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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. Cos n? + i sin n? (for all n integers)
Rules of Complex Arithmetic
Every complex number has the 'Standard Form': a + bi for some real a and b.
has a solution.
(cos? +isin?)n

2. (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
Any polynomial O(xn) - (n > 0)
complex
How to solve (2i+3)/(9-i)
z1 ^ (z2)

3. x / r
Polar Coordinates - cos?
point of inflection
Polar Coordinates - Arg(z*)
Complex numbers are points in the plane

4. A complex number and its conjugate
conjugate pairs
|z| = mod(z)
How to find any Power
Real Numbers

5. Derives z = a+bi
Euler Formula
How to find any Power
subtracting complex numbers
non-integers

6. To simplify a complex fraction
Complex Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
cos z
Polar Coordinates - r

7. (e^(iz) - e^(-iz)) / 2i
Liouville's Theorem -
Subfield
sin z
can't get out of the complex numbers by adding (or subtracting) or multiplying two

8. Have radical
multiply the numerator and the denominator by the complex conjugate of the denominator.
subtracting complex numbers
radicals
imaginary

9. (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
How to multiply complex nubers(2+i)(2i-3)
How to find any Power
cos iy
Complex Number

10. Multiply moduli and add arguments
adding complex numbers
Complex Number
the distance from z to the origin in the complex plane
Polar Coordinates - Multiplication

11. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Real Numbers
conjugate pairs
integers

12. Equivalent to an Imaginary Unit.
Imaginary number
(a + c) + ( b + d)i
interchangeable
transcendental

13. I^2 =
-1
Complex Conjugate
v(-1)
The Complex Numbers

14. ½(e^(iz) + e^(-iz))
Polar Coordinates - Arg(z*)
integers
cos z
z1 ^ (z2)

15. E^(ln r) e^(i?) e^(2pin)
integers
For real a and b - a + bi = 0 if and only if a = b = 0
z1 ^ (z2)
e^(ln z)

16. Given (4-2i) the complex conjugate would be (4+2i)
Complex numbers are points in the plane
Complex Conjugate
How to find any Power
Polar Coordinates - cos?

17. Divide moduli and subtract arguments
Polar Coordinates - Division
Complex Exponentiation
Polar Coordinates - z?¹
multiply the numerator and the denominator by the complex conjugate of the denominator.

18. A plot of complex numbers as points.
Argand diagram
Liouville's Theorem -
Subfield
Field

19. z1z2* / |z2|²
(a + c) + ( b + d)i
z1 / z2
multiplying complex numbers
the complex numbers

20. We see in this way that the distance between two points z and w in the complex plane is
the complex numbers
z + z*
Polar Coordinates - r
|z-w|

21. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Field
Roots of Unity
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 / z2

22. x + iy = r(cos? + isin?) = re^(i?)
How to solve (2i+3)/(9-i)
Polar Coordinates - z
|z-w|
Polar Coordinates - Multiplication by i

23. (a + bi)(c + bi) =
i^2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0

24. To simplify the square root of a negative number
natural
Imaginary Unit
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

25. 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
multiplying complex numbers
Field
Rules of Complex Arithmetic
adding complex numbers

26. 1
|z-w|
cosh²y - sinh²y
conjugate
a real number: (a + bi)(a - bi) = a² + b²

27. Has exactly n roots by the fundamental theorem of algebra
0 if and only if a = b = 0
complex
Irrational Number
Any polynomial O(xn) - (n > 0)

28. Rotates anticlockwise by p/2
Complex Conjugate
Polar Coordinates - Multiplication by i
complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i

29. 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.'
Complex Division
the distance from z to the origin in the complex plane
Complex Number
real

30. 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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31. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
real
x-axis in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate

32. 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
|z| = mod(z)
a real number: (a + bi)(a - bi) = a² + b²
Complex Numbers: Add & subtract
the distance from z to the origin in the complex plane

33. 2a
conjugate
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z + z*
Complex Subtraction

34. All numbers
Polar Coordinates - z?¹
complex
point of inflection
Complex Number Formula

35. V(zz*) = v(a² + b²)
v(-1)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
|z| = mod(z)

36. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - cos?
Field
has a solution.
interchangeable

37. The square root of -1.
Square Root
Imaginary Unit
Every complex number has the 'Standard Form': a + bi for some real a and b.
transcendental

38. The product of an imaginary number and its conjugate is
radicals
a real number: (a + bi)(a - bi) = a² + b²
z1 / z2
Complex Exponentiation

39. 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
(a + c) + ( b + d)i
sin iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers

40. R?¹(cos? - isin?)
a + bi for some real a and b.
Polar Coordinates - z?¹
Field
Subfield

41. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
adding complex numbers
Polar Coordinates - Arg(z*)
multiplying complex numbers
Subfield

42. (e^(-y) - e^(y)) / 2i = i sinh y
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - z
z + z*
sin iy

43. A² + b² - real and non negative
zz*
multiplying complex numbers
Polar Coordinates - Division
Argand diagram

44. The complex number z representing a+bi.
subtracting complex numbers
x-axis in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.
Affix

45. For real a and b - a + bi =
conjugate pairs
z - z*
Polar Coordinates - Multiplication by i
0 if and only if a = b = 0

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

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47. A + bi
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)
standard form of complex numbers
i^3

48. Imaginary number

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49. A subset within a field.
Polar Coordinates - sin?
cos iy
Subfield
v(-1)

50. Written as fractions - terminating + repeating decimals
Real and Imaginary Parts
rational
Square Root
x-axis in the complex plane

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CLEP General Mathematics: Complex Numbers

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