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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. Every complex number has the 'Standard Form':
a + bi for some real a and b.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)
i^4

2. 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
imaginary
Polar Coordinates - z
integers
subtracting complex numbers

3. 5th. Rule of Complex Arithmetic
i^3
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane
sin iy

4. (a + bi) = (c + bi) =
Imaginary Unit
z1 ^ (z2)
Complex Subtraction
(a + c) + ( b + d)i

5. A² + b² - real and non negative
i^3
i^2 = -1
Complex Addition
zz*

6. 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
i^1
four different numbers: i - -i - 1 - and -1.
Imaginary Unit
Real and Imaginary Parts

7. 1
i^1
i²
z1 / z2
Affix

8. A+bi
z1 ^ (z2)
Complex Number Formula
a + bi for some real a and b.
Absolute Value of a Complex Number

9. (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
z1 / z2
Complex Exponentiation
Any polynomial O(xn) - (n > 0)
How to multiply complex nubers(2+i)(2i-3)

10. 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
standard form of complex numbers
transcendental
Complex Numbers: Add & subtract
e^(ln z)

11. When two complex numbers are divided.
|z-w|
Polar Coordinates - r
Complex Division
real

12. Starts at 1 - does not include 0
i^4
Complex Exponentiation
natural
transcendental

13. z1z2* / |z2|²
Complex numbers are points in the plane
Imaginary Unit
i^2 = -1
z1 / z2

14. A complex number may be taken to the power of another complex number.
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Numbers: Add & subtract
Complex Exponentiation
Polar Coordinates - cos?

15. ½(e^(iz) + e^(-iz))
Integers
cos z
sin z
radicals

16. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Number Formula
v(-1)
Absolute Value of a Complex Number
z1 ^ (z2)

17. I
Complex Number
i^1
v(-1)
standard form of complex numbers

18. (e^(iz) - e^(-iz)) / 2i
We say that c+di and c-di are complex conjugates.
sin z
0 if and only if a = b = 0
Imaginary number

19. 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.
i^1
i²
v(-1)
Complex numbers are points in the plane

20. I = imaginary unit - i² = -1 or i = v-1
i^2
Imaginary Numbers
four different numbers: i - -i - 1 - and -1.
We say that c+di and c-di are complex conjugates.

21. 3
i^3
z + z*
How to add and subtract complex numbers (2-3i)-(4+6i)
interchangeable

22. For real a and b - a + bi =
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the complex numbers
Complex Conjugate
0 if and only if a = b = 0

23. When two complex numbers are added together.
has a solution.
De Moivre's Theorem
Complex Addition
i²

24. The complex number z representing a+bi.
integers
Affix
Complex Number
i^1

25. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Polar Coordinates - Arg(z*)
Euler's Formula
i^2

26. Imaginary number

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27. The product of an imaginary number and its conjugate is
point of inflection
a real number: (a + bi)(a - bi) = a² + b²
|z| = mod(z)
Complex Exponentiation

28. Where the curvature of the graph changes
Polar Coordinates - cos?
Argand diagram
point of inflection
sin z

29. The field of all rational and irrational numbers.
cosh²y - sinh²y
Real Numbers
rational
z + z*

30. E ^ (z2 ln z1)
Integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
subtracting complex numbers
z1 ^ (z2)

31. R^2 = x
Square Root
Complex Exponentiation
sin iy
|z-w|

32. Derives z = a+bi
De Moivre's Theorem
Euler Formula
Argand diagram
We say that c+di and c-di are complex conjugates.

33. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Euler Formula
Polar Coordinates - sin?
Complex numbers are points in the plane
multiplying complex numbers

34. Like pi
non-integers
transcendental
|z-w|
i^0

35. 1
i^4
Imaginary Unit
Imaginary Numbers
i^0

36. Not on the numberline
non-integers
i^3
Polar Coordinates - Arg(z*)
cosh²y - sinh²y

37. 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.'
multiply the numerator and the denominator by the complex conjugate of the denominator.
radicals
Euler Formula
Complex Number

38. V(x² + y²) = |z|
irrational
Polar Coordinates - r
Complex Numbers: Add & subtract
Rational Number

39. ½(e^(-y) +e^(y)) = cosh y
cos iy
four different numbers: i - -i - 1 - and -1.
irrational
i²

40. (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
Real Numbers
How to solve (2i+3)/(9-i)
Complex Division
zz*

41. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Argand diagram
Polar Coordinates - Arg(z*)
How to add and subtract complex numbers (2-3i)-(4+6i)

42. 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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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
cos iy
Imaginary number
How to add and subtract complex numbers (2-3i)-(4+6i)
the complex numbers

44. The square root of -1.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Roots of Unity
point of inflection
Imaginary Unit

45. Any number not rational
Polar Coordinates - Division
irrational
Complex Addition
i^2 = -1

46. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Addition
(cos? +isin?)n
Polar Coordinates - Arg(z*)

47. A number that cannot be expressed as a fraction for any integer.
Complex Numbers: Add & subtract
Irrational Number
rational
How to find any Power

48. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula
|z-w|
Polar Coordinates - r

49. R?¹(cos? - isin?)
standard form of complex numbers
x-axis in the complex plane
Polar Coordinates - z?¹
Complex numbers are points in the plane

50. A + bi
x-axis in the complex plane
How to multiply complex nubers(2+i)(2i-3)
standard form of complex numbers
(a + c) + ( b + d)i

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CLEP General Mathematics: Complex Numbers

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