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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. 1st. Rule of Complex Arithmetic
Complex Subtraction
i^2 = -1
i²
Polar Coordinates - Arg(z*)

2. x / r
Polar Coordinates - cos?
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.

3. 1
i^0
cosh²y - sinh²y
adding complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

4. 1
irrational
i^2
i^2 = -1
i^4

5. 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)
cosh²y - sinh²y
Complex Multiplication

6. 1
i²
How to find any Power
z1 ^ (z2)
sin iy

7. To simplify a complex fraction
integers
Argand diagram
sin iy
multiply the numerator and the denominator by the complex conjugate of the denominator.

8. Numbers on a numberline
z + z*
|z| = mod(z)
Polar Coordinates - sin?
integers

9. 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
adding complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to find any Power
i^1

10. (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
cos iy
Complex Number
conjugate
How to multiply complex nubers(2+i)(2i-3)

11. 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.
cos z
real
adding complex numbers
How to find any Power

12. In this amazing number field every algebraic equation in z with complex coefficients
Polar Coordinates - Multiplication by i
cos iy
The Complex Numbers
has a solution.

13. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Subfield
Real Numbers
Roots of Unity

14. Any number not rational
Imaginary number
irrational
cos z
How to solve (2i+3)/(9-i)

15. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Polar Coordinates - r
Real and Imaginary Parts
integers

16. E ^ (z2 ln z1)
Polar Coordinates - Division
z1 ^ (z2)
Polar Coordinates - z
Complex Numbers: Multiply

17. xpressions such as ``the complex number z'' - and ``the point z'' are now
|z| = mod(z)
interchangeable
Subfield
multiply the numerator and the denominator by the complex conjugate of the denominator.

18. Have radical
standard form of complex numbers
Complex Numbers: Add & subtract
radicals
subtracting complex numbers

19. I
|z-w|
cosh²y - sinh²y
conjugate pairs
i^1

20. 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.
For real a and b - a + bi = 0 if and only if a = b = 0
(cos? +isin?)n
Complex numbers are points in the plane
irrational

21. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Exponentiation
Integers

22. A² + b² - real and non negative
zz*
i^0
Polar Coordinates - Multiplication
Complex numbers are points in the plane

23. When two complex numbers are divided.
four different numbers: i - -i - 1 - and -1.
cosh²y - sinh²y
Complex Division
cos iy

24. Written as fractions - terminating + repeating decimals
Rational Number
integers
rational
cos z

25. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
v(-1)
The Complex Numbers
Complex Division
cos z

26. Has exactly n roots by the fundamental theorem of algebra
Complex Number
point of inflection
Complex Conjugate
Any polynomial O(xn) - (n > 0)

27. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
i^3
Complex numbers are points in the plane
i^4
Field

28. (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
irrational
Euler Formula
x-axis in the complex plane
How to solve (2i+3)/(9-i)

29. 2nd. Rule of Complex Arithmetic

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30. R^2 = x
Square Root
Affix
i^3
cos z

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

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32. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
transcendental
Integers
Field
Roots of Unity

33. For real a and b - a + bi =
Absolute Value of a Complex Number
0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
How to solve (2i+3)/(9-i)

34. Where the curvature of the graph changes
integers
Complex Exponentiation
sin iy
point of inflection

35. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Complex Number
cos iy
conjugate pairs

36. The reals are just the
Complex numbers are points in the plane
natural
Affix
x-axis in the complex plane

37. A+bi
transcendental
Complex Number Formula
Imaginary Unit
Complex Numbers: Multiply

38. 2a
Square Root
Argand diagram
Imaginary Numbers
z + z*

39. 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 Number
Polar Coordinates - Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
'i'

40. 1
transcendental
point of inflection
e^(ln z)
i^4

41. ? = -tan?
Irrational Number
Polar Coordinates - r
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Arg(z*)

42. I^2 =
transcendental
Roots of Unity
-1
Complex Numbers: Multiply

43. 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
subtracting complex numbers
the complex numbers
complex numbers
zz*

44. 5th. Rule of Complex Arithmetic
Complex Number Formula
Affix
How to find any Power
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

45. V(zz*) = v(a² + b²)
|z| = mod(z)
real
i^0
Imaginary Unit

46. Like pi
the complex numbers
the vector (a -b)
transcendental
i^0

47. To simplify the square root of a negative number
multiply the numerator and the denominator by the complex conjugate of the denominator.
rational
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Affix

48. Given (4-2i) the complex conjugate would be (4+2i)
the vector (a -b)
v(-1)
Complex Conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.

49. Root negative - has letter i
Polar Coordinates - Arg(z*)
Complex Exponentiation
imaginary
conjugate

50. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Real Numbers
x-axis in the complex plane
i^1

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CLEP General Mathematics: Complex Numbers

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