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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. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
complex
cos z
conjugate pairs
Integers

2. To simplify a complex fraction
Polar Coordinates - r
multiply the numerator and the denominator by the complex conjugate of the denominator.
Argand diagram
Euler's Formula

3. We see in this way that the distance between two points z and w in the complex plane is
the distance from z to the origin in the complex plane
|z-w|
Polar Coordinates - sin?
How to add and subtract complex numbers (2-3i)-(4+6i)

4. xpressions such as ``the complex number z'' - and ``the point z'' are now
standard form of complex numbers
complex numbers
has a solution.
interchangeable

5. Every complex number has the 'Standard Form':
|z| = mod(z)
a + bi for some real a and b.
sin z
standard form of complex numbers

6. I = imaginary unit - i² = -1 or i = v-1
sin iy
Imaginary Numbers
Rules of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

7. Have radical
ln z
Complex numbers are points in the plane
subtracting complex numbers
radicals

8. E ^ (z2 ln z1)
z1 ^ (z2)
Affix
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Addition

9. 1
Any polynomial O(xn) - (n > 0)
i^2
Square Root
z - z*

10. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z + z*
transcendental
cos iy

11. I
How to solve (2i+3)/(9-i)
real
i^1
z1 / z2

12. 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.
Square Root
the vector (a -b)
a + bi for some real a and b.
Complex Numbers: Multiply

13. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
zz*
Polar Coordinates - Multiplication by i
Imaginary number
conjugate

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

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15. To simplify the square root of a negative number
z1 / z2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
has a solution.
i^0

16. ½(e^(iz) + e^(-iz))
cos z
rational
Euler Formula
complex

17. 3
i^3
Complex Division
Complex Number Formula
cosh²y - sinh²y

18. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
zz*
Real and Imaginary Parts
Complex Conjugate
Polar Coordinates - Multiplication by i

19. Numbers on a numberline
Imaginary Unit
Complex Numbers: Add & subtract
Rules of Complex Arithmetic
integers

20. The field of all rational and irrational numbers.
De Moivre's Theorem
rational
Real Numbers
Polar Coordinates - z

21. The complex number z representing a+bi.
interchangeable
Affix
i^1
complex

22. When two complex numbers are divided.
cos z
Complex Division
z1 ^ (z2)
Polar Coordinates - z?¹

23. 2ib
Complex Subtraction
a + bi for some real a and b.
z - z*
i²

24. 3rd. Rule of Complex Arithmetic
|z| = mod(z)
-1
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary number

25. Like pi
the complex numbers
Any polynomial O(xn) - (n > 0)
Complex Numbers: Multiply
transcendental

26. For real a and b - a + bi =
0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to solve (2i+3)/(9-i)
has a solution.

27. V(x² + y²) = |z|
-1
Polar Coordinates - r
Polar Coordinates - z
Polar Coordinates - Arg(z*)

28. Equivalent to an Imaginary Unit.
Argand diagram
Polar Coordinates - sin?
sin z
Imaginary number

29. We can also think of the point z= a+ ib as
z - z*
zz*
the vector (a -b)
i²

30. A subset within a field.
a real number: (a + bi)(a - bi) = a² + b²
Any polynomial O(xn) - (n > 0)
Subfield
-1

31. x + iy = r(cos? + isin?) = re^(i?)
Complex numbers are points in the plane
i^0
Imaginary Unit
Polar Coordinates - z

32. V(zz*) = v(a² + b²)
|z| = mod(z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Argand diagram
Complex Conjugate

33. Cos n? + i sin n? (for all n integers)
Complex Conjugate
(cos? +isin?)n
i^2
Complex Number

34. 1
Euler's Formula
Polar Coordinates - Division
We say that c+di and c-di are complex conjugates.
cosh²y - sinh²y

35. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
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
Polar Coordinates - z

36. When two complex numbers are subtracted from one another.
point of inflection
the vector (a -b)
How to find any Power
Complex Subtraction

37. ? = -tan?
How to solve (2i+3)/(9-i)
Polar Coordinates - Arg(z*)
the distance from z to the origin in the complex plane
Complex Division

38. A² + b² - real and non negative
conjugate
zz*
Polar Coordinates - Multiplication by i
a real number: (a + bi)(a - bi) = a² + b²

39. 2nd. Rule of Complex Arithmetic

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40. y / r
multiply the numerator and the denominator by the complex conjugate of the denominator.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
Polar Coordinates - sin?

41. A+bi
interchangeable
Polar Coordinates - Multiplication by i
standard form of complex numbers
Complex Number Formula

42. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
sin iy
Imaginary Unit
real

43. (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
natural
0 if and only if a = b = 0
How to solve (2i+3)/(9-i)
the complex numbers

44. Real and imaginary numbers
complex numbers
a + bi for some real a and b.
How to multiply complex nubers(2+i)(2i-3)
imaginary

45. Any number not rational
Polar Coordinates - Multiplication
Polar Coordinates - Arg(z*)
Real Numbers
irrational

46. 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 Addition
Square Root
Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.

47. (e^(-y) - e^(y)) / 2i = i sinh y
Complex Addition
sin iy
Complex Number
natural

48. 2a
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + c) + ( b + d)i

49. (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
i^2
i^1
Subfield
How to multiply complex nubers(2+i)(2i-3)

50. When two complex numbers are multipiled together.
Complex Multiplication
z - z*
Irrational Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i

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CLEP General Mathematics: Complex Numbers

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