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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. 3rd. Rule of Complex Arithmetic
subtracting complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
radicals
Integers

2. I
v(-1)
integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane

3. xpressions such as ``the complex number z'' - and ``the point z'' are now
z - z*
natural
four different numbers: i - -i - 1 - and -1.
interchangeable

4. Where the curvature of the graph changes
point of inflection
i^0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers

5. No i
real
standard form of complex numbers
Polar Coordinates - Division
integers

6. A+bi
i^2
Complex Division
Complex Number Formula
Integers

7. Any number not rational
irrational
adding complex numbers
Argand diagram
Real Numbers

8. Every complex number has the 'Standard Form':
a + bi for some real a and b.
has a solution.
Any polynomial O(xn) - (n > 0)
the complex numbers

9. z1z2* / |z2|²
Complex Multiplication
imaginary
z1 / z2
a + bi for some real a and b.

10. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Conjugate
conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^2

11. (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
Complex Conjugate
How to multiply complex nubers(2+i)(2i-3)
multiplying complex numbers
ln z

12. (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
How to solve (2i+3)/(9-i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number Formula
Integers

13. Imaginary number

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14. A subset within a field.
Complex Number Formula
Subfield
Complex Number
Complex numbers are points in the plane

15. Root negative - has letter i
Imaginary Numbers
Square Root
imaginary
ln z

16. 1
adding complex numbers
Polar Coordinates - cos?
cos z
i^0

17. R^2 = x
Polar Coordinates - z?¹
Square Root
De Moivre's Theorem
Argand diagram

18. To simplify the square root of a negative number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Numbers: Add & subtract

19. The reals are just the
i^1
x-axis in the complex plane
Polar Coordinates - Multiplication by i
v(-1)

20. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
rational
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula

21. A + bi
Roots of Unity
Imaginary Numbers
standard form of complex numbers
Polar Coordinates - Multiplication

22. The complex number z representing a+bi.
Affix
Polar Coordinates - sin?
Euler Formula
How to find any Power

23. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Field
Polar Coordinates - Multiplication by i
Euler's Formula

24. 2ib
z - z*
v(-1)
|z| = mod(z)
i^0

25. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0
ln z

26. A complex number may be taken to the power of another complex number.
the vector (a -b)
Integers
Complex Exponentiation
z - z*

27. Starts at 1 - does not include 0
cosh²y - sinh²y
has a solution.
natural
|z-w|

28. R?¹(cos? - isin?)
z + z*
Subfield
Polar Coordinates - z?¹
(a + c) + ( b + d)i

29. Cos n? + i sin n? (for all n integers)
De Moivre's Theorem
How to add and subtract complex numbers (2-3i)-(4+6i)
Argand diagram
(cos? +isin?)n

30. Not on the numberline
non-integers
(cos? +isin?)n
Rules of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two

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

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32. 1
i^4
i^2 = -1
z1 ^ (z2)
i²

33. When two complex numbers are added together.
Euler Formula
Complex Addition
complex numbers
Imaginary Unit

34. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
Polar Coordinates - Arg(z*)
conjugate pairs
the vector (a -b)

35. 3
i^3
i^1
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Subtraction

36. For real a and b - a + bi =
Polar Coordinates - Multiplication by i
Imaginary number
0 if and only if a = b = 0
Complex Number

37. I^2 =
(a + bi) = (c + bi) = (a + c) + ( b + d)i
-1
e^(ln z)
Complex Number

38. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
irrational
ln z
Polar Coordinates - Division
complex numbers

39. E^(ln r) e^(i?) e^(2pin)
Euler's Formula
e^(ln z)
Polar Coordinates - z
conjugate

40. Derives z = a+bi
Euler Formula
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - r
conjugate

41. I
i^1
(cos? +isin?)n
Complex Conjugate
Polar Coordinates - z?¹

42. x + iy = r(cos? + isin?) = re^(i?)
z - z*
point of inflection
natural
Polar Coordinates - z

43. In this amazing number field every algebraic equation in z with complex coefficients
Complex Multiplication
cos z
For real a and b - a + bi = 0 if and only if a = b = 0
has a solution.

44. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
How to multiply complex nubers(2+i)(2i-3)
Real and Imaginary Parts
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers

45. A² + b² - real and non negative
0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
zz*
Complex Exponentiation

46. 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.
De Moivre's Theorem
|z| = mod(z)
Imaginary Unit
Complex numbers are points in the plane

47. (e^(-y) - e^(y)) / 2i = i sinh y
(a + c) + ( b + d)i
sin iy
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler Formula

48. 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.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
point of inflection
Complex Conjugate
Complex Numbers: Multiply

49. The modulus of the complex number z= a + ib now can be interpreted as
a + bi for some real a and b.
sin z
the distance from z to the origin in the complex plane
z + z*

50. Given (4-2i) the complex conjugate would be (4+2i)
Complex Number Formula
cosh²y - sinh²y
multiplying complex numbers
Complex Conjugate