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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Imaginary Unit
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0

2. V(x² + y²) = |z|
Polar Coordinates - r
ln z
z1 / z2
has a solution.

3. When two complex numbers are subtracted from one another.
Liouville's Theorem -
Complex Subtraction
ln z
Imaginary number

4. y / r
Polar Coordinates - sin?
Complex Conjugate
Affix
De Moivre's Theorem

5. In this amazing number field every algebraic equation in z with complex coefficients
imaginary
has a solution.
Polar Coordinates - z?¹
Complex Subtraction

6. Every complex number has the 'Standard Form':
a + bi for some real a and b.
i^3
Absolute Value of a Complex Number
multiplying complex numbers

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

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8. A² + b² - real and non negative
zz*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex
Imaginary number

9. Multiply moduli and add arguments
Polar Coordinates - Multiplication
natural
i^1
-1

10. 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
Polar Coordinates - z
subtracting complex numbers
Complex numbers are points in the plane
can't get out of the complex numbers by adding (or subtracting) or multiplying two

11. Given (4-2i) the complex conjugate would be (4+2i)
z - z*
Complex Conjugate
Rules of Complex Arithmetic
radicals

12. All numbers
complex
Polar Coordinates - Multiplication by i
sin z
irrational

13. Where the curvature of the graph changes
point of inflection
How to solve (2i+3)/(9-i)
real
integers

14. The field of all rational and irrational numbers.
sin iy
Real Numbers
point of inflection
Subfield

15. 1
transcendental
The Complex Numbers
(a + c) + ( b + d)i
i²

16. 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
Integers
Complex Conjugate
Real and Imaginary Parts
i^4

17. When two complex numbers are divided.
How to find any Power
Complex Division
imaginary
(cos? +isin?)n

18. ½(e^(iz) + e^(-iz))
cos z
Integers
a real number: (a + bi)(a - bi) = a² + b²
z - z*

19. (a + bi) = (c + bi) =
How to solve (2i+3)/(9-i)
radicals
'i'
(a + c) + ( b + d)i

20. 1st. Rule of Complex Arithmetic
i^2 = -1
multiplying complex numbers
Complex Numbers: Add & subtract
i^3

21. 3
Complex numbers are points in the plane
zz*
Imaginary Numbers
i^3

22. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
z1 / z2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers

23. A plot of complex numbers as points.
Any polynomial O(xn) - (n > 0)
Irrational Number
Argand diagram
Polar Coordinates - Division

24. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
the vector (a -b)
transcendental
integers
multiplying complex numbers

25. E ^ (z2 ln z1)
Absolute Value of a Complex Number
For real a and b - a + bi = 0 if and only if a = b = 0
z1 ^ (z2)
|z| = mod(z)

26. Real and imaginary numbers
complex numbers
Imaginary number
(cos? +isin?)n
cos iy

27. 2nd. Rule of Complex Arithmetic

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28. 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.'
the vector (a -b)
Complex Multiplication
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number

29. Imaginary number

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30. 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
Polar Coordinates - z
the distance from z to the origin in the complex plane
Rules of Complex Arithmetic
Field

31. (e^(iz) - e^(-iz)) / 2i
sin z
adding complex numbers
Real Numbers
multiplying complex numbers

32. I
-1
subtracting complex numbers
Field
v(-1)

33. 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
Subfield
Complex Numbers: Add & subtract
adding complex numbers
Roots of Unity

34. The reals are just the
i^2
x-axis in the complex plane
the complex numbers
non-integers

35. A+bi
Argand diagram
a real number: (a + bi)(a - bi) = a² + b²
cos iy
Complex Number Formula

36. R?¹(cos? - isin?)
Polar Coordinates - sin?
Complex Number Formula
Polar Coordinates - z?¹
How to solve (2i+3)/(9-i)

37. 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.
Liouville's Theorem -
e^(ln z)
Every complex number has the 'Standard Form': a + bi for some real a and b.

38. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
subtracting complex numbers
z - z*
the complex numbers

39. The complex number z representing a+bi.
Affix
ln z
Roots of Unity
i^3

40. No i
Roots of Unity
Complex Number
conjugate pairs
real

41. x / r
Complex Numbers: Multiply
Polar Coordinates - cos?
ln z
Euler's Formula

42. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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43. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
i^4
i^2 = -1
Complex Numbers: Multiply

44. Divide moduli and subtract arguments
Polar Coordinates - Division
The Complex Numbers
Polar Coordinates - z?¹
Every complex number has the 'Standard Form': a + bi for some real a and b.

45. When two complex numbers are multipiled together.
Complex Multiplication
non-integers
|z| = mod(z)
Argand diagram

46. 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
z1 / z2
non-integers
Polar Coordinates - z

47. A + bi
point of inflection
Polar Coordinates - Arg(z*)
standard form of complex numbers
i^3

48. Like pi
conjugate
De Moivre's Theorem
real
transcendental

49. For real a and b - a + bi =
Euler's Formula
z1 / z2
Irrational Number
0 if and only if a = b = 0

50. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
complex
i^4
Complex Multiplication

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CLEP General Mathematics: Complex Numbers

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