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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. V(zz*) = v(a² + b²)
|z| = mod(z)
Complex Division
|z-w|
De Moivre's Theorem

2. When two complex numbers are divided.
z + z*
Complex Number Formula
Imaginary Numbers
Complex Division

3. To simplify the square root of a negative number
i^2 = -1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0
(a + bi) = (c + bi) = (a + c) + ( b + d)i

4. Root negative - has letter i
How to multiply complex nubers(2+i)(2i-3)
imaginary
cos z
non-integers

5. E ^ (z2 ln z1)
Argand diagram
standard form of complex numbers
z1 ^ (z2)
sin z

6. 5th. Rule of Complex Arithmetic
x-axis in the complex plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^1

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

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8. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
z - z*
x-axis in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Exponentiation

9. We see in this way that the distance between two points z and w in the complex plane is
a + bi for some real a and b.
x-axis in the complex plane
Complex Numbers: Multiply
|z-w|

10. When two complex numbers are added together.
irrational
cosh²y - sinh²y
Absolute Value of a Complex Number
Complex Addition

11. Have radical
|z-w|
conjugate
radicals
subtracting complex numbers

12. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
We say that c+di and c-di are complex conjugates.
(a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)

13. Like pi
(cos? +isin?)n
transcendental
Absolute Value of a Complex Number
cos z

14. Not on the numberline
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Irrational Number
non-integers
imaginary

15. 3rd. Rule of Complex Arithmetic
How to multiply complex nubers(2+i)(2i-3)
i^2
For real a and b - a + bi = 0 if and only if a = b = 0
z + z*

16. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Imaginary Numbers
Absolute Value of a Complex Number
the distance from z to the origin in the complex plane
Euler Formula

17. (a + bi) = (c + bi) =
(a + bi) = (c + bi) = (a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0
z + z*
(a + c) + ( b + d)i

18. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

19. 1
Liouville's Theorem -
Polar Coordinates - cos?
v(-1)
i^2

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

21. When two complex numbers are subtracted from one another.
The Complex Numbers
Complex Subtraction
z1 / z2
non-integers

22. Multiply moduli and add arguments
i^2 = -1
Polar Coordinates - Multiplication
Polar Coordinates - Division
De Moivre's Theorem

23. Numbers on a numberline
non-integers
Complex Number
integers
How to multiply complex nubers(2+i)(2i-3)

24. z1z2* / |z2|²
z + z*
Liouville's Theorem -
cos z
z1 / z2

25. I
Any polynomial O(xn) - (n > 0)
point of inflection
rational
v(-1)

26. (a + bi)(c + bi) =
How to find any Power
De Moivre's Theorem
Polar Coordinates - sin?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. Given (4-2i) the complex conjugate would be (4+2i)
x-axis in the complex plane
natural
(a + c) + ( b + d)i
Complex Conjugate

28. All the powers of i can be written as
Subfield
Polar Coordinates - z?¹
four different numbers: i - -i - 1 - and -1.
the complex numbers

29. Starts at 1 - does not include 0
radicals
Euler Formula
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

30. Where the curvature of the graph changes
point of inflection
has a solution.
0 if and only if a = b = 0
Polar Coordinates - z?¹

31. V(x² + y²) = |z|
x-axis in the complex plane
Polar Coordinates - r
i^0
Polar Coordinates - cos?

32. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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33. 2nd. Rule of Complex Arithmetic

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34. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
transcendental
Liouville's Theorem -
Complex Numbers: Multiply

35. A plot of complex numbers as points.
Argand diagram
Complex Exponentiation
(a + c) + ( b + d)i
the complex numbers

36. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Real Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^1

37. I
Roots of Unity
i^1
v(-1)
a real number: (a + bi)(a - bi) = a² + b²

38. 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
Complex Conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.
Absolute Value of a Complex Number

39. (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
the complex numbers
cosh²y - sinh²y
How to multiply complex nubers(2+i)(2i-3)
a real number: (a + bi)(a - bi) = a² + b²

40. The complex number z representing a+bi.
Affix
rational
Field
Complex Multiplication

41. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
-1
Field
ln z
complex

42. Equivalent to an Imaginary Unit.
Absolute Value of a Complex Number
Complex Numbers: Multiply
integers
Imaginary number

43. R?¹(cos? - isin?)
Integers
Subfield
Polar Coordinates - z?¹
Complex Conjugate

44. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Euler's Formula
a + bi for some real a and b.
zz*

45. No i
How to find any Power
the complex numbers
Polar Coordinates - Division
real

46. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Number Formula
Complex Numbers: Add & subtract
The Complex Numbers
i^0

47. A subset within a field.
i^1
(cos? +isin?)n
Euler Formula
Subfield

48. Derives z = a+bi
conjugate pairs
standard form of complex numbers
v(-1)
Euler Formula

49. 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
subtracting complex numbers
complex
Field
Euler Formula

50. I^2 =
rational
the vector (a -b)
Complex Multiplication
-1