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# help with complex numbers

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i need to find the square root of 3+4i, which is a complex number, i thought the answer was just 2+i because when you square that you get 3+4i, but that's wrong :( are there other square roots of 3+4i that i don't know about?

Feb 15, 2020

#1
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Who says 2 + i is wrong? That is the sqrt of 3 + 4i, because it factors into: (2 +i)^2 and sqrt(2 +i)^2 =2 + i

Feb 15, 2020
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yeah, but i was thinking what if there are more solutions that i don't know about?

atlas9  Feb 15, 2020
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Roots of complex numbers are multi valued.

A complex number has two square roots, three cube roots, four fourth roots and so on.

You will already have come across this with the square roots of positive numbers. The square root of 4 for example is plus or minus 2.

For your example, the second root is -2 - i .

Feb 15, 2020
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ah ok thank you!

atlas9  Feb 15, 2020
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No, not in this case, because both sqrt are the same: 2 + i and 2 + i.

Feb 15, 2020
#6
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2 + ì and   -2 - i  are different complex numbers, as are plus and minus 2 as the square roots of 4.

Feb 15, 2020