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CLEP General Mathematics: Complex Numbers

Subjects : clep, math
Instructions:
  • Answer 50 questions in 15 minutes.
  • If you are not ready to take this test, you can study here.
  • Match each statement with the correct term.
  • Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. (e^(iz) - e^(-iz)) / 2i






2. (a + bi)(c + bi) =






3. The reals are just the






4. A+bi






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






6. The field of all rational and irrational numbers.






7. 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.'






8. A² + b² - real and non negative






9. Starts at 1 - does not include 0






10. z1z2* / |z2|²






11. V(zz*) = v(a² + b²)






12. (a + bi) = (c + bi) =






13. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of






14. 1






15. 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






16. 1






17. The square root of -1.






18. Numbers on a numberline






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






20. 4th. Rule of Complex Arithmetic






21. V(x² + y²) = |z|






22. A subset within a field.






23. ½(e^(-y) +e^(y)) = cosh y






24. Equivalent to an Imaginary Unit.






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






26. 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.






27. 2a






28. 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






29. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.






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






31. We can also think of the point z= a+ ib as






32. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i






33. 3






34. Every complex number has the 'Standard Form':






35. Imaginary number

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36. When two complex numbers are added together.






37. All numbers






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






39. Have radical






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






41. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1






42. R^2 = x






43. Multiply moduli and add arguments






44. Has exactly n roots by the fundamental theorem of algebra






45. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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






47. Like pi






48. ? = -tan?






49. Derives z = a+bi






50. x / r