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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. 2a
z + z*
complex
Polar Coordinates - Arg(z*)
Every complex number has the 'Standard Form': a + bi for some real a and b.

2. Derives z = a+bi
Euler Formula
interchangeable
'i'
i^4

3. A subset within a field.
integers
z1 ^ (z2)
Subfield
Any polynomial O(xn) - (n > 0)

4. 1st. Rule of Complex Arithmetic
i^2 = -1
sin z
conjugate
natural

5. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
z1 / z2
We say that c+di and c-di are complex conjugates.
Complex Subtraction
How to add and subtract complex numbers (2-3i)-(4+6i)

6. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Addition
Integers
Complex Numbers: Add & subtract
v(-1)

7. (e^(iz) - e^(-iz)) / 2i
sin z
conjugate
Subfield
Rules of Complex Arithmetic

8. A complex number and its conjugate
Imaginary Numbers
'i'
i^2
conjugate pairs

9. 1
integers
Subfield
i^4
sin z

10. Imaginary number

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11. Any number not rational
irrational
i^4
Polar Coordinates - sin?
standard form of complex numbers

12. xpressions such as ``the complex number z'' - and ``the point z'' are now
0 if and only if a = b = 0
interchangeable
non-integers
Polar Coordinates - Multiplication

13. Cos n? + i sin n? (for all n integers)
Real Numbers
adding complex numbers
(cos? +isin?)n
Polar Coordinates - z

14. All numbers
complex
adding complex numbers
transcendental
cosh²y - sinh²y

15. Every complex number has the 'Standard Form':
cosh²y - sinh²y
zz*
a + bi for some real a and b.
i^1

16. A number that can be expressed as a fraction p/q where q is not equal to 0.
e^(ln z)
Rational Number
Complex Division
Polar Coordinates - Arg(z*)

17. The reals are just the
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Subtraction
x-axis in the complex plane
Real Numbers

18. E ^ (z2 ln z1)
z1 ^ (z2)
complex numbers
Integers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

19. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Field
Rules of Complex Arithmetic
x-axis in the complex plane

20. 1
cosh²y - sinh²y
z - z*
Subfield
Polar Coordinates - Multiplication

21. 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.
the vector (a -b)
real
How to find any Power
Rules of Complex Arithmetic

22. 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.'
the complex numbers
interchangeable
adding complex numbers
Complex Number

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

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24. ? = -tan?
Polar Coordinates - Arg(z*)
sin iy
|z-w|
Polar Coordinates - r

25. To simplify a complex fraction
cos z
complex numbers
Integers
multiply the numerator and the denominator by the complex conjugate of the denominator.

26. (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)
transcendental
Polar Coordinates - r
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. 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)
zz*
Polar Coordinates - z?¹
standard form of complex numbers

28. Root negative - has letter i
Complex Numbers: Add & subtract
v(-1)
How to multiply complex nubers(2+i)(2i-3)
imaginary

29. Not on the numberline
e^(ln z)
x-axis in the complex plane
non-integers
Polar Coordinates - Arg(z*)

30. R?¹(cos? - isin?)
subtracting complex numbers
Polar Coordinates - z?¹
standard form of complex numbers
the distance from z to the origin in the complex plane

31. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
the vector (a -b)
integers
De Moivre's Theorem

32. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Rational Number
Field
Polar Coordinates - Multiplication
Polar Coordinates - Division

33. I^2 =
Imaginary Numbers
Real Numbers
interchangeable
-1

34. (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
How to multiply complex nubers(2+i)(2i-3)
Complex Subtraction
How to solve (2i+3)/(9-i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

35. The square root of -1.
Polar Coordinates - r
Imaginary Unit
Polar Coordinates - z
Integers

36. 2ib
z - z*
integers
cosh²y - sinh²y
the complex numbers

37. 3rd. Rule of Complex Arithmetic
i²
Polar Coordinates - Multiplication
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

38. When two complex numbers are subtracted from one another.
How to find any Power
Complex Subtraction
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Liouville's Theorem -

39. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Imaginary Numbers
real
Roots of Unity
Polar Coordinates - z

40. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Complex Addition
i^4
Complex Number Formula

41. ½(e^(-y) +e^(y)) = cosh y
For real a and b - a + bi = 0 if and only if a = b = 0
complex
cos iy
Absolute Value of a Complex Number

42. Equivalent to an Imaginary Unit.
Complex Addition
sin z
Imaginary number
Polar Coordinates - Multiplication by i

43. 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
has a solution.
Every complex number has the 'Standard Form': a + bi for some real a and b.
(cos? +isin?)n
adding complex numbers

44. 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
Complex Exponentiation
Polar Coordinates - z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Numbers: Add & subtract

45. A² + b² - real and non negative
the complex numbers
Polar Coordinates - Division
i^3
zz*

46. A + bi
z - z*
Euler's Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
standard form of complex numbers

47. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
z1 ^ (z2)

48. 5th. Rule of Complex Arithmetic
real
Liouville's Theorem -
Complex Conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

49. (a + bi)(c + bi) =
subtracting complex numbers
ln z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)

50. Have radical
i^1
Polar Coordinates - Division
radicals
Field

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CLEP General Mathematics: Complex Numbers

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