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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. Not on the numberline
The Complex Numbers
-1
non-integers
De Moivre's Theorem

2. Real and imaginary numbers
the vector (a -b)
Polar Coordinates - sin?
four different numbers: i - -i - 1 - and -1.
complex numbers

3. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
How to multiply complex nubers(2+i)(2i-3)
(a + c) + ( b + d)i
Integers
Polar Coordinates - r

4. A + bi
conjugate pairs
standard form of complex numbers
Any polynomial O(xn) - (n > 0)
Complex Addition

5. E ^ (z2 ln z1)
the vector (a -b)
z1 ^ (z2)
i²
How to add and subtract complex numbers (2-3i)-(4+6i)

6. Rotates anticlockwise by p/2
e^(ln z)
i^2 = -1
Polar Coordinates - Multiplication by i
-1

7. When two complex numbers are added together.
Complex Numbers: Multiply
0 if and only if a = b = 0
Complex Addition
(cos? +isin?)n

8. A plot of complex numbers as points.
(a + c) + ( b + d)i
Polar Coordinates - Division
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Argand diagram

9. 1
Polar Coordinates - z?¹
i^2
z1 / z2
i^4

10. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
e^(ln z)
a real number: (a + bi)(a - bi) = a² + b²
adding complex numbers

11. y / r
complex
How to solve (2i+3)/(9-i)
The Complex Numbers
Polar Coordinates - sin?

12. When two complex numbers are subtracted from one another.
Polar Coordinates - sin?
irrational
Complex Subtraction
Any polynomial O(xn) - (n > 0)

13. 1
x-axis in the complex plane
e^(ln z)
i^4
point of inflection

14. 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 Numbers: Add & subtract
i^4
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Euler Formula

15. x / r
real
Complex Addition
subtracting complex numbers
Polar Coordinates - cos?

16. For real a and b - a + bi =
Rules of Complex Arithmetic
How to find any Power
multiplying complex numbers
0 if and only if a = b = 0

17. x + iy = r(cos? + isin?) = re^(i?)
conjugate
Polar Coordinates - z
Absolute Value of a Complex Number
Complex Numbers: Multiply

18. 1
Complex Numbers: Add & subtract
rational
cosh²y - sinh²y
multiplying complex numbers

19. A number that can be expressed as a fraction p/q where q is not equal to 0.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^3
radicals
Rational Number

20. 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.
Euler's Formula
Complex Numbers: Multiply
sin z
x-axis in the complex plane

21. Like pi
standard form of complex numbers
interchangeable
z1 / z2
transcendental

22. Multiply moduli and add arguments
a + bi for some real a and b.
Polar Coordinates - Multiplication
point of inflection
i^2

23. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Real Numbers
Polar Coordinates - Arg(z*)
Field
adding complex numbers

24. Have radical
radicals
Polar Coordinates - Arg(z*)
Absolute Value of a Complex Number
Integers

25. 4th. Rule of Complex Arithmetic
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)
a real number: (a + bi)(a - bi) = a² + b²
(a + bi) = (c + bi) = (a + c) + ( b + d)i

26. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
De Moivre's Theorem
conjugate
irrational
z1 ^ (z2)

27. 5th. Rule of Complex Arithmetic
has a solution.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
natural
conjugate pairs

28. Written as fractions - terminating + repeating decimals
z1 ^ (z2)
Field
rational
Absolute Value of a Complex Number

29. In this amazing number field every algebraic equation in z with complex coefficients
Real Numbers
has a solution.
-1
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

30. Has exactly n roots by the fundamental theorem of algebra
multiplying complex numbers
Any polynomial O(xn) - (n > 0)
How to solve (2i+3)/(9-i)
the complex numbers

31. 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.'
Argand diagram
(a + c) + ( b + d)i
Complex Number
Real Numbers

32. Where the curvature of the graph changes
point of inflection
Complex Number
cosh²y - sinh²y
the complex numbers

33. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Absolute Value of a Complex Number
The Complex Numbers
Roots of Unity
(a + c) + ( b + d)i

34. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
z + z*
Polar Coordinates - Multiplication
De Moivre's Theorem

35. We can also think of the point z= a+ ib as
irrational
How to multiply complex nubers(2+i)(2i-3)
the vector (a -b)
Square Root

36. A+bi
multiplying complex numbers
i^2
Imaginary number
Complex Number Formula

37. 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
sin z
multiplying complex numbers
the complex numbers
Polar Coordinates - Multiplication

38. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
irrational
subtracting complex numbers
has a solution.
point of inflection

39. z1z2* / |z2|²
i^2 = -1
integers
Euler's Formula
z1 / z2

40. ½(e^(iz) + e^(-iz))
cos z
Imaginary Numbers
Polar Coordinates - Arg(z*)
Complex Conjugate

41. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Real and Imaginary Parts
ln z
a real number: (a + bi)(a - bi) = a² + b²
How to find any Power

42. Starts at 1 - does not include 0
Roots of Unity
'i'
natural
Euler Formula

43. 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.
z1 ^ (z2)
Complex numbers are points in the plane
Polar Coordinates - sin?
Any polynomial O(xn) - (n > 0)

44. 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
Imaginary number
complex
How to find any Power

45. (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
Liouville's Theorem -
How to solve (2i+3)/(9-i)
interchangeable
Complex Subtraction

46. When two complex numbers are multipiled together.
Euler Formula
Real and Imaginary Parts
Complex Multiplication
standard form of complex numbers

47. 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
conjugate
We say that c+di and c-di are complex conjugates.
|z| = mod(z)
Rules of Complex Arithmetic

48. ? = -tan?
Euler Formula
Polar Coordinates - Arg(z*)
cos iy
z - z*

49. All numbers
complex
Complex Subtraction
Euler Formula
Complex Addition

50. Root negative - has letter i
Subfield
Real and Imaginary Parts
imaginary
i^2 = -1

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CLEP General Mathematics: Complex Numbers

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