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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. 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
v(-1)
Liouville's Theorem -
Rules of Complex Arithmetic
ln z

2. Not on the numberline
irrational
z1 / z2
(a + c) + ( b + d)i
non-integers

3. (e^(iz) - e^(-iz)) / 2i
sin z
real
Rules of Complex Arithmetic
Complex Exponentiation

4. ½(e^(iz) + e^(-iz))
Complex Number
Polar Coordinates - z?¹
cos z
natural

5. Every complex number has the 'Standard Form':
Imaginary Unit
i^3
a + bi for some real a and b.
conjugate

6. 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
Liouville's Theorem -
How to multiply complex nubers(2+i)(2i-3)
Complex Conjugate
We say that c+di and c-di are complex conjugates.

7. We can also think of the point z= a+ ib as
0 if and only if a = b = 0
Complex Multiplication
the vector (a -b)
Subfield

8. x + iy = r(cos? + isin?) = re^(i?)
x-axis in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
conjugate
Polar Coordinates - z

9. ? = -tan?
Polar Coordinates - Arg(z*)
Imaginary number
(a + c) + ( b + d)i
Polar Coordinates - Multiplication

10. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Euler's Formula
conjugate
|z-w|
multiplying complex numbers

11. In this amazing number field every algebraic equation in z with complex coefficients
a real number: (a + bi)(a - bi) = a² + b²
the vector (a -b)
sin z
has a solution.

12. 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
Absolute Value of a Complex Number
Polar Coordinates - r
Complex Numbers: Add & subtract
transcendental

13. Where the curvature of the graph changes
point of inflection
i^2
a + bi for some real a and b.
How to find any Power

14. 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.'
Field
irrational
z + z*
Complex Number

15. Real and imaginary numbers
irrational
complex numbers
z1 / z2
the complex numbers

16. Numbers on a numberline
integers
cosh²y - sinh²y
irrational
'i'

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

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18. I
(cos? +isin?)n
i^1
point of inflection
the complex numbers

19. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Complex Addition
sin z
Any polynomial O(xn) - (n > 0)

20. Any number not rational
Polar Coordinates - sin?
sin iy
irrational
Irrational Number

21. Divide moduli and subtract arguments
i^0
cosh²y - sinh²y
The Complex Numbers
Polar Coordinates - Division

22. A + bi
standard form of complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to solve (2i+3)/(9-i)
Argand diagram

23. The complex number z representing a+bi.
transcendental
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Numbers: Multiply
Affix

24. 1st. Rule of Complex Arithmetic
Absolute Value of a Complex Number
Polar Coordinates - z
cosh²y - sinh²y
i^2 = -1

25. The field of all rational and irrational numbers.
cos iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ln z
Real Numbers

26. V(zz*) = v(a² + b²)
i²
Polar Coordinates - z
a + bi for some real a and b.
|z| = mod(z)

27. When two complex numbers are multipiled together.
Complex Multiplication
Polar Coordinates - z
-1
radicals

28. All numbers
i^2 = -1
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex

29. 2a
x-axis in the complex plane
Polar Coordinates - z?¹
Square Root
z + z*

30. Written as fractions - terminating + repeating decimals
a real number: (a + bi)(a - bi) = a² + b²
rational
i^3
imaginary

31. I
sin iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z?¹
v(-1)

32. When two complex numbers are added together.
Polar Coordinates - cos?
Complex Addition
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - r

33. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
i^0
Complex Conjugate
Complex numbers are points in the plane
Complex Number

34. 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
conjugate
the complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
Roots of Unity

35. Like pi
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Field
transcendental
Irrational Number

36. To simplify the square root of a negative number
i^3
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z - z*

37. A+bi
the complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Number Formula
Imaginary Unit

38. No i
Polar Coordinates - Multiplication by i
i²
Polar Coordinates - r
real

39. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Complex numbers are points in the plane
sin z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

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

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41. A plot of complex numbers as points.
i²
x-axis in the complex plane
Polar Coordinates - Arg(z*)
Argand diagram

42. Equivalent to an Imaginary Unit.
Polar Coordinates - cos?
Imaginary number
non-integers
Euler's Formula

43. Root negative - has letter i
imaginary
Square Root
Affix
The Complex Numbers

44. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Argand diagram
i^4
(a + c) + ( b + d)i

45. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Complex Numbers: Multiply
natural
Polar Coordinates - z?¹
|z-w|

46. 2ib
z - z*
complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^2

47. 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
The Complex Numbers
multiplying complex numbers
four different numbers: i - -i - 1 - and -1.

48. z1z2* / |z2|²
z1 / z2
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
De Moivre's Theorem
Square Root

49. y / r
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - sin?
zz*
rational

50. I^2 =
-1
Complex Division
Argand diagram
Polar Coordinates - Multiplication by i

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CLEP General Mathematics: Complex Numbers

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