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CLEP General Mathematics: Complex Numbers

Subjects : clep, math
Instructions:
  • Answer 50 questions in 15 minutes.
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  • Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.
1. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1






2. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'






3. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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4. I = imaginary unit - i² = -1 or i = v-1






5. R?¹(cos? - isin?)






6. 1st. Rule of Complex Arithmetic






7. V(x² + y²) = |z|






8. The complex number z representing a+bi.






9. A + bi






10. Cos n? + i sin n? (for all n integers)






11. E^(ln r) e^(i?) e^(2pin)






12. xpressions such as ``the complex number z'' - and ``the point z'' are now






13. A subset within a field.






14. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc






15. A² + b² - real and non negative






16. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i






17. For real a and b - a + bi =






18. Divide moduli and subtract arguments






19. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i






20. A+bi






21. Derives z = a+bi






22. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17






23. 1






24. The field of all rational and irrational numbers.






25. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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26. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.






27. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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28. x + iy = r(cos? + isin?) = re^(i?)






29. E ^ (z2 ln z1)






30. ½(e^(-y) +e^(y)) = cosh y






31. 2a






32. Starts at 1 - does not include 0






33. To simplify a complex fraction






34. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi






35. 5th. Rule of Complex Arithmetic






36. No i






37. To simplify the square root of a negative number






38. ½(e^(iz) + e^(-iz))






39. 1






40. Real and imaginary numbers






41. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.






42. The reals are just the






43. Numbers on a numberline






44. I






45. We see in this way that the distance between two points z and w in the complex plane is






46. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.






47. Given (4-2i) the complex conjugate would be (4+2i)






48. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n






49. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that






50. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called