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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. 4th. Rule of Complex Arithmetic
transcendental
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate pairs
The Complex Numbers

2. A plot of complex numbers as points.
Euler Formula
Argand diagram
Affix
Imaginary Numbers

3. 1
i²
Complex Number Formula
interchangeable
|z| = mod(z)

4. Root negative - has letter i
imaginary
standard form of complex numbers
Square Root
the vector (a -b)

5. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Imaginary number
x-axis in the complex plane
Roots of Unity
Real and Imaginary Parts

6. Numbers on a numberline
i^4
integers
x-axis in the complex plane
z1 / z2

7. Imaginary number

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8. ? = -tan?
interchangeable
Polar Coordinates - Arg(z*)
i^3
Complex Multiplication

9. I^2 =
z - z*
Euler Formula
-1
Square Root

10. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
We say that c+di and c-di are complex conjugates.
Complex Exponentiation
Real and Imaginary Parts
Irrational Number

11. 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
sin iy
Complex Numbers: Add & subtract
standard form of complex numbers
z - z*

12. Has exactly n roots by the fundamental theorem of algebra
interchangeable
Integers
i^2
Any polynomial O(xn) - (n > 0)

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

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14. (e^(iz) - e^(-iz)) / 2i
radicals
i^0
sin z
Irrational Number

15. The field of all rational and irrational numbers.
We say that c+di and c-di are complex conjugates.
Euler's Formula
point of inflection
Real Numbers

16. When two complex numbers are subtracted from one another.
Complex Subtraction
complex numbers
the distance from z to the origin in the complex plane
cosh²y - sinh²y

17. 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)
-1
multiplying complex numbers

18. 3
i^2
e^(ln z)
i^3
cos z

19. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Complex Numbers: Multiply
a + bi for some real a and b.
Every complex number has the 'Standard Form': a + bi for some real a and b.

20. (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
Complex Exponentiation
transcendental
radicals
How to solve (2i+3)/(9-i)

21. The product of an imaginary number and its conjugate is
x-axis in the complex plane
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Any polynomial O(xn) - (n > 0)
a real number: (a + bi)(a - bi) = a² + b²

22. 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
Affix
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers
cos iy

23. A subset within a field.
complex
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Subfield
zz*

24. Derives z = a+bi
Polar Coordinates - Division
Euler Formula
Subfield
Every complex number has the 'Standard Form': a + bi for some real a and b.

25. z1z2* / |z2|²
Polar Coordinates - Arg(z*)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z + z*
z1 / z2

26. A complex number and its conjugate
For real a and b - a + bi = 0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - cos?
conjugate pairs

27. R?¹(cos? - isin?)
Polar Coordinates - z?¹
sin iy
interchangeable
v(-1)

28. 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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29. ½(e^(iz) + e^(-iz))
Polar Coordinates - Multiplication by i
cos z
z - z*
radicals

30. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
z1 ^ (z2)
sin z
conjugate
Polar Coordinates - sin?

31. For real a and b - a + bi =
Complex Conjugate
0 if and only if a = b = 0
Complex Multiplication
has a solution.

32. No i
Polar Coordinates - Multiplication
non-integers
real
Polar Coordinates - sin?

33. Written as fractions - terminating + repeating decimals
rational
Imaginary Numbers
Complex Subtraction
transcendental

34. Divide moduli and subtract arguments
Any polynomial O(xn) - (n > 0)
The Complex Numbers
Polar Coordinates - Division
How to add and subtract complex numbers (2-3i)-(4+6i)

35. Like pi
transcendental
standard form of complex numbers
a real number: (a + bi)(a - bi) = a² + b²
v(-1)

36. E ^ (z2 ln z1)
'i'
For real a and b - a + bi = 0 if and only if a = b = 0
radicals
z1 ^ (z2)

37. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Complex Numbers: Multiply
How to multiply complex nubers(2+i)(2i-3)
radicals

38. In this amazing number field every algebraic equation in z with complex coefficients
De Moivre's Theorem
Complex numbers are points in the plane
has a solution.
ln z

39. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Conjugate
the distance from z to the origin in the complex plane
|z| = mod(z)

40. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
irrational
transcendental
Polar Coordinates - r

41. Have radical
ln z
Liouville's Theorem -
radicals
Imaginary number

42. The square root of -1.
De Moivre's Theorem
multiplying complex numbers
interchangeable
Imaginary Unit

43. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
conjugate pairs
Imaginary number
i^2

44. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
z1 / z2
How to add and subtract complex numbers (2-3i)-(4+6i)
|z| = mod(z)

45. R^2 = x
Square Root
i²
Complex Multiplication
Real Numbers

46. I = imaginary unit - i² = -1 or i = v-1
real
For real a and b - a + bi = 0 if and only if a = b = 0
sin z
Imaginary Numbers

47. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
Polar Coordinates - Multiplication by i
v(-1)
i^1

48. 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.
Complex Numbers: Multiply
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex numbers are points in the plane
Irrational Number

49. 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
Complex Number
the distance from z to the origin in the complex plane
(a + c) + ( b + d)i

50. 2ib
point of inflection
Polar Coordinates - sin?
z - z*
standard form of complex numbers

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CLEP General Mathematics: Complex Numbers

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