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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. 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)
standard form of complex numbers
Irrational Number
sin z

2. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
real
Complex Addition
De Moivre's Theorem
ln z

3. 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
imaginary
subtracting complex numbers
|z-w|

4. 1
transcendental
cos iy
real
i^0

5. V(zz*) = v(a² + b²)
Complex Number Formula
|z| = mod(z)
Complex numbers are points in the plane
z1 ^ (z2)

6. 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
Real Numbers
Rules of Complex Arithmetic
'i'
point of inflection

7. The modulus of the complex number z= a + ib now can be interpreted as
standard form of complex numbers
i^1
the distance from z to the origin in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i

8. ½(e^(iz) + e^(-iz))
Irrational Number
z1 ^ (z2)
cos z
-1

9. Derives z = a+bi
rational
Euler Formula
Field
Polar Coordinates - Division

10. (e^(-y) - e^(y)) / 2i = i sinh y
z + z*
Complex Exponentiation
Polar Coordinates - cos?
sin iy

11. x + iy = r(cos? + isin?) = re^(i?)
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
imaginary

12. A subset within a field.
conjugate pairs
How to solve (2i+3)/(9-i)
x-axis in the complex plane
Subfield

13. 4th. Rule of Complex Arithmetic
Polar Coordinates - z
Imaginary number
sin z
(a + bi) = (c + bi) = (a + c) + ( b + d)i

14. Cos n? + i sin n? (for all n integers)
complex
Polar Coordinates - Multiplication
point of inflection
(cos? +isin?)n

15. A² + b² - real and non negative
Irrational Number
Euler Formula
Polar Coordinates - cos?
zz*

16. 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 Number
standard form of complex numbers
How to find any Power
(cos? +isin?)n

17. All numbers
cosh²y - sinh²y
How to solve (2i+3)/(9-i)
complex
transcendental

18. E ^ (z2 ln z1)
z1 / z2
z - z*
z1 ^ (z2)
Roots of Unity

19. 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.
natural
|z-w|
For real a and b - a + bi = 0 if and only if a = b = 0
Complex numbers are points in the plane

20. A + bi
For real a and b - a + bi = 0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
standard form of complex numbers
Rules of Complex Arithmetic

21. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Irrational Number
the distance from z to the origin in the complex plane
conjugate pairs
multiplying complex numbers

22. In this amazing number field every algebraic equation in z with complex coefficients
Roots of Unity
ln z
The Complex Numbers
has a solution.

23. When two complex numbers are divided.
transcendental
Complex Division
Euler Formula
De Moivre's Theorem

24. Have radical
radicals
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Exponentiation
We say that c+di and c-di are complex conjugates.

25. Every complex number has the 'Standard Form':
ln z
a + bi for some real a and b.
rational
i^3

26. V(x² + y²) = |z|
Irrational Number
Polar Coordinates - r
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
We say that c+di and c-di are complex conjugates.

27. Root negative - has letter i
a real number: (a + bi)(a - bi) = a² + b²
imaginary
Complex Number
the distance from z to the origin in the complex plane

28. ? = -tan?
ln z
Polar Coordinates - Arg(z*)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Roots of Unity

29. The reals are just the
x-axis in the complex plane
a real number: (a + bi)(a - bi) = a² + b²
cos iy
radicals

30. The product of an imaginary number and its conjugate is
rational
Real and Imaginary Parts
sin iy
a real number: (a + bi)(a - bi) = a² + b²

31. Divide moduli and subtract arguments
complex
Polar Coordinates - Division
conjugate
radicals

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

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33. 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
z1 ^ (z2)
point of inflection
i^4
We say that c+di and c-di are complex conjugates.

34. A+bi
interchangeable
Complex Number Formula
v(-1)
Polar Coordinates - Division

35. Given (4-2i) the complex conjugate would be (4+2i)
real
z1 ^ (z2)
Complex Conjugate
|z| = mod(z)

36. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
|z| = mod(z)
Integers
Imaginary Unit

37. When two complex numbers are added together.
Polar Coordinates - sin?
Complex Addition
Polar Coordinates - Division
cos z

38. Any number not rational
z1 ^ (z2)
has a solution.
radicals
irrational

39. Like pi
natural
the complex numbers
transcendental
Complex Number

40. (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
ln z
non-integers
How to multiply complex nubers(2+i)(2i-3)
real

41. 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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42. ½(e^(-y) +e^(y)) = cosh y
cos iy
ln z
the complex numbers
Polar Coordinates - z

43. A complex number and its conjugate
Absolute Value of a Complex Number
x-axis in the complex plane
conjugate pairs
i^2 = -1

44. Rotates anticlockwise by p/2
Euler Formula
Complex numbers are points in the plane
the vector (a -b)
Polar Coordinates - Multiplication by i

45. When two complex numbers are multipiled together.
Complex Multiplication
transcendental
z1 / z2
the complex numbers

46. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
complex
non-integers
Any polynomial O(xn) - (n > 0)
How to add and subtract complex numbers (2-3i)-(4+6i)

47. All the powers of i can be written as
a + bi for some real a and b.
The Complex Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.

48. The field of all rational and irrational numbers.
(a + c) + ( b + d)i
Real Numbers
conjugate
For real a and b - a + bi = 0 if and only if a = b = 0

49. A number that cannot be expressed as a fraction for any integer.
Field
Integers
x-axis in the complex plane
Irrational Number

50. 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
integers
Rules of Complex Arithmetic
Polar Coordinates - r
the complex numbers

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CLEP General Mathematics: Complex Numbers

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