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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
the distance from z to the origin in the complex plane
Roots of Unity
Field
subtracting complex numbers

2. When two complex numbers are added together.
sin iy
Imaginary Numbers
Complex Addition
sin z

3. 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.
standard form of complex numbers
Complex Number
z1 ^ (z2)

4. All the powers of i can be written as
Imaginary Numbers
|z| = mod(z)
four different numbers: i - -i - 1 - and -1.
can't get out of the complex numbers by adding (or subtracting) or multiplying two

5. (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
Polar Coordinates - Arg(z*)
adding complex numbers
How to solve (2i+3)/(9-i)
Square Root

6. (a + bi) = (c + bi) =
0 if and only if a = b = 0
(a + c) + ( b + d)i
z + z*
a + bi for some real a and b.

7. The complex number z representing a+bi.
i^4
multiply the numerator and the denominator by the complex conjugate of the denominator.
Affix
De Moivre's Theorem

8. When two complex numbers are multipiled together.
Complex Multiplication
Argand diagram
zz*
Affix

9. A complex number may be taken to the power of another complex number.
The Complex Numbers
Complex Exponentiation
multiplying complex numbers
Complex Number

10. 1
natural
zz*
i^0
non-integers

11. I
Polar Coordinates - r
zz*
i^1
point of inflection

12. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
|z| = mod(z)
Field
How to solve (2i+3)/(9-i)
Euler Formula

13. Written as fractions - terminating + repeating decimals
|z-w|
Polar Coordinates - Arg(z*)
rational
Rules of Complex Arithmetic

14. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
How to find any Power
How to add and subtract complex numbers (2-3i)-(4+6i)
i^2

15. 2ib
-1
De Moivre's Theorem
z - z*
Complex Multiplication

16. We can also think of the point z= a+ ib as
Complex Multiplication
the vector (a -b)
Polar Coordinates - Arg(z*)
How to multiply complex nubers(2+i)(2i-3)

17. Any number not rational
irrational
z - z*
-1
Affix

18. A subset within a field.
Polar Coordinates - r
Subfield
radicals
i^0

19. A² + b² - real and non negative
zz*
natural
Imaginary number
i^2 = -1

20. 1
the distance from z to the origin in the complex plane
i^2
Complex Addition
We say that c+di and c-di are complex conjugates.

21. ? = -tan?
Complex Multiplication
'i'
Polar Coordinates - Arg(z*)
|z-w|

22. (e^(iz) - e^(-iz)) / 2i
sin z
Polar Coordinates - r
Complex Addition
i^0

23. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Imaginary Unit
cos iy
Rules of Complex Arithmetic

24. V(x² + y²) = |z|
Argand diagram
Polar Coordinates - cos?
Polar Coordinates - r
Euler's Formula

25. V(zz*) = v(a² + b²)
|z| = mod(z)
sin z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
four different numbers: i - -i - 1 - and -1.

26. E^(ln r) e^(i?) e^(2pin)
Argand diagram
cos z
e^(ln z)
Polar Coordinates - sin?

27. A complex number and its conjugate
Any polynomial O(xn) - (n > 0)
conjugate pairs
De Moivre's Theorem
point of inflection

28. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
How to solve (2i+3)/(9-i)
i^1
Field

29. Where the curvature of the graph changes
Complex Number
point of inflection
Complex Division
Complex numbers are points in the plane

30. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Division
complex
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate

31. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
i²
Polar Coordinates - z?¹
sin z
subtracting complex numbers

32. Root negative - has letter i
imaginary
complex
natural
adding complex numbers

33. To simplify the square root of a negative number
Complex Numbers: Multiply
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number Formula
Polar Coordinates - z

34. Like pi
Imaginary Numbers
(a + c) + ( b + d)i
transcendental
Complex Subtraction

35. x / r
multiplying complex numbers
conjugate pairs
Polar Coordinates - cos?
ln z

36. 2nd. Rule of Complex Arithmetic

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37. A number that can be expressed as a fraction p/q where q is not equal to 0.
the complex numbers
Rational Number
i²
point of inflection

38. (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)
i^4
standard form of complex numbers
Complex Division

39. Every complex number has the 'Standard Form':
a + bi for some real a and b.
transcendental
rational
(cos? +isin?)n

40. x + iy = r(cos? + isin?) = re^(i?)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - sin?
Polar Coordinates - z
z1 / z2

41. A number that cannot be expressed as a fraction for any integer.
Irrational Number
(cos? +isin?)n
zz*
adding complex numbers

42. 3rd. Rule of Complex Arithmetic
Polar Coordinates - sin?
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Numbers: Multiply
Real and Imaginary Parts

43. A + bi
standard form of complex numbers
transcendental
i^3
rational

44. 1
Polar Coordinates - r
rational
i^4
Argand diagram

45. (a + bi)(c + bi) =
i^3
x-axis in the complex plane
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ln z

46. The square root of -1.
Imaginary Unit
radicals
subtracting complex numbers
Polar Coordinates - Arg(z*)

47. The product of an imaginary number and its conjugate is
Field
a real number: (a + bi)(a - bi) = a² + b²
|z-w|
i²

48. ½(e^(-y) +e^(y)) = cosh y
Polar Coordinates - z
Real Numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
cos iy

49. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
We say that c+di and c-di are complex conjugates.
ln z
Irrational Number
Complex Subtraction

50. The field of all rational and irrational numbers.
Real Numbers
real
Liouville's Theorem -
has a solution.