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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. ½(e^(-y) +e^(y)) = cosh y
complex
cos z
0 if and only if a = b = 0
cos iy

2. When two complex numbers are subtracted from one another.
x-axis in the complex plane
z - z*
Complex Subtraction
sin iy

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

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4. All numbers
|z| = mod(z)
Polar Coordinates - Multiplication by i
zz*
complex

5. A subset within a field.
z - z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication
Subfield

6. A + bi
standard form of complex numbers
Complex Multiplication
cos iy
0 if and only if a = b = 0

7. Root negative - has letter i
imaginary
a real number: (a + bi)(a - bi) = a² + b²
non-integers
Polar Coordinates - sin?

8. When two complex numbers are multipiled together.
For real a and b - a + bi = 0 if and only if a = b = 0
natural
Irrational Number
Complex Multiplication

9. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
natural
sin z
z + z*

10. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Division
multiplying complex numbers
Argand diagram
Complex Addition

11. 2nd. Rule of Complex Arithmetic

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12. E ^ (z2 ln z1)
z1 / z2
sin z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z1 ^ (z2)

13. (e^(iz) - e^(-iz)) / 2i
a + bi for some real a and b.
z - z*
Complex Addition
sin z

14. I
integers
How to add and subtract complex numbers (2-3i)-(4+6i)
How to solve (2i+3)/(9-i)
i^1

15. Written as fractions - terminating + repeating decimals
Complex Numbers: Multiply
rational
Complex Subtraction
multiplying complex numbers

16. 3
We say that c+di and c-di are complex conjugates.
i^3
Complex Division
Polar Coordinates - Arg(z*)

17. I^2 =
standard form of complex numbers
-1
Polar Coordinates - z?¹
Rules of Complex Arithmetic

18. Has exactly n roots by the fundamental theorem of algebra
can't get out of the complex numbers by adding (or subtracting) or multiplying two
four different numbers: i - -i - 1 - and -1.
adding complex numbers
Any polynomial O(xn) - (n > 0)

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

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20. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
e^(ln z)
conjugate
(a + c) + ( b + d)i
irrational

21. A² + b² - real and non negative
a + bi for some real a and b.
i^0
zz*
imaginary

22. For real a and b - a + bi =
0 if and only if a = b = 0
Irrational Number
i^1
can't get out of the complex numbers by adding (or subtracting) or multiplying two

23. (a + bi) = (c + bi) =
|z-w|
a real number: (a + bi)(a - bi) = a² + b²
(a + c) + ( b + d)i
Complex Exponentiation

24. ½(e^(iz) + e^(-iz))
Rational Number
imaginary
cos z
Integers

25. The complex number z representing a+bi.
adding complex numbers
Polar Coordinates - Multiplication
Affix
How to add and subtract complex numbers (2-3i)-(4+6i)

26. Where the curvature of the graph changes
real
point of inflection
Complex Conjugate
For real a and b - a + bi = 0 if and only if a = b = 0

27. 1
Affix
point of inflection
cosh²y - sinh²y
Complex Number

28. No i
sin iy
integers
Complex numbers are points in the plane
real

29. 1
i^2
conjugate
zz*
How to find any Power

30. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
How to find any Power
Roots of Unity
Field
'i'

31. 1
Euler's Formula
0 if and only if a = b = 0
radicals
i²

32. Rotates anticlockwise by p/2
(a + c) + ( b + d)i
Euler's Formula
Polar Coordinates - Multiplication by i
Field

33. y / r
Polar Coordinates - Multiplication
Polar Coordinates - sin?
Roots of Unity
complex

34. 1st. Rule of Complex Arithmetic
i^2 = -1
non-integers
Complex Number Formula
ln z

35. Cos n? + i sin n? (for all n integers)
irrational
(cos? +isin?)n
Polar Coordinates - Multiplication
i²

36. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Complex Addition
multiplying complex numbers
Polar Coordinates - z?¹

37. Divide moduli and subtract arguments
cos z
Polar Coordinates - Division
Euler's Formula
integers

38. z1z2* / |z2|²
|z-w|
Polar Coordinates - z?¹
Any polynomial O(xn) - (n > 0)
z1 / z2

39. When two complex numbers are divided.
Complex Division
Imaginary number
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Unit

40. In this amazing number field every algebraic equation in z with complex coefficients
imaginary
has a solution.
Complex Numbers: Add & subtract
Complex Division

41. E^(ln r) e^(i?) e^(2pin)
has a solution.
e^(ln z)
Imaginary number
i^2 = -1

42. V(x² + y²) = |z|
We say that c+di and c-di are complex conjugates.
Polar Coordinates - r
Real and Imaginary Parts
subtracting complex numbers

43. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
Polar Coordinates - cos?
natural

44. V(zz*) = v(a² + b²)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers
Roots of Unity
|z| = mod(z)

45. (a + bi)(c + bi) =
Affix
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers
Polar Coordinates - r

46. 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
Polar Coordinates - Multiplication by i
subtracting complex numbers
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

47. xpressions such as ``the complex number z'' - and ``the point z'' are now
conjugate pairs
interchangeable
Complex Multiplication
Polar Coordinates - Division

48. Have radical
radicals
Every complex number has the 'Standard Form': a + bi for some real a and b.
Affix
For real a and b - a + bi = 0 if and only if a = b = 0

49. Starts at 1 - does not include 0
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
natural
Roots of Unity

50. All the powers of i can be written as
De Moivre's Theorem
sin z
four different numbers: i - -i - 1 - and -1.
Rational Number