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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. A complex number may be taken to the power of another complex number.
Complex Exponentiation
sin z
real
Complex Conjugate

2. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
integers
z1 / z2
Real and Imaginary Parts

3. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
cos iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i

4. The product of an imaginary number and its conjugate is
Polar Coordinates - cos?
Real Numbers
a real number: (a + bi)(a - bi) = a² + b²
sin iy

5. Like pi
transcendental
rational
i^0
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

6. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - Multiplication by i
How to add and subtract complex numbers (2-3i)-(4+6i)
i^3
sin iy

7. Real and imaginary numbers
complex numbers
z - z*
complex
Real and Imaginary Parts

8. I
i^1
point of inflection
'i'
z - z*

9. 1
adding complex numbers
i²
multiply the numerator and the denominator by the complex conjugate of the denominator.
conjugate pairs

10. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
(a + c) + ( b + d)i
complex
i^4

11. 2ib
Complex Conjugate
z - z*
Polar Coordinates - z
zz*

12. xpressions such as ``the complex number z'' - and ``the point z'' are now
Argand diagram
interchangeable
Polar Coordinates - Arg(z*)
The Complex Numbers

13. 5th. Rule of Complex Arithmetic
Complex Number
a + bi for some real a and b.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable

14. 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.
Irrational Number
|z-w|
How to find any Power
cos z

15. Numbers on a numberline
We say that c+di and c-di are complex conjugates.
integers
irrational
Complex Number Formula

16. Where the curvature of the graph changes
point of inflection
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
0 if and only if a = b = 0

17. 3
i^3
Polar Coordinates - sin?
Complex Number
four different numbers: i - -i - 1 - and -1.

18. Imaginary number

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19. V(zz*) = v(a² + b²)
|z| = mod(z)
imaginary
conjugate pairs
0 if and only if a = b = 0

20. x / r
Complex numbers are points in the plane
Polar Coordinates - cos?
Square Root
Polar Coordinates - Division

21. ½(e^(iz) + e^(-iz))
rational
Every complex number has the 'Standard Form': a + bi for some real a and b.
cos z
e^(ln z)

22. (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)
-1
Complex numbers are points in the plane
x-axis in the complex plane

23. I^2 =
-1
adding complex numbers
Complex Exponentiation
the distance from z to the origin in the complex plane

24. Have radical
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
radicals
Complex Numbers: Multiply
Argand diagram

25. A subset within a field.
integers
point of inflection
Subfield
the distance from z to the origin in the complex plane

26. x + iy = r(cos? + isin?) = re^(i?)
Absolute Value of a Complex Number
cos iy
Polar Coordinates - z
Euler Formula

27. (a + bi)(c + bi) =
standard form of complex numbers
Polar Coordinates - r
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
-1

28. 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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29. We can also think of the point z= a+ ib as
Any polynomial O(xn) - (n > 0)
sin z
The Complex Numbers
the vector (a -b)

30. A number that cannot be expressed as a fraction for any integer.
i^4
Irrational Number
point of inflection
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

31. 1
natural
real
the distance from z to the origin in the complex plane
i^4

32. To simplify a complex fraction
Every complex number has the 'Standard Form': a + bi for some real a and b.
Integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
0 if and only if a = b = 0

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

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34. The field of all rational and irrational numbers.
Rational Number
Real Numbers
Imaginary Numbers
Complex Number Formula

35. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Polar Coordinates - r
conjugate
z1 / z2
Complex Number Formula

36. To simplify the square root of a negative number
z - z*
point of inflection
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
non-integers

37. 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 Multiplication
Complex Numbers: Multiply
Polar Coordinates - r
The Complex Numbers

38. (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
Polar Coordinates - Arg(z*)
How to solve (2i+3)/(9-i)
Absolute Value of a Complex Number
subtracting complex numbers

39. Cos n? + i sin n? (for all n integers)
i^3
cosh²y - sinh²y
(cos? +isin?)n
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

40. Not on the numberline
Complex Conjugate
Complex Multiplication
non-integers
Euler's Formula

41. Written as fractions - terminating + repeating decimals
rational
i^0
Rational Number
Complex Subtraction

42. 2nd. Rule of Complex Arithmetic

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43. When two complex numbers are added together.
Polar Coordinates - z?¹
Complex Addition
Irrational Number
Roots of Unity

44. When two complex numbers are divided.
Polar Coordinates - Multiplication by i
How to solve (2i+3)/(9-i)
Imaginary Numbers
Complex Division

45. 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
four different numbers: i - -i - 1 - and -1.
the complex numbers
Liouville's Theorem -
For real a and b - a + bi = 0 if and only if a = b = 0

46. The complex number z representing a+bi.
Polar Coordinates - z?¹
complex
Affix
i^3

47. A + bi
(cos? +isin?)n
|z-w|
v(-1)
standard form of complex numbers

48. V(x² + y²) = |z|
rational
|z-w|
Polar Coordinates - r
Subfield

49. 1
i^1
Polar Coordinates - sin?
cosh²y - sinh²y
'i'

50. A complex number and its conjugate
Complex numbers are points in the plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate pairs
For real a and b - a + bi = 0 if and only if a = b = 0

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CLEP General Mathematics: Complex Numbers

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