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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. Any number not rational
irrational
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^4
imaginary

2. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
Integers
Complex Multiplication
adding complex numbers
Complex Numbers: Add & subtract

3. x / r
Liouville's Theorem -
Polar Coordinates - cos?
adding complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.

4. Starts at 1 - does not include 0
Complex Number Formula
natural
complex
conjugate

5. Every complex number has the 'Standard Form':
a + bi for some real a and b.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication by i
Argand diagram

6. The product of an imaginary number and its conjugate is
Complex Exponentiation
a real number: (a + bi)(a - bi) = a² + b²
Rules of Complex Arithmetic
point of inflection

7. Written as fractions - terminating + repeating decimals
Complex Multiplication
natural
0 if and only if a = b = 0
rational

8. Root negative - has letter i
Any polynomial O(xn) - (n > 0)
i^2
zz*
imaginary

9. 3rd. Rule of Complex Arithmetic
(cos? +isin?)n
(a + bi) = (c + bi) = (a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Numbers: Add & subtract

10. When two complex numbers are multipiled together.
Polar Coordinates - cos?
Complex Subtraction
Complex Multiplication
rational

11. A complex number and its conjugate
has a solution.
conjugate pairs
sin z
i^2

12. 2nd. Rule of Complex Arithmetic

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13. A plot of complex numbers as points.
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 / z2
We say that c+di and c-di are complex conjugates.
Argand diagram

14. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z
Polar Coordinates - cos?
0 if and only if a = b = 0

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

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16. Equivalent to an Imaginary Unit.
Roots of Unity
Complex Number Formula
point of inflection
Imaginary number

17. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Rules of Complex Arithmetic
multiplying complex numbers
Roots of Unity
Imaginary Unit

18. A subset within a field.
We say that c+di and c-di are complex conjugates.
Subfield
Polar Coordinates - Division
De Moivre's Theorem

19. The field of all rational and irrational numbers.
Real Numbers
How to multiply complex nubers(2+i)(2i-3)
ln z
Polar Coordinates - cos?

20. E ^ (z2 ln z1)
z1 ^ (z2)
conjugate pairs
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Multiplication by i

21. 2a
x-axis in the complex plane
z + z*
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

22. The complex number z representing a+bi.
z - z*
Affix
Polar Coordinates - sin?
i^2 = -1

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

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24. ½(e^(iz) + e^(-iz))
De Moivre's Theorem
cos z
x-axis in the complex plane
Polar Coordinates - Arg(z*)

25. V(x² + y²) = |z|
How to multiply complex nubers(2+i)(2i-3)
sin iy
Polar Coordinates - r
natural

26. 1
Complex numbers are points in the plane
Imaginary Unit
i^2
i^4

27. 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
(a + c) + ( b + d)i
Polar Coordinates - sin?
irrational

28. ? = -tan?
Polar Coordinates - Arg(z*)
How to find any Power
x-axis in the complex plane
Complex Addition

29. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
How to multiply complex nubers(2+i)(2i-3)
Roots of Unity
i^1
multiplying complex numbers

30. We see in this way that the distance between two points z and w in the complex plane is
0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
real
|z-w|

31. (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*)
Square Root
Polar Coordinates - cos?
How to solve (2i+3)/(9-i)

32. All the powers of i can be written as
Euler Formula
the distance from z to the origin in the complex plane
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)

33. A number that cannot be expressed as a fraction for any integer.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Euler's Formula
Irrational Number
transcendental

34. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Complex Division
z + z*
has a solution.

35. A+bi
zz*
Complex Number Formula
Field
sin iy

36. 5th. Rule of Complex Arithmetic
Complex Conjugate
subtracting complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i

37. A number that can be expressed as a fraction p/q where q is not equal to 0.
complex numbers
Rational Number
Argand diagram
Liouville's Theorem -

38. y / r
Complex Addition
standard form of complex numbers
Polar Coordinates - sin?
Real and Imaginary Parts

39. xpressions such as ``the complex number z'' - and ``the point z'' are now
Roots of Unity
Square Root
How to multiply complex nubers(2+i)(2i-3)
interchangeable

40. z1z2* / |z2|²
i^2 = -1
z1 / z2
i^3
Imaginary Unit

41. Rotates anticlockwise by p/2
Euler's Formula
Polar Coordinates - Multiplication by i
cos z
Field

42. I^2 =
-1
Polar Coordinates - z?¹
standard form of complex numbers
Rational Number

43. V(zz*) = v(a² + b²)
complex numbers
the distance from z to the origin in the complex plane
non-integers
|z| = mod(z)

44. No i
conjugate
real
Polar Coordinates - cos?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

45. 1
Complex Subtraction
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z1 / z2
i^0

46. (a + bi) = (c + bi) =
Complex Number
(a + c) + ( b + d)i
Imaginary Numbers
sin z

47. Multiply moduli and add arguments
real
|z| = mod(z)
Polar Coordinates - Multiplication
z - z*

48. I
Complex Multiplication
Liouville's Theorem -
Complex Number
v(-1)

49. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 / z2
Polar Coordinates - z
Rational Number

50. Derives z = a+bi
Euler Formula
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Every complex number has the 'Standard Form': a + bi for some real a and b.
the distance from z to the origin in the complex plane

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CLEP General Mathematics: Complex Numbers

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