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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. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
conjugate pairs
For real a and b - a + bi = 0 if and only if a = b = 0
Roots of Unity
'i'

2. When two complex numbers are subtracted from one another.
Complex Subtraction
Polar Coordinates - cos?
complex
the complex numbers

3. To simplify a complex fraction
Polar Coordinates - r
-1
multiply the numerator and the denominator by the complex conjugate of the denominator.
conjugate pairs

4. In this amazing number field every algebraic equation in z with complex coefficients
Roots of Unity
has a solution.
Subfield
real

5. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
complex
the distance from z to the origin in the complex plane
i^2

6. E ^ (z2 ln z1)
e^(ln z)
z1 ^ (z2)
irrational
complex numbers

7. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

8. E^(ln r) e^(i?) e^(2pin)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
0 if and only if a = b = 0
real

9. 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
De Moivre's Theorem
i^2 = -1
adding complex numbers
Real Numbers

10. We can also think of the point z= a+ ib as
the vector (a -b)
the complex numbers
0 if and only if a = b = 0
i^1

11. I
Complex Addition
i^2 = -1
Argand diagram
v(-1)

12. 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
De Moivre's Theorem
adding complex numbers
Complex numbers are points in the plane

13. 3rd. Rule of Complex Arithmetic
non-integers
For real a and b - a + bi = 0 if and only if a = b = 0
can't get out of the complex numbers by adding (or subtracting) or multiplying two
natural

14. A complex number and its conjugate
conjugate pairs
Rules of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)

15. Equivalent to an Imaginary Unit.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication
Imaginary number
e^(ln z)

16. xpressions such as ``the complex number z'' - and ``the point z'' are now
We say that c+di and c-di are complex conjugates.
a + bi for some real a and b.
interchangeable
Field

17. 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
Any polynomial O(xn) - (n > 0)
i^2
Rules of Complex Arithmetic
Complex Division

18. 1
the vector (a -b)
i²
Complex numbers are points in the plane
i^4

19. (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
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Multiplication by i
Complex Exponentiation
Complex Number Formula

20. Cos n? + i sin n? (for all n integers)
Polar Coordinates - r
(cos? +isin?)n
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Multiplication by i

21. 4th. Rule of Complex Arithmetic
i^2 = -1
Complex Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary Numbers

22. 2ib
-1
z - z*
Affix
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

23. Imaginary number

24. Root negative - has letter i
Complex Exponentiation
cosh²y - sinh²y
Every complex number has the 'Standard Form': a + bi for some real a and b.
imaginary

25. Rotates anticlockwise by p/2
the distance from z to the origin in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication by i
Absolute Value of a Complex Number

26. The square root of -1.
Imaginary Unit
Polar Coordinates - sin?
-1
|z-w|

27. When two complex numbers are added together.
imaginary
standard form of complex numbers
Complex Addition
i^3

28. Have radical
a + bi for some real a and b.
(cos? +isin?)n
radicals
Polar Coordinates - Multiplication by i

29. Like pi
The Complex Numbers
sin z
cosh²y - sinh²y
transcendental

30. 1
i^2
Roots of Unity
The Complex Numbers
Imaginary Numbers

31. The field of all rational and irrational numbers.
Real Numbers
z - z*
conjugate pairs
Euler Formula

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

33. 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
v(-1)
Complex Division
We say that c+di and c-di are complex conjugates.
Roots of Unity

34. Written as fractions - terminating + repeating decimals
complex numbers
i²
multiplying complex numbers
rational

35. (e^(iz) - e^(-iz)) / 2i
cos z
i^2
sin z
v(-1)

36. I
Euler's Formula
Complex Conjugate
Polar Coordinates - sin?
i^1

37. When two complex numbers are multipiled together.
Complex Multiplication
How to find any Power
Square Root
z + z*

38. For real a and b - a + bi =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate
0 if and only if a = b = 0
zz*

39. Derives z = a+bi
Euler Formula
Polar Coordinates - sin?
(a + c) + ( b + d)i
ln z

40. Not on the numberline
non-integers
Rules of Complex Arithmetic
Complex Numbers: Multiply
transcendental

41. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
adding complex numbers
Absolute Value of a Complex Number
radicals
natural

42. R^2 = x
subtracting complex numbers
Rules of Complex Arithmetic
Square Root
rational

43. 1
integers
i²
v(-1)
x-axis in the complex plane

44. 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
cos iy
De Moivre's Theorem
Complex Numbers: Add & subtract
Polar Coordinates - z?¹

45. V(x² + y²) = |z|
Any polynomial O(xn) - (n > 0)
Complex Number
Polar Coordinates - r
Subfield

46. 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
interchangeable
radicals
(cos? +isin?)n
the complex numbers

47. A + bi
standard form of complex numbers
Polar Coordinates - Multiplication
e^(ln z)
a real number: (a + bi)(a - bi) = a² + b²

48. All numbers
conjugate pairs
Complex Multiplication
complex
For real a and b - a + bi = 0 if and only if a = b = 0

49. 1st. Rule of Complex Arithmetic
The Complex Numbers
Complex Subtraction
sin z
i^2 = -1

50. (a + bi)(c + bi) =
-1
Complex Numbers: Multiply
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)