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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Absolute Value of a Complex Number
e^(ln z)
Complex Number
Euler Formula

2. For real a and b - a + bi =
Polar Coordinates - r
0 if and only if a = b = 0
Field
i^2 = -1

3. 1
Affix
four different numbers: i - -i - 1 - and -1.
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^2

4. 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.
i^2
Polar Coordinates - z
Every complex number has the 'Standard Form': a + bi for some real a and b.

5. We see in this way that the distance between two points z and w in the complex plane is
ln z
How to find any Power
Imaginary Numbers
|z-w|

6. ½(e^(-y) +e^(y)) = cosh y
has a solution.
i^0
Absolute Value of a Complex Number
cos iy

7. The modulus of the complex number z= a + ib now can be interpreted as
Complex Addition
imaginary
the distance from z to the origin in the complex plane
point of inflection

8. In this amazing number field every algebraic equation in z with complex coefficients
x-axis in the complex plane
has a solution.
transcendental
i^1

9. When two complex numbers are divided.
The Complex Numbers
Argand diagram
rational
Complex Division

10. Where the curvature of the graph changes
Complex Number
Complex Numbers: Add & subtract
Polar Coordinates - sin?
point of inflection

11. 1
i^4
Complex Conjugate
Polar Coordinates - r
irrational

12. I
|z| = mod(z)
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin z
i^1

13. Numbers on a numberline
(cos? +isin?)n
interchangeable
integers
real

14. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
conjugate
multiplying complex numbers
complex numbers
Affix

15. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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16. (e^(-y) - e^(y)) / 2i = i sinh y
Absolute Value of a Complex Number
sin iy
i^2
imaginary

17. ½(e^(iz) + e^(-iz))
e^(ln z)
complex
cos z
Imaginary Numbers

18. E ^ (z2 ln z1)
complex
multiplying complex numbers
cosh²y - sinh²y
z1 ^ (z2)

19. Divide moduli and subtract arguments
Any polynomial O(xn) - (n > 0)
i^3
i^4
Polar Coordinates - Division

20. (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
v(-1)
How to multiply complex nubers(2+i)(2i-3)
the distance from z to the origin in the complex plane
Imaginary Numbers

21. (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
the vector (a -b)
integers
How to solve (2i+3)/(9-i)
Integers

22. All numbers
the complex numbers
imaginary
complex
Polar Coordinates - z

23. 1st. Rule of Complex Arithmetic
How to multiply complex nubers(2+i)(2i-3)
i^2 = -1
Real Numbers
integers

24. To simplify a complex fraction
i^3
Imaginary Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^1

25. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
a + bi for some real a and b.
Polar Coordinates - sin?
Complex Numbers: Add & subtract
z + z*

26. 3rd. Rule of Complex Arithmetic
How to solve (2i+3)/(9-i)
Absolute Value of a Complex Number
For real a and b - a + bi = 0 if and only if a = b = 0
the distance from z to the origin in the complex plane

27. 1
i^0
e^(ln z)
Integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

28. A plot of complex numbers as points.
radicals
Argand diagram
Complex Numbers: Add & subtract
e^(ln z)

29. 4th. Rule of Complex Arithmetic
Real Numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
The Complex Numbers
a + bi for some real a and b.

30. Imaginary number

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31. 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.
subtracting complex numbers
How to find any Power
Integers
Affix

32. I^2 =
sin iy
-1
Complex Number Formula
has a solution.

33. Written as fractions - terminating + repeating decimals
'i'
Any polynomial O(xn) - (n > 0)
rational
Euler's Formula

34. Have radical
Complex Division
e^(ln z)
radicals
Roots of Unity

35. (e^(iz) - e^(-iz)) / 2i
sin z
interchangeable
Complex Number
the distance from z to the origin in the complex plane

36. (a + bi) = (c + bi) =
z1 / z2
Polar Coordinates - Multiplication by i
(a + c) + ( b + d)i
imaginary

37. A complex number and its conjugate
Complex Division
cos iy
i²
conjugate pairs

38. V(zz*) = v(a² + b²)
|z| = mod(z)
multiplying complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Subfield

39. When two complex numbers are multipiled together.
Imaginary number
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Multiplication by i
Complex Multiplication

40. Not on the numberline
Liouville's Theorem -
non-integers
Real and Imaginary Parts
Complex Numbers: Add & subtract

41. A + bi
-1
standard form of complex numbers
z - z*
Complex Numbers: Multiply

42. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
the complex numbers
(a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Multiplication

43. Has exactly n roots by the fundamental theorem of algebra
point of inflection
Any polynomial O(xn) - (n > 0)
standard form of complex numbers
cos z

44. A complex number may be taken to the power of another complex number.
has a solution.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
conjugate
Complex Exponentiation

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

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46. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Conjugate
conjugate
x-axis in the complex plane

47. The field of all rational and irrational numbers.
Real Numbers
Absolute Value of a Complex Number
interchangeable
We say that c+di and c-di are complex conjugates.

48. A subset within a field.
cos z
i^2
Subfield
radicals

49. Like pi
0 if and only if a = b = 0
|z-w|
transcendental
adding complex numbers

50. When two complex numbers are added together.
Complex Conjugate
Square Root
Complex Addition
Polar Coordinates - Multiplication by i

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CLEP General Mathematics: Complex Numbers

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