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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. 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
subtracting complex numbers
i^2
zz*
ln z

2. Have radical
radicals
i^2 = -1
Complex Addition
conjugate pairs

3. 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: Multiply
Complex numbers are points in the plane
sin z
Imaginary Unit

4. Not on the numberline
i^2
e^(ln z)
non-integers
integers

5. (e^(-y) - e^(y)) / 2i = i sinh y
z1 / z2
sin iy
|z-w|
De Moivre's Theorem

6. Derives z = a+bi
Euler Formula
The Complex Numbers
Complex Numbers: Add & subtract
non-integers

7. We can also think of the point z= a+ ib as
cos iy
the vector (a -b)
Argand diagram
Real Numbers

8. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Integers
How to multiply complex nubers(2+i)(2i-3)
Complex Multiplication

9. Equivalent to an Imaginary Unit.
Imaginary number
z - z*
zz*
Field

10. No i
real
subtracting complex numbers
Rational Number
'i'

11. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
natural
Polar Coordinates - cos?
(cos? +isin?)n
Roots of Unity

12. Written as fractions - terminating + repeating decimals
rational
multiply the numerator and the denominator by the complex conjugate of the denominator.
|z-w|
z1 / z2

13. We see in this way that the distance between two points z and w in the complex plane is
z + z*
|z-w|
Polar Coordinates - z
(cos? +isin?)n

14. 2nd. Rule of Complex Arithmetic

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15. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
Polar Coordinates - Division
Polar Coordinates - r
cos z

16. 1
(a + c) + ( b + d)i
i^2
z1 ^ (z2)
Rational Number

17. x / r
(cos? +isin?)n
Polar Coordinates - cos?
Imaginary Unit
Polar Coordinates - r

18. Cos n? + i sin n? (for all n integers)
Polar Coordinates - sin?
Complex Numbers: Multiply
(cos? +isin?)n
0 if and only if a = b = 0

19. Where the curvature of the graph changes
conjugate pairs
point of inflection
Polar Coordinates - cos?
Field

20. A number that can be expressed as a fraction p/q where q is not equal to 0.
four different numbers: i - -i - 1 - and -1.
conjugate pairs
De Moivre's Theorem
Rational Number

21. E^(ln r) e^(i?) e^(2pin)
sin z
How to add and subtract complex numbers (2-3i)-(4+6i)
z1 / z2
e^(ln z)

22. A+bi
Polar Coordinates - sin?
cos iy
i²
Complex Number Formula

23. I
v(-1)
i^0
Any polynomial O(xn) - (n > 0)
Euler Formula

24. The square root of -1.
Imaginary Unit
Real Numbers
z - z*
Complex Division

25. 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
Polar Coordinates - Division
Complex Numbers: Add & subtract
We say that c+di and c-di are complex conjugates.
sin z

26. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
the vector (a -b)
irrational

27. (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
i^1
How to solve (2i+3)/(9-i)
(a + c) + ( b + d)i
Rational Number

28. Imaginary number

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29. 2a
z + z*
How to solve (2i+3)/(9-i)
Polar Coordinates - z
the vector (a -b)

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

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31. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
z1 ^ (z2)
natural
Complex Division

32. When two complex numbers are subtracted from one another.
Polar Coordinates - cos?
Complex Subtraction
Affix
How to multiply complex nubers(2+i)(2i-3)

33. 1
i^4
a real number: (a + bi)(a - bi) = a² + b²
The Complex Numbers
Euler Formula

34. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
We say that c+di and c-di are complex conjugates.
Real Numbers
Field
Absolute Value of a Complex Number

35. 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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36. Given (4-2i) the complex conjugate would be (4+2i)
Polar Coordinates - z
Polar Coordinates - cos?
i²
Complex Conjugate

37. A² + b² - real and non negative
sin iy
radicals
zz*
Polar Coordinates - Multiplication

38. Any number not rational
e^(ln z)
Polar Coordinates - z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
irrational

39. Real and imaginary numbers
sin iy
complex numbers
i^0
|z| = mod(z)

40. z1z2* / |z2|²
-1
Rational Number
z1 / z2
four different numbers: i - -i - 1 - and -1.

41. y / r
cos z
point of inflection
conjugate pairs
Polar Coordinates - sin?

42. (a + bi)(c + bi) =
Polar Coordinates - cos?
Imaginary Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Irrational Number

43. Rotates anticlockwise by p/2
x-axis in the complex plane
Polar Coordinates - Multiplication by i
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Multiply

44. 1
Complex Conjugate
(a + c) + ( b + d)i
Polar Coordinates - Arg(z*)
i^0

45. Numbers on a numberline
How to find any Power
Complex Number
integers
multiply the numerator and the denominator by the complex conjugate of the denominator.

46. I^2 =
-1
z1 / z2
Imaginary Unit
Real and Imaginary Parts

47. 1st. Rule of Complex Arithmetic
a real number: (a + bi)(a - bi) = a² + b²
i^2 = -1
Field
Complex Addition

48. ½(e^(iz) + e^(-iz))
Subfield
(cos? +isin?)n
z + z*
cos z

49. 3rd. Rule of Complex Arithmetic
subtracting complex numbers
four different numbers: i - -i - 1 - and -1.
For real a and b - a + bi = 0 if and only if a = b = 0
natural

50. V(x² + y²) = |z|
Polar Coordinates - r
Complex Conjugate
(cos? +isin?)n
natural