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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

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Answer 50 questions in 15 minutes.
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1. 2a
Liouville's Theorem -
z + z*
i^4
(a + bi) = (c + bi) = (a + c) + ( b + d)i

2. Equivalent to an Imaginary Unit.
sin iy
real
i^3
Imaginary number

3. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Field
Any polynomial O(xn) - (n > 0)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

4. For real a and b - a + bi =
irrational
Imaginary Unit
0 if and only if a = b = 0
(cos? +isin?)n

5. Imaginary number

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6. 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
Complex Division
De Moivre's Theorem
Complex Numbers: Add & subtract
transcendental

7. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Euler's Formula
Complex Exponentiation
Rules of Complex Arithmetic

8. 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
Real and Imaginary Parts
Field
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
has a solution.

9. 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
Complex Subtraction
Polar Coordinates - sin?
sin iy
adding complex numbers

10. x / r
i^1
Imaginary Numbers
Polar Coordinates - cos?
Complex Number Formula

11. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - r
Any polynomial O(xn) - (n > 0)
multiply the numerator and the denominator by the complex conjugate of the denominator.
cosh²y - sinh²y

12. To simplify the square root of a negative number
Subfield
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
non-integers
0 if and only if a = b = 0

13. The complex number z representing a+bi.
Rules of Complex Arithmetic
0 if and only if a = b = 0
Affix
(a + c) + ( b + d)i

14. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
complex
the complex numbers
Polar Coordinates - Division
Complex Numbers: Add & subtract

15. Starts at 1 - does not include 0
natural
-1
i²
Roots of Unity

16. 5th. Rule of Complex Arithmetic
|z| = mod(z)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Division
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. The modulus of the complex number z= a + ib now can be interpreted as
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)
the distance from z to the origin in the complex plane
Complex Numbers: Add & subtract

18. 4th. Rule of Complex Arithmetic
the distance from z to the origin in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)

19. 1
cosh²y - sinh²y
Polar Coordinates - sin?
Affix
imaginary

20. 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.'
x-axis in the complex plane
Complex Number
the distance from z to the origin in the complex plane
Imaginary Numbers

21. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
z - z*
conjugate
Argand diagram
sin iy

22. Like pi
Imaginary Numbers
Polar Coordinates - r
transcendental
Complex Number Formula

23. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Rules of Complex Arithmetic

24. Divide moduli and subtract arguments
conjugate
Imaginary Numbers
Imaginary number
Polar Coordinates - Division

25. I
subtracting complex numbers
z + z*
i^1
Argand diagram

26. 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
Complex Number
a real number: (a + bi)(a - bi) = a² + b²
adding complex numbers
Rules of Complex Arithmetic

27. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
subtracting complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
rational

28. ½(e^(iz) + e^(-iz))
cos z
four different numbers: i - -i - 1 - and -1.
'i'
Polar Coordinates - Multiplication

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

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30. V(zz*) = v(a² + b²)
|z| = mod(z)
Imaginary number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^3

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
Imaginary number
subtracting complex numbers
Polar Coordinates - Multiplication
the vector (a -b)

32. 1
has a solution.
Complex Conjugate
Polar Coordinates - z?¹
i^4

33. Rotates anticlockwise by p/2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
e^(ln z)
Complex numbers are points in the plane
Polar Coordinates - Multiplication by i

34. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
cos z
real
Polar Coordinates - z
Integers

35. I^2 =
i^3
-1
Liouville's Theorem -
Complex Number Formula

36. R^2 = x
Square Root
Imaginary Numbers
adding complex numbers
a real number: (a + bi)(a - bi) = a² + b²

37. y / r
point of inflection
x-axis in the complex plane
Polar Coordinates - sin?
Complex numbers are points in the plane

38. Written as fractions - terminating + repeating decimals
Square Root
How to find any Power
rational
Polar Coordinates - r

39. (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
Integers
How to solve (2i+3)/(9-i)
conjugate
Square Root

40. 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.
Complex Subtraction
'i'
How to find any Power
Polar Coordinates - r

41. To simplify a complex fraction
x-axis in the complex plane
a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.
multiply the numerator and the denominator by the complex conjugate of the denominator.

42. R?¹(cos? - isin?)
x-axis in the complex plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
De Moivre's Theorem
Polar Coordinates - z?¹

43. The reals are just the
cos iy
x-axis in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)
subtracting complex numbers

44. (e^(iz) - e^(-iz)) / 2i
a + bi for some real a and b.
Imaginary number
sin z
i^4

45. (a + bi) = (c + bi) =
i^1
(a + c) + ( b + d)i
i^3
Rules of Complex Arithmetic

46. 2ib
Any polynomial O(xn) - (n > 0)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin iy
z - z*

47. When two complex numbers are multipiled together.
i^0
Complex Multiplication
transcendental
How to solve (2i+3)/(9-i)

48. When two complex numbers are subtracted from one another.
Polar Coordinates - sin?
Polar Coordinates - Arg(z*)
Complex Subtraction
Roots of Unity

49. A + bi
standard form of complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
real
(a + c) + ( b + d)i

50. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Complex Subtraction
Complex Number
How to add and subtract complex numbers (2-3i)-(4+6i)

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CLEP General Mathematics: Complex Numbers

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