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# If z=a+bi, w=x+iy for a,b,x,y real, then what is the real part of zw?

### 3 Answers

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- PhilipLv 61 month ago
z = (a+ib), w = (x+iy). zw = (a+ib)(x+iy) = (ax-by) + i(bx+ay). Real part of zw is

(ax-by).

- llafferLv 71 month ago
z = a + bi and w = x + yi

So:

zw

Substitute and simplify:

(a + bi)(x + yi)

Multiply:

ax + ayi + bxi + byi²

i² = -1, so:

ax + ayi + bxi - by

Group the terms and factor out the "i":

ax - by + ayi + bxi

ax - by + (ay + bx)i

the real part of "zw" is (ax - by)

The imaginary part of "zw" is (ay + bx)

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