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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.
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. 4th. Rule of Complex Arithmetic
i^2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Euler Formula
Field

2. (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
How to solve (2i+3)/(9-i)
Absolute Value of a Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.
ln z

3. E^(ln r) e^(i?) e^(2pin)
cosh²y - sinh²y
e^(ln z)
radicals
Any polynomial O(xn) - (n > 0)

4. 2nd. Rule of Complex Arithmetic

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5. ? = -tan?
i^2 = -1
standard form of complex numbers
Complex Subtraction
Polar Coordinates - Arg(z*)

6. Divide moduli and subtract arguments
Polar Coordinates - r
complex numbers
conjugate
Polar Coordinates - Division

7. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Imaginary Unit
Absolute Value of a Complex Number
Euler's Formula
(a + bi) = (c + bi) = (a + c) + ( b + d)i

8. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|
Roots of Unity
Polar Coordinates - cos?

9. Any number not rational
irrational
Imaginary Unit
subtracting complex numbers
imaginary

10. x / r
imaginary
conjugate
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0

11. In this amazing number field every algebraic equation in z with complex coefficients
Square Root
has a solution.
Affix
the distance from z to the origin in the complex plane

12. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + bi) = (c + bi) = (a + c) + ( b + d)i

13. x + iy = r(cos? + isin?) = re^(i?)
non-integers
Polar Coordinates - z
z - z*
How to add and subtract complex numbers (2-3i)-(4+6i)

14. Equivalent to an Imaginary Unit.
Imaginary number
standard form of complex numbers
Polar Coordinates - Arg(z*)
De Moivre's Theorem

15. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Imaginary number
Imaginary Numbers
a + bi for some real a and b.

16. A complex number may be taken to the power of another complex number.
Complex Exponentiation
(cos? +isin?)n
sin iy
Any polynomial O(xn) - (n > 0)

17. Derives z = a+bi
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler Formula
radicals
multiplying complex numbers

18. To simplify a complex fraction
z1 / z2
-1
cos z
multiply the numerator and the denominator by the complex conjugate of the denominator.

19. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
sin z
-1

20. ½(e^(-y) +e^(y)) = cosh y
z1 / z2
cos iy
i²
z1 ^ (z2)

21. 1
conjugate
Roots of Unity
non-integers
i^0

22. Real and imaginary numbers
Complex Number Formula
complex numbers
Absolute Value of a Complex Number
Complex Numbers: Multiply

23. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
has a solution.
i^2 = -1
Complex Numbers: Add & subtract

24. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
Complex Multiplication

25. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
multiplying complex numbers
Imaginary number
zz*
The Complex Numbers

26. The field of all rational and irrational numbers.
Polar Coordinates - sin?
Real Numbers
Imaginary Unit
Liouville's Theorem -

27. A + bi
interchangeable
Polar Coordinates - cos?
standard form of complex numbers
(cos? +isin?)n

28. All the powers of i can be written as
Irrational Number
four different numbers: i - -i - 1 - and -1.
rational
Any polynomial O(xn) - (n > 0)

29. No i
Any polynomial O(xn) - (n > 0)
subtracting complex numbers
i^0
real

30. 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
sin z
subtracting complex numbers
Affix
How to find any Power

31. 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
Real Numbers
Irrational Number
Imaginary number
the complex numbers

32. I
0 if and only if a = b = 0
i^1
x-axis in the complex plane
point of inflection

33. V(x² + y²) = |z|
Square Root
Polar Coordinates - r
Field
cosh²y - sinh²y

34. Imaginary number

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35. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Addition
How to solve (2i+3)/(9-i)
Rational Number
multiplying complex numbers

36. 1
i^4
radicals
Liouville's Theorem -
'i'

37. 2a
z + z*
Polar Coordinates - Division
i^1
Complex Numbers: Multiply

38. A number that cannot be expressed as a fraction for any integer.
imaginary
a real number: (a + bi)(a - bi) = a² + b²
z - z*
Irrational Number

39. Every complex number has the 'Standard Form':
How to solve (2i+3)/(9-i)
multiplying complex numbers
a + bi for some real a and b.
z + z*

40. The complex number z representing a+bi.
cos iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
Affix
real

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

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42. Have radical
cos z
radicals
the vector (a -b)
multiply the numerator and the denominator by the complex conjugate of the denominator.

43. (e^(iz) - e^(-iz)) / 2i
sin z
Absolute Value of a Complex Number
conjugate pairs
(a + c) + ( b + d)i

44. 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
Complex Numbers: Multiply
has a solution.
Complex Addition
Rules of Complex Arithmetic

45. Like pi
Imaginary Numbers
transcendental
How to add and subtract complex numbers (2-3i)-(4+6i)
Subfield

46. The square root of -1.
radicals
Any polynomial O(xn) - (n > 0)
Imaginary Unit
irrational

47. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
imaginary
How to find any Power
i^3
standard form of complex numbers

48. E ^ (z2 ln z1)
point of inflection
i^3
z1 ^ (z2)
Complex Number Formula

49. y / r
Polar Coordinates - sin?
Polar Coordinates - z
i^2 = -1
(cos? +isin?)n

50. z1z2* / |z2|²
Liouville's Theorem -
complex
z1 / z2
zz*