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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. (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
multiply the numerator and the denominator by the complex conjugate of the denominator.
a real number: (a + bi)(a - bi) = a² + b²
Complex Division
How to solve (2i+3)/(9-i)

2. Divide moduli and subtract arguments
Polar Coordinates - Division
point of inflection
rational
Absolute Value of a Complex Number

3. Numbers on a numberline
integers
z - z*
Complex Addition
Any polynomial O(xn) - (n > 0)

4. Any number not rational
a real number: (a + bi)(a - bi) = a² + b²
e^(ln z)
z1 / z2
irrational

5. I^2 =
i^3
Affix
-1
a real number: (a + bi)(a - bi) = a² + b²

6. Not on the numberline
non-integers
i^2
z1 ^ (z2)
rational

7. Written as fractions - terminating + repeating decimals
Liouville's Theorem -
Complex Numbers: Multiply
rational
Every complex number has the 'Standard Form': a + bi for some real a and b.

8. No i
real
can't get out of the complex numbers by adding (or subtracting) or multiplying two
e^(ln z)
The Complex Numbers

9. The modulus of the complex number z= a + ib now can be interpreted as
Imaginary Numbers
the distance from z to the origin in the complex plane
standard form of complex numbers
integers

10. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Complex Number Formula
Complex numbers are points in the plane
Field
Real Numbers

11. I
i^2
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Subfield
i^1

12. A plot of complex numbers as points.
Imaginary Numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^1
Argand diagram

13. Real and imaginary numbers
z1 ^ (z2)
integers
Integers
complex numbers

14. Have radical
(a + c) + ( b + d)i
sin z
Complex Number
radicals

15. A+bi
We say that c+di and c-di are complex conjugates.
Complex Number Formula
z1 ^ (z2)
cosh²y - sinh²y

16. 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
(cos? +isin?)n
Complex Numbers: Add & subtract
i^1
i^0

17. x / r
-1
'i'
z + z*
Polar Coordinates - cos?

18. Where the curvature of the graph changes
point of inflection
Complex Number Formula
integers
Imaginary Unit

19. 2nd. Rule of Complex Arithmetic

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20. 3rd. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
For real a and b - a + bi = 0 if and only if a = b = 0
Liouville's Theorem -
integers

21. The square root of -1.
Polar Coordinates - z?¹
Imaginary Unit
0 if and only if a = b = 0
Complex Numbers: Add & subtract

22. A complex number may be taken to the power of another complex number.
ln z
Affix
Complex Exponentiation
i^3

23. 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.
Polar Coordinates - Multiplication by i
0 if and only if a = b = 0
Polar Coordinates - sin?
How to find any Power

24. ½(e^(iz) + e^(-iz))
cos z
x-axis in the complex plane
How to solve (2i+3)/(9-i)
multiply the numerator and the denominator by the complex conjugate of the denominator.

25. Every complex number has the 'Standard Form':
Polar Coordinates - Multiplication by i
multiply the numerator and the denominator by the complex conjugate of the denominator.
a + bi for some real a and b.
i^1

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

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27. E^(ln r) e^(i?) e^(2pin)
point of inflection
Absolute Value of a Complex Number
e^(ln z)
complex numbers

28. 1
Every complex number has the 'Standard Form': a + bi for some real a and b.
v(-1)
Integers
i^0

29. 4th. Rule of Complex Arithmetic
(cos? +isin?)n
i^0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
adding complex numbers

30. (a + bi) = (c + bi) =
sin z
Polar Coordinates - Arg(z*)
(a + c) + ( b + d)i
Complex Multiplication

31. The field of all rational and irrational numbers.
Complex Exponentiation
How to multiply complex nubers(2+i)(2i-3)
Real Numbers
radicals

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

33. Like pi
Euler's Formula
i^3
The Complex Numbers
transcendental

34. ½(e^(-y) +e^(y)) = cosh y
i^1
a real number: (a + bi)(a - bi) = a² + b²
cos iy
sin z

35. 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
|z-w|
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Arg(z*)
subtracting complex numbers

36. (e^(iz) - e^(-iz)) / 2i
integers
Complex Subtraction
sin z
point of inflection

37. 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
Affix
Field
Complex Subtraction
Real and Imaginary Parts

38. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
a real number: (a + bi)(a - bi) = a² + b²
i^1
Complex Numbers: Add & subtract
Integers

39. When two complex numbers are divided.
i²
Imaginary number
Complex Division
|z-w|

40. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex
How to add and subtract complex numbers (2-3i)-(4+6i)
has a solution.

41. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
radicals
the distance from z to the origin in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
ln z

42. 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 Division
Rules of Complex Arithmetic
Complex Numbers: Multiply

43. A number that cannot be expressed as a fraction for any integer.
i^3
Argand diagram
Irrational Number
Polar Coordinates - Arg(z*)

44. 3
adding complex numbers
i^3
complex numbers
z1 / z2

45. Imaginary number

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46. In this amazing number field every algebraic equation in z with complex coefficients
v(-1)
Euler's Formula
The Complex Numbers
has a solution.

47. (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
Real Numbers
Euler's Formula
How to multiply complex nubers(2+i)(2i-3)
cosh²y - sinh²y

48. Equivalent to an Imaginary Unit.
Imaginary number
z + z*
i^2 = -1
x-axis in the complex plane

49. (a + bi)(c + bi) =
i^1
Subfield
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rational Number

50. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
integers
i^1
the distance from z to the origin in the complex plane

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CLEP General Mathematics: Complex Numbers

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