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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. z1z2* / |z2|²
Subfield
Polar Coordinates - cos?
the vector (a -b)
z1 / z2

2. (a + bi)(c + bi) =
Euler Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
imaginary
Complex numbers are points in the plane

3. 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.
real
Polar Coordinates - z?¹
subtracting complex numbers
Complex Numbers: Multiply

4. Starts at 1 - does not include 0
four different numbers: i - -i - 1 - and -1.
cos z
Roots of Unity
natural

5. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to add and subtract complex numbers (2-3i)-(4+6i)
the vector (a -b)
x-axis in the complex plane

6. (e^(-y) - e^(y)) / 2i = i sinh y
How to add and subtract complex numbers (2-3i)-(4+6i)
Argand diagram
sin iy
Liouville's Theorem -

7. A plot of complex numbers as points.
Polar Coordinates - cos?
i^0
Complex Numbers: Multiply
Argand diagram

8. A number that cannot be expressed as a fraction for any integer.
Imaginary Unit
De Moivre's Theorem
Polar Coordinates - Division
Irrational Number

9. Any number not rational
Polar Coordinates - Multiplication
i^2 = -1
z1 / z2
irrational

10. 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
Polar Coordinates - z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Real and Imaginary Parts
Complex Subtraction

11. I^2 =
Square Root
Polar Coordinates - Arg(z*)
four different numbers: i - -i - 1 - and -1.
-1

12. 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.
How to find any Power
natural
i^0
Any polynomial O(xn) - (n > 0)

13. (e^(iz) - e^(-iz)) / 2i
Roots of Unity
zz*
sin z
i²

14. Has exactly n roots by the fundamental theorem of algebra
-1
Liouville's Theorem -
Any polynomial O(xn) - (n > 0)
e^(ln z)

15. 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.
(cos? +isin?)n
Complex numbers are points in the plane
Subfield
Complex Subtraction

16. Have radical
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler's Formula
Polar Coordinates - Arg(z*)
radicals

17. V(zz*) = v(a² + b²)
Liouville's Theorem -
|z| = mod(z)
i^0
Subfield

18. (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)
four different numbers: i - -i - 1 - and -1.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number

19. R^2 = x
Imaginary Numbers
Euler Formula
Square Root
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

20. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Numbers: Multiply
Rational Number
x-axis in the complex plane
i²

21. y / r
Polar Coordinates - sin?
Euler Formula
0 if and only if a = b = 0
the complex numbers

22. I
v(-1)
adding complex numbers
Complex Number Formula
z - z*

23. 1
Real Numbers
i²
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)

24. When two complex numbers are subtracted from one another.
Complex Subtraction
The Complex Numbers
-1
Polar Coordinates - Multiplication

25. Written as fractions - terminating + repeating decimals
(cos? +isin?)n
Complex Number Formula
Polar Coordinates - Multiplication
rational

26. 2nd. Rule of Complex Arithmetic

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27. All numbers
complex
Complex Number
Liouville's Theorem -
Roots of Unity

28. We see in this way that the distance between two points z and w in the complex plane is
Complex Numbers: Add & subtract
Imaginary number
Argand diagram
|z-w|

29. Real and imaginary numbers
complex numbers
How to find any Power
cos z
Complex Multiplication

30. 3rd. Rule of Complex Arithmetic
non-integers
For real a and b - a + bi = 0 if and only if a = b = 0
|z-w|
Euler's Formula

31. 1
i^0
cosh²y - sinh²y
Polar Coordinates - sin?
(a + bi) = (c + bi) = (a + c) + ( b + d)i

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

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33. The complex number z representing a+bi.
Complex Subtraction
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Addition
Affix

34. A complex number and its conjugate
(cos? +isin?)n
conjugate pairs
Complex Number
Polar Coordinates - cos?

35. 3
a + bi for some real a and b.
i^3
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
imaginary

36. R?¹(cos? - isin?)
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z?¹
Complex Addition
real

37. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Exponentiation
i²
conjugate
adding complex numbers

38. 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
the complex numbers
De Moivre's Theorem
v(-1)
(a + bi) = (c + bi) = (a + c) + ( b + d)i

39. 1
cos z
Polar Coordinates - Arg(z*)
i^2
Complex Addition

40. We can also think of the point z= a+ ib as
can't get out of the complex numbers by adding (or subtracting) or multiplying two
rational
the vector (a -b)
conjugate

41. 2a
Square Root
Complex Numbers: Multiply
For real a and b - a + bi = 0 if and only if a = b = 0
z + z*

42. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Rules of Complex Arithmetic
point of inflection
Euler Formula
multiplying complex numbers

43. The reals are just the
x-axis in the complex plane
i^2
natural
conjugate

44. Given (4-2i) the complex conjugate would be (4+2i)
adding complex numbers
Complex Conjugate
imaginary
conjugate pairs

45. No i
real
a real number: (a + bi)(a - bi) = a² + b²
How to find any Power
Affix

46. 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
|z| = mod(z)
Rules of Complex Arithmetic
Roots of Unity
interchangeable

47. When two complex numbers are added together.
How to multiply complex nubers(2+i)(2i-3)
z - z*
Any polynomial O(xn) - (n > 0)
Complex Addition

48. The field of all rational and irrational numbers.
real
Complex Addition
Polar Coordinates - r
Real Numbers

49. Like pi
z - z*
transcendental
i^0
x-axis in the complex plane

50. A² + b² - real and non negative
z1 ^ (z2)
zz*
How to solve (2i+3)/(9-i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i