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CLEP General Mathematics: Complex Numbers

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1. 1
Rational Number
standard form of complex numbers
i^3
i²

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

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3. We can also think of the point z= a+ ib as
Square Root
non-integers
multiplying complex numbers
the vector (a -b)

4. ½(e^(-y) +e^(y)) = cosh y
Rational Number
Every complex number has the 'Standard Form': a + bi for some real a and b.
cos iy
non-integers

5. 1
z1 ^ (z2)
i^2
x-axis in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

6. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a real number: (a + bi)(a - bi) = a² + b²
Liouville's Theorem -
z1 / z2

7. E ^ (z2 ln z1)
Subfield
i^4
z1 ^ (z2)
natural

8. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^3
interchangeable

9. 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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10. I
Real Numbers
complex numbers
Complex Number
i^1

11. 4th. Rule of Complex Arithmetic
Rules of Complex Arithmetic
Complex Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers

12. We see in this way that the distance between two points z and w in the complex plane is
How to multiply complex nubers(2+i)(2i-3)
zz*
i^2 = -1
|z-w|

13. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
|z-w|
radicals
the distance from z to the origin in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)

14. R?¹(cos? - isin?)
interchangeable
Polar Coordinates - z?¹
Polar Coordinates - cos?
Roots of Unity

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

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16. The reals are just the
|z-w|
x-axis in the complex plane
i^1
i^3

17. (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
For real a and b - a + bi = 0 if and only if a = b = 0
cos iy
How to multiply complex nubers(2+i)(2i-3)
Complex Exponentiation

18. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
real
ln z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Rational Number

19. 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 Conjugate
Euler Formula
Complex Number

20. Have radical
radicals
conjugate
Euler's Formula
ln z

21. 3rd. Rule of Complex Arithmetic
Any polynomial O(xn) - (n > 0)
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0
natural

22. Root negative - has letter i
natural
imaginary
i^2 = -1
cosh²y - sinh²y

23. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Liouville's Theorem -
Imaginary number
De Moivre's Theorem

24. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
adding complex numbers
Complex numbers are points in the plane
-1

25. The complex number z representing a+bi.
(cos? +isin?)n
Affix
Complex Conjugate
Roots of Unity

26. To simplify a complex fraction
i^2
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - sin?
z + z*

27. x + iy = r(cos? + isin?) = re^(i?)
Rules of Complex Arithmetic
Polar Coordinates - z
adding complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

28. All numbers
e^(ln z)
complex
Integers
Polar Coordinates - Multiplication by i

29. Any number not rational
adding complex numbers
ln z
irrational
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

30. I = imaginary unit - i² = -1 or i = v-1
real
Complex Exponentiation
Imaginary Numbers
complex numbers

31. No i
Argand diagram
real
radicals
i^4

32. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
Field
z1 / z2
can't get out of the complex numbers by adding (or subtracting) or multiplying two

33. When two complex numbers are multipiled together.
Rules of Complex Arithmetic
Complex Multiplication
Polar Coordinates - sin?
complex numbers

34. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Number
cosh²y - sinh²y
Rational Number
Roots of Unity

35. I
Polar Coordinates - sin?
Imaginary number
v(-1)
z + z*

36. ½(e^(iz) + e^(-iz))
cos z
real
has a solution.
Affix

37. x / r
Complex Exponentiation
integers
Complex numbers are points in the plane
Polar Coordinates - cos?

38. Divide moduli and subtract arguments
How to find any Power
Polar Coordinates - Division
multiplying complex numbers
|z-w|

39. ? = -tan?
How to solve (2i+3)/(9-i)
i^2
Polar Coordinates - Arg(z*)
sin iy

40. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Complex Multiplication
subtracting complex numbers
i^2

41. (e^(iz) - e^(-iz)) / 2i
Affix
sin z
sin iy
subtracting complex numbers

42. z1z2* / |z2|²
(a + bi) = (c + bi) = (a + c) + ( b + d)i
natural
i^3
z1 / z2

43. (a + bi)(c + bi) =
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)

44. 1
Subfield
v(-1)
point of inflection
i^0

45. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
De Moivre's Theorem
i^2
Complex Conjugate

46. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Integers
Field
Rules of Complex Arithmetic
the complex numbers

47. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Numbers: Add & subtract
Complex Multiplication
multiplying complex numbers
transcendental

48. Given (4-2i) the complex conjugate would be (4+2i)
interchangeable
Complex Conjugate
cosh²y - sinh²y
Integers

49. 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 Exponentiation
Integers
Complex Numbers: Multiply
the distance from z to the origin in the complex plane

50. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Argand diagram
Polar Coordinates - sin?
Every complex number has the 'Standard Form': a + bi for some real a and b.

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CLEP General Mathematics: Complex Numbers

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