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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. A subset within a field.
Subfield
Complex Number Formula
rational
The Complex Numbers

2. 1
(cos? +isin?)n
Complex Subtraction
transcendental
i^0

3. When two complex numbers are added together.
z1 / z2
i^2 = -1
Complex Addition
Polar Coordinates - sin?

4. 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
adding complex numbers
complex
Rules of Complex Arithmetic
The Complex Numbers

5. Written as fractions - terminating + repeating decimals
We say that c+di and c-di are complex conjugates.
rational
z + z*
cos iy

6. A + bi
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
standard form of complex numbers
the vector (a -b)

7. When two complex numbers are subtracted from one another.
Irrational Number
the distance from z to the origin in the complex plane
Liouville's Theorem -
Complex Subtraction

8. xpressions such as ``the complex number z'' - and ``the point z'' are now
non-integers
interchangeable
Polar Coordinates - Division
conjugate

9. Every complex number has the 'Standard Form':
z - z*
a + bi for some real a and b.
Rules of Complex Arithmetic
i^2

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.
i^1
Complex numbers are points in the plane
z - z*
cos iy

11. 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
irrational
the complex numbers
non-integers
i^2

12. 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
real
i^1
Complex Division
adding complex numbers

13. 1
Polar Coordinates - Arg(z*)
Imaginary Numbers
Affix
i²

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

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15. 2a
-1
z + z*
standard form of complex numbers
De Moivre's Theorem

16. A number that can be expressed as a fraction p/q where q is not equal to 0.
sin iy
Complex Numbers: Multiply
Rational Number
complex numbers

17. When two complex numbers are multipiled together.
Real and Imaginary Parts
Complex Multiplication
a real number: (a + bi)(a - bi) = a² + b²
Irrational Number

18. R?¹(cos? - isin?)
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z?¹
adding complex numbers
Complex Addition

19. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Addition
Polar Coordinates - r

20. A complex number may be taken to the power of another complex number.
v(-1)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Numbers: Add & subtract
Complex Exponentiation

21. ? = -tan?
Polar Coordinates - cos?
non-integers
sin iy
Polar Coordinates - Arg(z*)

22. In this amazing number field every algebraic equation in z with complex coefficients
Irrational Number
has a solution.
adding complex numbers
(cos? +isin?)n

23. Derives z = a+bi
Complex Division
multiplying complex numbers
Euler Formula
conjugate pairs

24. 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.
Complex Numbers: Multiply
z - z*
the complex numbers
has a solution.

25. The reals are just the
x-axis in the complex plane
0 if and only if a = b = 0
Complex Addition
can't get out of the complex numbers by adding (or subtracting) or multiplying two

26. Multiply moduli and add arguments
Square Root
z - z*
Polar Coordinates - Multiplication
z + z*

27. 3
multiply the numerator and the denominator by the complex conjugate of the denominator.
cos z
i^3
i²

28. Like pi
Absolute Value of a Complex Number
Integers
the vector (a -b)
transcendental

29. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
a real number: (a + bi)(a - bi) = a² + b²
Roots of Unity
Imaginary number
point of inflection

30. I
Complex numbers are points in the plane
Field
v(-1)
Imaginary Numbers

31. E ^ (z2 ln z1)
z1 ^ (z2)
the distance from z to the origin in the complex plane
point of inflection
Polar Coordinates - r

32. R^2 = x
sin iy
the vector (a -b)
Square Root
non-integers

33. All numbers
i^2
Polar Coordinates - sin?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex

34. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
0 if and only if a = b = 0
integers
We say that c+di and c-di are complex conjugates.

35. Numbers on a numberline
How to find any Power
imaginary
point of inflection
integers

36. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
integers
i^0
Imaginary Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)

37. 1
Euler Formula
i^4
a real number: (a + bi)(a - bi) = a² + b²
irrational

38. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
natural
z1 ^ (z2)
ln z
For real a and b - a + bi = 0 if and only if a = b = 0

39. A complex number and its conjugate
conjugate pairs
Polar Coordinates - Arg(z*)
four different numbers: i - -i - 1 - and -1.
(a + c) + ( b + d)i

40. 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
Complex Numbers: Add & subtract
standard form of complex numbers
adding complex numbers
z1 ^ (z2)

41. Equivalent to an Imaginary Unit.
z1 / z2
Imaginary number
v(-1)
|z| = mod(z)

42. Imaginary number

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43. We can also think of the point z= a+ ib as
the vector (a -b)
Subfield
e^(ln z)
Rules of Complex Arithmetic

44. y / r
Field
i^4
Polar Coordinates - sin?
How to find any Power

45. 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
multiplying complex numbers
We say that c+di and c-di are complex conjugates.
interchangeable
subtracting complex numbers

46. We see in this way that the distance between two points z and w in the complex plane is
adding complex numbers
subtracting complex numbers
|z-w|
can't get out of the complex numbers by adding (or subtracting) or multiplying two

47. Starts at 1 - does not include 0
natural
irrational
z - z*
(cos? +isin?)n

48. ½(e^(iz) + e^(-iz))
cos z
cosh²y - sinh²y
integers
i^1

49. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Complex Numbers: Add & subtract
Rules of Complex Arithmetic
Complex Number
Complex Number Formula

50. (e^(iz) - e^(-iz)) / 2i
sin z
Polar Coordinates - Multiplication
i^2 = -1
Euler Formula