# Help with imaginary numbers?

I need help with imaginary numbers like with "i". I know i=sqr-1...but how do I do problems like:

1) 1/2+5i ?

I'm supposed to simplify it. How do I find out what 5i is?

2) 3+ isqr2/ 7-isqr2 ?

Relevance

key point: the product of a complex number a + bi and its conjugate a - bi is the real number (a^2 + b^2)-- this technique is analogous to rationalizing a denominator--it makes denominators real

1) is this 1/(2 + 5i)?

if so, multiply the top and the bottom by the complex conjugate (2 - 5i) to give you: (2 - 5i) / 29, or 2/29 - 5i/29

If it's (1/2) + 5i, then this is already standard form... you could rearrange it to (1 + 10i)/2, but that isn't really simplifying

simplifying complex numbers usually means putting it into standard form a + bi

2) (3 + isqrt(2)) / (7 - isqrt(2))

multiply top and bottom by the conjugate of the denominator: (7 + isqrt(2))

(7 - isqrt(2))(7 + isqrt(2)) = 49 - i^2 (sqrt(2))^2 = 49 + 2 = 51

(3 + isqrt(2))(7 + isqrt(2)) / (51) =

[21 + 3isqrt(2) + 7isqrt(2) +i^2 (sqrt(2))^2 ] / 51=

(21 + 10 i sqrt(2) - 2 ) / 51 =

[19 + 10 i sqrt(2) ] / 51 =

19/51 + i10sqrt(2)/51

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• specific, there's a reason. specifically to be waiting to exhibit the concepts of equations like x^2 + a million = 0, which has no actual concepts. Equations and formula that bring about imaginary numbers arise commonly in the sciences and engineering. Voltage, present day, and skill, for example, are somewhat some the parts which may be expressed in terms of imaginary numbers.

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