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CLEP General Mathematics: Complex Numbers

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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 plot of complex numbers as points.
Argand diagram
|z| = mod(z)
multiply the numerator and the denominator by the complex conjugate of the denominator.
real

2. 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.'
Imaginary number
transcendental
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number

3. 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.
rational
How to find any Power
|z-w|
transcendental

4. 1st. Rule of Complex Arithmetic
i^2 = -1
Complex Number
conjugate
zz*

5. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)
Imaginary Numbers
How to multiply complex nubers(2+i)(2i-3)

6. I
Real Numbers
natural
sin iy
v(-1)

7. For real a and b - a + bi =
imaginary
0 if and only if a = b = 0
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers

8. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)
How to solve (2i+3)/(9-i)
irrational

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

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10. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
z - z*
standard form of complex numbers
multiplying complex numbers
cosh²y - sinh²y

11. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
i^0
Complex Number
(a + c) + ( b + d)i

12. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Imaginary Numbers
Integers
Complex Number
Rational Number

13. 3
non-integers
i^3
'i'
the complex numbers

14. 4th. Rule of Complex Arithmetic
radicals
can't get out of the complex numbers by adding (or subtracting) or multiplying two
|z-w|
(a + bi) = (c + bi) = (a + c) + ( b + d)i

15. 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
rational
the distance from z to the origin in the complex plane
e^(ln z)
Rules of Complex Arithmetic

16. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
z1 ^ (z2)
Polar Coordinates - r
De Moivre's Theorem

17. 3rd. Rule of Complex Arithmetic
cos iy
For real a and b - a + bi = 0 if and only if a = b = 0
(cos? +isin?)n
conjugate

18. When two complex numbers are subtracted from one another.
Field
Complex Subtraction
Polar Coordinates - Arg(z*)
ln z

19. Divide moduli and subtract arguments
conjugate
Complex Division
sin iy
Polar Coordinates - Division

20. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - cos?
Complex Number
Roots of Unity
Rational Number

21. Written as fractions - terminating + repeating decimals
rational
Polar Coordinates - Arg(z*)
Affix
z1 / z2

22. 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
z1 ^ (z2)
Polar Coordinates - z
Irrational Number

23. No i
Square Root
the vector (a -b)
real
Any polynomial O(xn) - (n > 0)

24. (a + bi)(c + bi) =
We say that c+di and c-di are complex conjugates.
radicals
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|

25. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Real Numbers
Polar Coordinates - Multiplication

26. V(x² + y²) = |z|
i^3
sin iy
Polar Coordinates - r
Imaginary Unit

27. R^2 = x
Square Root
v(-1)
has a solution.
Polar Coordinates - z

28. Given (4-2i) the complex conjugate would be (4+2i)
Complex Number
multiplying complex numbers
How to find any Power
Complex Conjugate

29. The complex number z representing a+bi.
Affix
Complex Numbers: Add & subtract
multiply the numerator and the denominator by the complex conjugate of the denominator.
the complex numbers

30. A + bi
standard form of complex numbers
Polar Coordinates - z?¹
For real a and b - a + bi = 0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)

31. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - Division
How to multiply complex nubers(2+i)(2i-3)
Rules of Complex Arithmetic
the distance from z to the origin in the complex plane

32. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
rational
Rational Number
the complex numbers

33. Root negative - has letter i
imaginary
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - r

34. When two complex numbers are multipiled together.
Complex Multiplication
Polar Coordinates - sin?
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex numbers are points in the plane

35. Any number not rational
Imaginary number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
irrational
i^2

36. (e^(-y) - e^(y)) / 2i = i sinh y
non-integers
sin iy
Euler Formula
i^3

37. (e^(iz) - e^(-iz)) / 2i
non-integers
z1 ^ (z2)
Euler's Formula
sin z

38. A complex number and its conjugate
Real and Imaginary Parts
i^3
conjugate pairs
the complex numbers

39. 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
Polar Coordinates - z
Absolute Value of a Complex Number
the complex numbers
Polar Coordinates - z?¹

40. Has exactly n roots by the fundamental theorem of algebra
Imaginary number
'i'
Complex Addition
Any polynomial O(xn) - (n > 0)

41. 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
We say that c+di and c-di are complex conjugates.
Rules of Complex Arithmetic
natural
conjugate pairs

42. Have radical
Complex numbers are points in the plane
Complex Subtraction
radicals
multiply the numerator and the denominator by the complex conjugate of the denominator.

43. 1
i^0
-1
We say that c+di and c-di are complex conjugates.
De Moivre's Theorem

44. 2a
Imaginary number
How to add and subtract complex numbers (2-3i)-(4+6i)
z + z*
Polar Coordinates - Multiplication

45. 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
irrational
four different numbers: i - -i - 1 - and -1.
Complex Exponentiation

46. All numbers
natural
complex
Polar Coordinates - z?¹
imaginary

47. xpressions such as ``the complex number z'' - and ``the point z'' are now
can't get out of the complex numbers by adding (or subtracting) or multiplying two
interchangeable
i^0
For real a and b - a + bi = 0 if and only if a = b = 0

48. Not on the numberline
subtracting complex numbers
rational
non-integers
the complex numbers

49. E ^ (z2 ln z1)
i^2 = -1
z1 ^ (z2)
(cos? +isin?)n
'i'

50. z1z2* / |z2|²
the vector (a -b)
z1 / z2
Euler's Formula
complex

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CLEP General Mathematics: Complex Numbers

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