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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. 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
Field
adding complex numbers
Polar Coordinates - sin?
e^(ln z)

2. 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.
How to solve (2i+3)/(9-i)
Complex Subtraction
a + bi for some real a and b.
Complex Numbers: Multiply

3. ½(e^(-y) +e^(y)) = cosh y
Polar Coordinates - sin?
cos iy
can't get out of the complex numbers by adding (or subtracting) or multiplying two
multiply the numerator and the denominator by the complex conjugate of the denominator.

4. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Absolute Value of a Complex Number
e^(ln z)
i^0

5. A+bi
cos iy
z - z*
Complex Number Formula
four different numbers: i - -i - 1 - and -1.

6. I
point of inflection
v(-1)
multiplying complex numbers
Real and Imaginary Parts

7. x + iy = r(cos? + isin?) = re^(i?)
subtracting complex numbers
Polar Coordinates - z?¹
Polar Coordinates - z
How to find any Power

8. 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
Real Numbers
How to solve (2i+3)/(9-i)
Irrational Number

9. In this amazing number field every algebraic equation in z with complex coefficients
the distance from z to the origin in the complex plane
Polar Coordinates - z?¹
has a solution.
Field

10. 1st. Rule of Complex Arithmetic
v(-1)
Complex Numbers: Add & subtract
i^2 = -1
point of inflection

11. 3rd. Rule of Complex Arithmetic
Irrational Number
Complex Number
conjugate pairs
For real a and b - a + bi = 0 if and only if a = b = 0

12. Given (4-2i) the complex conjugate would be (4+2i)
We say that c+di and c-di are complex conjugates.
Any polynomial O(xn) - (n > 0)
Polar Coordinates - sin?
Complex Conjugate

13. Equivalent to an Imaginary Unit.
Complex Subtraction
Imaginary number
-1
Euler Formula

14. R^2 = x
ln z
Irrational Number
Square Root
Polar Coordinates - z

15. Rotates anticlockwise by p/2
cosh²y - sinh²y
Polar Coordinates - Multiplication by i
z + z*
i^2

16. Divide moduli and subtract arguments
standard form of complex numbers
Polar Coordinates - Division
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Multiplication

17. 4th. Rule of Complex Arithmetic
-1
Real and Imaginary Parts
imaginary
(a + bi) = (c + bi) = (a + c) + ( b + d)i

18. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
natural
The Complex Numbers
Imaginary Numbers
Complex Conjugate

19. Not on the numberline
non-integers
How to solve (2i+3)/(9-i)
i^0
i²

20. 1
Complex Addition
Rules of Complex Arithmetic
non-integers
i²

21. Have radical
How to multiply complex nubers(2+i)(2i-3)
radicals
We say that c+di and c-di are complex conjugates.
Imaginary Unit

22. Multiply moduli and add arguments
Subfield
Polar Coordinates - Multiplication
subtracting complex numbers
has a solution.

23. 2a
z + z*
Complex Number Formula
Imaginary Numbers
De Moivre's Theorem

24. 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
has a solution.
conjugate pairs
the complex numbers
(cos? +isin?)n

25. Real and imaginary numbers
Complex Number
complex numbers
ln z
irrational

26. ½(e^(iz) + e^(-iz))
We say that c+di and c-di are complex conjugates.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
Euler Formula

27. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Complex Numbers: Multiply
Roots of Unity
How to find any Power

28. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Numbers: Multiply
subtracting complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
'i'

29. (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
Euler's Formula
Subfield
How to add and subtract complex numbers (2-3i)-(4+6i)
How to multiply complex nubers(2+i)(2i-3)

30. The reals are just the
imaginary
x-axis in the complex plane
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - cos?

31. z1z2* / |z2|²
standard form of complex numbers
the complex numbers
Complex Subtraction
z1 / z2

32. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Arg(z*)
standard form of complex numbers
Imaginary Unit

33. E^(ln r) e^(i?) e^(2pin)
For real a and b - a + bi = 0 if and only if a = b = 0
e^(ln z)
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

34. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Complex numbers are points in the plane
Roots of Unity
Polar Coordinates - z
subtracting complex numbers

35. V(x² + y²) = |z|
Complex Subtraction
Affix
Polar Coordinates - r
standard form of complex numbers

36. A subset within a field.
Polar Coordinates - Division
Subfield
the vector (a -b)
integers

37. I^2 =
z - z*
-1
Polar Coordinates - z
Polar Coordinates - cos?

38. A + bi
standard form of complex numbers
Argand diagram
interchangeable
Imaginary Numbers

39. (e^(iz) - e^(-iz)) / 2i
Field
sin z
transcendental
Polar Coordinates - Division

40. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
i^4
Subfield
Complex Number Formula

41. ? = -tan?
Field
cos z
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Arg(z*)

42. When two complex numbers are divided.
Complex Division
Polar Coordinates - z?¹
imaginary
sin z

43. 1
i^4
How to solve (2i+3)/(9-i)
conjugate pairs
multiply the numerator and the denominator by the complex conjugate of the denominator.

44. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ln z

45. (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
Argand diagram
How to solve (2i+3)/(9-i)
point of inflection
can't get out of the complex numbers by adding (or subtracting) or multiplying two

46. 2ib
z - z*
The Complex Numbers
cos iy
Complex Numbers: Multiply

47. Where the curvature of the graph changes
Complex Conjugate
imaginary
point of inflection
i^2

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

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49. A number that cannot be expressed as a fraction for any integer.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the distance from z to the origin in the complex plane
Complex Numbers: Multiply
Irrational Number

50. E ^ (z2 ln z1)
z1 ^ (z2)
-1
v(-1)
complex