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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. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Subtraction
can't get out of the complex numbers by adding (or subtracting) or multiplying two
point of inflection

2. We see in this way that the distance between two points z and w in the complex plane is
z - z*
Roots of Unity
the complex numbers
|z-w|

3. A+bi
Integers
Complex Number Formula
Argand diagram
(cos? +isin?)n

4. Starts at 1 - does not include 0
z - z*
-1
Argand diagram
natural

5. ½(e^(iz) + e^(-iz))
Complex Conjugate
cos z
Polar Coordinates - r
Roots of Unity

6. The complex number z representing a+bi.
Euler's Formula
'i'
Affix
(a + c) + ( b + d)i

7. The reals are just the
x-axis in the complex plane
cosh²y - sinh²y
For real a and b - a + bi = 0 if and only if a = b = 0
zz*

8. Every complex number has the 'Standard Form':
a + bi for some real a and b.
i^4
Polar Coordinates - Multiplication by i
a real number: (a + bi)(a - bi) = a² + b²

9. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
real
Roots of Unity
Polar Coordinates - cos?
conjugate pairs

10. I
Absolute Value of a Complex Number
v(-1)
Subfield
i²

11. Derives z = a+bi
Euler Formula
Rational Number
cos iy
cosh²y - sinh²y

12. The product of an imaginary number and its conjugate is
i^4
zz*
irrational
a real number: (a + bi)(a - bi) = a² + b²

13. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
the complex numbers
conjugate
Irrational Number
(cos? +isin?)n

14. A plot of complex numbers as points.
zz*
Argand diagram
Polar Coordinates - Division
Complex numbers are points in the plane

15. Imaginary number

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16. (e^(-y) - e^(y)) / 2i = i sinh y
Imaginary number
cos iy
sin iy
Polar Coordinates - Division

17. All numbers
Roots of Unity
complex
Complex Multiplication
|z-w|

18. (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
irrational
Polar Coordinates - Division
z1 / z2
How to solve (2i+3)/(9-i)

19. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
v(-1)
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.
multiplying complex numbers

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

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21. I^2 =
Complex numbers are points in the plane
a + bi for some real a and b.
i^0
-1

22. I
subtracting complex numbers
Absolute Value of a Complex Number
Euler Formula
i^1

23. Like pi
cos z
transcendental
(cos? +isin?)n
v(-1)

24. To simplify the square root of a negative number
imaginary
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Affix
i^2

25. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
'i'
Polar Coordinates - Arg(z*)
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field

26. 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.
point of inflection
Real Numbers
ln z
Complex Numbers: Multiply

27. 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
Argand diagram
Complex Numbers: Multiply
cosh²y - sinh²y
subtracting complex numbers

28. x + iy = r(cos? + isin?) = re^(i?)
i^3
conjugate
Polar Coordinates - z
cos z

29. Numbers on a numberline
the vector (a -b)
irrational
integers
Any polynomial O(xn) - (n > 0)

30. 1
cosh²y - sinh²y
a real number: (a + bi)(a - bi) = a² + b²
Euler's Formula
subtracting complex numbers

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

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32. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication
The Complex Numbers
z1 ^ (z2)

33. Any number not rational
Field
four different numbers: i - -i - 1 - and -1.
irrational
Euler's Formula

34. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
i^4
De Moivre's Theorem
Complex Addition

35. Have radical
i²
Rules of Complex Arithmetic
Complex Numbers: Add & subtract
radicals

36. 1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
The Complex Numbers
Complex Number Formula
i^2

37. 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.
irrational
Polar Coordinates - Arg(z*)
the complex numbers
How to find any Power

38. Equivalent to an Imaginary Unit.
Complex Multiplication
Imaginary number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication

39. x / r
i^2
ln z
Polar Coordinates - cos?
How to find any Power

40. When two complex numbers are subtracted from one another.
Complex Subtraction
Imaginary Unit
How to multiply complex nubers(2+i)(2i-3)
cosh²y - sinh²y

41. Divide moduli and subtract arguments
Polar Coordinates - Division
e^(ln z)
Complex Addition
Polar Coordinates - z

42. V(x² + y²) = |z|
Complex Number Formula
i^2 = -1
Polar Coordinates - r
Polar Coordinates - Arg(z*)

43. Written as fractions - terminating + repeating decimals
How to multiply complex nubers(2+i)(2i-3)
rational
sin iy
Polar Coordinates - cos?

44. E^(ln r) e^(i?) e^(2pin)
Real Numbers
(cos? +isin?)n
e^(ln z)
(a + c) + ( b + d)i

45. 2nd. Rule of Complex Arithmetic

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46. Root negative - has letter i
ln z
multiplying complex numbers
imaginary
How to add and subtract complex numbers (2-3i)-(4+6i)

47. Not on the numberline
non-integers
Rules of Complex Arithmetic
zz*
radicals

48. 1
non-integers
interchangeable
i²
the vector (a -b)

49. A number that can be expressed as a fraction p/q where q is not equal to 0.
sin z
Rational Number
Real and Imaginary Parts
Complex Multiplication

50. y / r
Polar Coordinates - sin?
z + z*
(cos? +isin?)n
Complex Multiplication