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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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1. A subset within a field.
Complex Multiplication
i^2
Subfield
i²

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

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3. 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
irrational
De Moivre's Theorem
Complex Subtraction
We say that c+di and c-di are complex conjugates.

4. Where the curvature of the graph changes
conjugate pairs
point of inflection
Polar Coordinates - cos?
the complex numbers

5. ? = -tan?
Complex Number Formula
a + bi for some real a and b.
Polar Coordinates - Arg(z*)
Square Root

6. When two complex numbers are divided.
ln z
Complex Division
For real a and b - a + bi = 0 if and only if a = b = 0
non-integers

7. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
How to add and subtract complex numbers (2-3i)-(4+6i)
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Numbers

8. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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9. The reals are just the
x-axis in the complex plane
Polar Coordinates - r
zz*
i^3

10. 1
Complex Multiplication
The Complex Numbers
-1
i^0

11. Like pi
transcendental
real
Polar Coordinates - r
cos iy

12. Real and imaginary numbers
real
Complex Numbers: Add & subtract
non-integers
complex numbers

13. The product of an imaginary number and its conjugate is
Complex Subtraction
x-axis in the complex plane
Polar Coordinates - Division
a real number: (a + bi)(a - bi) = a² + b²

14. Equivalent to an Imaginary Unit.
Imaginary number
Complex Division
interchangeable
natural

15. 2a
We say that c+di and c-di are complex conjugates.
z + z*
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - sin?

16. ½(e^(iz) + e^(-iz))
cos z
Complex Addition
Polar Coordinates - Arg(z*)
Complex Numbers: Add & subtract

17. (e^(iz) - e^(-iz)) / 2i
subtracting complex numbers
sin z
irrational
(a + bi) = (c + bi) = (a + c) + ( b + d)i

18. 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
Real and Imaginary Parts
sin iy
0 if and only if a = b = 0
subtracting complex numbers

19. A number that cannot be expressed as a fraction for any integer.
z - z*
Irrational Number
transcendental
Subfield

20. A+bi
Complex Number Formula
cos z
Polar Coordinates - Arg(z*)
Integers

21. For real a and b - a + bi =
subtracting complex numbers
0 if and only if a = b = 0
transcendental
e^(ln z)

22. Written as fractions - terminating + repeating decimals
Complex Subtraction
rational
Polar Coordinates - z
has a solution.

23. 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
natural
adding complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
De Moivre's Theorem

24. A number that can be expressed as a fraction p/q where q is not equal to 0.
the distance from z to the origin in the complex plane
complex
Rational Number
0 if and only if a = b = 0

25. Have radical
Real Numbers
radicals
v(-1)
x-axis in the complex plane

26. The square root of -1.
Any polynomial O(xn) - (n > 0)
adding complex numbers
Imaginary Unit
How to solve (2i+3)/(9-i)

27. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
subtracting complex numbers
imaginary
(a + c) + ( b + d)i
Integers

28. All the powers of i can be written as
|z| = mod(z)
Every complex number has the 'Standard Form': a + bi for some real a and b.
four different numbers: i - -i - 1 - and -1.
Complex Division

29. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
|z| = mod(z)
rational
i^0

30. I^2 =
-1
(cos? +isin?)n
Square Root
sin z

31. 2ib
v(-1)
irrational
How to multiply complex nubers(2+i)(2i-3)
z - z*

32. 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.
a + bi for some real a and b.
Complex Division
Real Numbers
How to find any Power

33. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
ln z
interchangeable
multiplying complex numbers

34. Derives z = a+bi
the vector (a -b)
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler Formula
Subfield

35. V(x² + y²) = |z|
four different numbers: i - -i - 1 - and -1.
i^4
the vector (a -b)
Polar Coordinates - r

36. The complex number z representing a+bi.
Any polynomial O(xn) - (n > 0)
0 if and only if a = b = 0
Affix
cos z

37. To simplify the square root of a negative number
z - z*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
has a solution.
conjugate

38. Not on the numberline
z1 / z2
z1 ^ (z2)
Polar Coordinates - sin?
non-integers

39. 1st. Rule of Complex Arithmetic
Complex Exponentiation
Polar Coordinates - Multiplication by i
a real number: (a + bi)(a - bi) = a² + b²
i^2 = -1

40. 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
complex numbers
Real and Imaginary Parts
Liouville's Theorem -
the complex numbers

41. ½(e^(-y) +e^(y)) = cosh y
point of inflection
v(-1)
cos iy
Integers

42. 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
z - z*
Rules of Complex Arithmetic
Complex Subtraction
Roots of Unity

43. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - Arg(z*)
Affix
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.

44. 1
radicals
imaginary
i²
Imaginary Numbers

45. 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
Polar Coordinates - cos?
Real and Imaginary Parts
Integers
a real number: (a + bi)(a - bi) = a² + b²

46. Rotates anticlockwise by p/2
radicals
Roots of Unity
Polar Coordinates - Multiplication by i
has a solution.

47. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex Exponentiation
(a + c) + ( b + d)i
Complex Numbers: Multiply

48. R^2 = x
multiplying complex numbers
Square Root
cos iy
z1 / z2

49. E ^ (z2 ln z1)
Real Numbers
Polar Coordinates - z?¹
z1 ^ (z2)
the vector (a -b)

50. I = imaginary unit - i² = -1 or i = v-1
adding complex numbers
Rational Number
Argand diagram
Imaginary Numbers

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CLEP General Mathematics: Complex Numbers

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