat butts allowed sign on my keyboard

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