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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. All numbers
complex
complex numbers
real
i^3

2. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
ln z
conjugate
Polar Coordinates - r
Affix

3. I^2 =
i^2
e^(ln z)
-1
i²

4. ½(e^(iz) + e^(-iz))
How to add and subtract complex numbers (2-3i)-(4+6i)
cos z
Polar Coordinates - Division
Any polynomial O(xn) - (n > 0)

5. Starts at 1 - does not include 0
Complex Number
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate

6. Equivalent to an Imaginary Unit.
conjugate
Imaginary number
complex numbers
Complex Multiplication

7. 5th. Rule of Complex Arithmetic
Real Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)

8. Real and imaginary numbers
complex numbers
Integers
Imaginary Numbers
a real number: (a + bi)(a - bi) = a² + b²

9. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
integers
Complex Number Formula
-1
adding complex numbers

10. 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.
Complex numbers are points in the plane
natural
imaginary
For real a and b - a + bi = 0 if and only if a = b = 0

11. Imaginary number

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12. The square root of -1.
z - z*
Imaginary Unit
cosh²y - sinh²y
Euler's Formula

13. The modulus of the complex number z= a + ib now can be interpreted as
Square Root
Polar Coordinates - Division
How to solve (2i+3)/(9-i)
the distance from z to the origin in the complex plane

14. V(x² + y²) = |z|
a real number: (a + bi)(a - bi) = a² + b²
Euler Formula
Polar Coordinates - r
has a solution.

15. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition

16. 3
i^3
the vector (a -b)
complex numbers
complex

17. Root negative - has letter i
Polar Coordinates - Arg(z*)
imaginary
Complex Number Formula
For real a and b - a + bi = 0 if and only if a = b = 0

18. R?¹(cos? - isin?)
Polar Coordinates - z?¹
e^(ln z)
Square Root
Euler Formula

19. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
a real number: (a + bi)(a - bi) = a² + b²
(cos? +isin?)n
conjugate

20. x + iy = r(cos? + isin?) = re^(i?)
Roots of Unity
Euler's Formula
'i'
Polar Coordinates - z

21. Divide moduli and subtract arguments
Polar Coordinates - Division
Complex Multiplication
Complex Addition
z - z*

22. y / r
x-axis in the complex plane
sin iy
Real Numbers
Polar Coordinates - sin?

23. In this amazing number field every algebraic equation in z with complex coefficients
Complex numbers are points in the plane
has a solution.
cos z
Polar Coordinates - sin?

24. Like pi
transcendental
four different numbers: i - -i - 1 - and -1.
Complex Subtraction
Rational Number

25. For real a and b - a + bi =
0 if and only if a = b = 0
radicals
Complex Numbers: Multiply
How to multiply complex nubers(2+i)(2i-3)

26. When two complex numbers are added together.
Complex Addition
the vector (a -b)
rational
De Moivre's Theorem

27. 3rd. Rule of Complex Arithmetic
multiplying complex numbers
Polar Coordinates - Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
For real a and b - a + bi = 0 if and only if a = b = 0

28. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
radicals
multiplying complex numbers
has a solution.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

29. When two complex numbers are divided.
the complex numbers
irrational
Polar Coordinates - Division
Complex Division

30. 1st. Rule of Complex Arithmetic
Polar Coordinates - sin?
Polar Coordinates - r
integers
i^2 = -1

31. 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.
real
How to find any Power
i²
|z| = mod(z)

32. A subset within a field.
Subfield
multiply the numerator and the denominator by the complex conjugate of the denominator.
interchangeable
Square Root

33. (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
Polar Coordinates - Multiplication by i
Absolute Value of a Complex Number
z - z*
How to solve (2i+3)/(9-i)

34. V(zz*) = v(a² + b²)
|z| = mod(z)
Complex Number
Real Numbers
'i'

35. A + bi
Polar Coordinates - z
Irrational Number
standard form of complex numbers
irrational

36. Cos n? + i sin n? (for all n integers)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
real
(cos? +isin?)n
Absolute Value of a Complex Number

37. 2ib
rational
complex
z - z*
Irrational Number

38. (e^(iz) - e^(-iz)) / 2i
Complex Division
sin z
multiply the numerator and the denominator by the complex conjugate of the denominator.
0 if and only if a = b = 0

39. A complex number and its conjugate
irrational
conjugate pairs
How to solve (2i+3)/(9-i)
the distance from z to the origin in the complex plane

40. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Euler Formula
point of inflection
Imaginary Unit
How to add and subtract complex numbers (2-3i)-(4+6i)

41. Have radical
radicals
standard form of complex numbers
Integers
Liouville's Theorem -

42. ? = -tan?
Polar Coordinates - Arg(z*)
a real number: (a + bi)(a - bi) = a² + b²
How to solve (2i+3)/(9-i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i

43. E ^ (z2 ln z1)
0 if and only if a = b = 0
z1 ^ (z2)
conjugate
i^2

44. R^2 = x
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Square Root
i^0
Liouville's Theorem -

45. I
Real and Imaginary Parts
Complex Number Formula
interchangeable
i^1

46. The complex number z representing a+bi.
Polar Coordinates - Multiplication
Affix
real
(a + bi) = (c + bi) = (a + c) + ( b + d)i

47. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Real and Imaginary Parts
Real Numbers
i^3

48. E^(ln r) e^(i?) e^(2pin)
Euler's Formula
i²
cosh²y - sinh²y
e^(ln z)

49. 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
the complex numbers
non-integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

50. The product of an imaginary number and its conjugate is
Complex Addition
a real number: (a + bi)(a - bi) = a² + b²
radicals
a + bi for some real a and b.