# quiz1 - 0 or what is the same 1 0 i In the same way...

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MAT334 QUIZ No.1 out of 10 First Name: Franklin Last Name: VP Student number: 000000000 There is space in the front for the solutions (remember it is not just the an- swers). Use only the back for scrap work so that your solutions are nice and readable. Show oﬀ that you know what you are doing. 1. Add 1 + 2 i and - 6 i + 3 + 8 i Solution: (1 + 2 i ) + ( - 6 i + 3 + 8 i ) = 4 + 4 i 2. Multiply 2 e i π 3 and 1 2 e i π 5 Solution: 2 e i π 3 1 2 e i π 5 = e i 8 π 15 3. Divide 1 - i by 1 + i Solution: (1 - i ) 1+ i = (1 - i )(1 - i ) (1+ i )(1 - i ) = - 2 i 2 = - i 4. Translate the point 1 + i one unit to the right, then the result two units down, and rotate what you get around the origin π 6 radians (counterclock- wise). Solution: Translation correspond to addition of the complex num- ber that represents the vector along which the translation is done. That is why translating one unit to the right correspond to adding 1 since that translation is represented by the vector (1
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Unformatted text preview: , 0) or what is the same 1+0 i . In the same way, translating two units down corresponds to adding-2 i . Now, rotation corresponds to multiplication, in this case, by e i π 6 . Therefore we need to compute ((1 + i ) + 1 + (-2 i )) e i π 6 . This multiplication is easier to perform in binomial form. We have that e i π 6 = cos π 6 + i sin π 6 = √ 3 2 + i 1 2 and therefore we need to multiply (2-i )( √ 3 2 + i 1 2 ) = ( √ 3 2 +1)+ i (1-√ 3 2 ). (Notice that we get the exact answer explicitly. The computation in polar form is less convenient in this case to do the multiplication.) 5. Compute 3 √ 1 + i Solution: 3 √ 1 + i = 3 p √ 2 e i π 4 = 2 1 / 6 e i π/ 4+2 kπ 3 for k = , 1 , 2....
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