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IrishBoy123

  • one year ago

complex number

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  1. IrishBoy123
    • one year ago
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    for \( |z| = 2\), prove: \[|z^2 - 4 z - 3 | \le 15\]

  2. freckles
    • one year ago
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    \[-15 \le z^2-4z-3 \le 15 \\ -15+3 \le z^2-4z \le 15+3 \\ -12 \le z^2-4z \le 18 \\ -12+4 \le z^2-4z+4 \le 18+4 \\ -8 \le (z-2)^2 \le 22 \\ 0 \le (z-2)^2 \le 22 \\ (z-2)^2 \le 22 \\ -\sqrt{22} \le z-2 \le \sqrt{22} \\ \] I wonder if it has anything to do with this I'm still thinking

  3. Loser66
    • one year ago
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    I wonder if I take it easy?? Using Triangle in equality, I have \(|z^2-4z -3|\leq |z^2| +|-4z|+|-3|= |z|^2 +4|z| +|-3)= 2^2 +4*2 +3 =15\)

  4. freckles
    • one year ago
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    I think loser's way works awesome

  5. Loser66
    • one year ago
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    Thank you. I was doubt myself.

  6. IrishBoy123
    • one year ago
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    @Loser66 which are the sides of the triangle??

  7. IrishBoy123
    • one year ago
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    can you draw it pls?

  8. Loser66
    • one year ago
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    |dw:1440896633171:dw|

  9. Loser66
    • one year ago
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    That is the triangle inequality theorem.

  10. Loser66
    • one year ago
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    \(|x+y|\leq |x| +|y|\)

  11. IrishBoy123
    • one year ago
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    yes, i can see that, it is absolutely marvellous however, i cannot see how we apply it to \[z^2−4z−3\]

  12. freckles
    • one year ago
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    \[|z^2-4z-3|=|z^2+(-4z-3)| \le |z^2|+|(-4z-3)| =|z|^2+|(-1)(4z+3)| \\ =2^2+|-1| \cdot |4z+3|=4+1 \cdot |4z+3| \\= 4+|4z+3| \le 4+|4z|+|3| =4+|4| \cdot |z|+3 \\ =4+4(2)+3=4+8+3=15\] is it the three term thing @IrishBoy123 ?

  13. Loser66
    • one year ago
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    Yes! same. :)

  14. freckles
    • one year ago
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    http://mathworld.wolfram.com/TriangleInequality.html triangle inequality can hold for more than two term

  15. IrishBoy123
    • one year ago
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    yep @freckles it is the 3 term thing i can see the basic triangle thing but i am staring at a quadratic in z and it is not so clear

  16. freckles
    • one year ago
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    Well you could apply the triangle inequality twice as I did above

  17. freckles
    • one year ago
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    |dw:1440897294917:dw|

  18. IrishBoy123
    • one year ago
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    right, i need to read some more brilliant, @freckles and @Loser66 much appreciated! i will revert :p

  19. IrishBoy123
    • one year ago
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    and by read, i mean read 'this thread'

  20. freckles
    • one year ago
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    loser was the brilliant one he was like oh lets use triangle inequality

  21. freckles
    • one year ago
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    oops she

  22. freckles
    • one year ago
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    I'm so sorry loser

  23. IrishBoy123
    • one year ago
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    "I'm so sorry loser" you could not make this up! good night lovely people! sleep well!

  24. IrishBoy123
    • one year ago
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    well, depends on what time zone you are in, i suppose, but......zzzzzzzzz.

  25. freckles
    • one year ago
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    goodnight

  26. IrishBoy123
    • one year ago
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    got there! posting a related question. i think i know how to do it now but i am interested in seeing what people have to say.

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