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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 complex number and its conjugate
irrational
conjugate pairs
cos z
How to multiply complex nubers(2+i)(2i-3)

2. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix
standard form of complex numbers
Complex Multiplication

3. No i
real
Imaginary Numbers
complex
The Complex Numbers

4. 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.
interchangeable
Complex Numbers: Multiply
i^3
Complex Number Formula

5. 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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6. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
non-integers
Polar Coordinates - z?¹
Imaginary Unit

7. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Numbers: Add & subtract
Subfield
interchangeable
Real Numbers

8. 2nd. Rule of Complex Arithmetic

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9. Numbers on a numberline
Polar Coordinates - Multiplication
a real number: (a + bi)(a - bi) = a² + b²
integers
-1

10. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)
Complex Numbers: Add & subtract
i^2

11. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
transcendental
Complex Number
The Complex Numbers
|z-w|

12. Given (4-2i) the complex conjugate would be (4+2i)
Any polynomial O(xn) - (n > 0)
Complex Conjugate
x-axis in the complex plane
standard form of complex numbers

13. Where the curvature of the graph changes
Complex Numbers: Add & subtract
point of inflection
multiplying complex numbers
|z| = mod(z)

14. Has exactly n roots by the fundamental theorem of algebra
Complex Addition
subtracting complex numbers
Any polynomial O(xn) - (n > 0)
Real Numbers

15. V(x² + y²) = |z|
Polar Coordinates - r
Any polynomial O(xn) - (n > 0)
sin z
real

16. 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
Imaginary Unit
x-axis in the complex plane
Rules of Complex Arithmetic
Irrational Number

17. A number that cannot be expressed as a fraction for any integer.
Polar Coordinates - Arg(z*)
Irrational Number
Polar Coordinates - Multiplication by i
How to multiply complex nubers(2+i)(2i-3)

18. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Complex numbers are points in the plane
conjugate pairs
How to add and subtract complex numbers (2-3i)-(4+6i)

19. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
|z-w|
De Moivre's Theorem
Complex Addition
ln z

20. We see in this way that the distance between two points z and w in the complex plane is
Integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Numbers: Add & subtract
|z-w|

21. R^2 = x
Square Root
How to add and subtract complex numbers (2-3i)-(4+6i)
Liouville's Theorem -
sin iy

22. (e^(iz) - e^(-iz)) / 2i
Complex Exponentiation
(cos? +isin?)n
Subfield
sin z

23. (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
z + z*
Absolute Value of a Complex Number
four different numbers: i - -i - 1 - and -1.
How to solve (2i+3)/(9-i)

24. 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
Liouville's Theorem -
conjugate
the complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)

25. Real and imaginary numbers
Polar Coordinates - z
Complex Multiplication
complex numbers
cosh²y - sinh²y

26. x + iy = r(cos? + isin?) = re^(i?)
x-axis in the complex plane
Complex Exponentiation
Polar Coordinates - Division
Polar Coordinates - z

27. Root negative - has letter i
interchangeable
Roots of Unity
Polar Coordinates - Division
imaginary

28. Multiply moduli and add arguments
Polar Coordinates - Arg(z*)
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication
i²

29. y / r
Polar Coordinates - Multiplication
Polar Coordinates - sin?
(a + c) + ( b + d)i
natural

30. 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
Real and Imaginary Parts
standard form of complex numbers
the distance from z to the origin in the complex plane
Irrational Number

31. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(cos? +isin?)n
Roots of Unity
Field
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

32. ½(e^(-y) +e^(y)) = cosh y
cos iy
i^1
ln z
radicals

33. E ^ (z2 ln z1)
z1 ^ (z2)
(cos? +isin?)n
Integers
four different numbers: i - -i - 1 - and -1.

34. 1st. Rule of Complex Arithmetic
Square Root
radicals
i^2 = -1
multiply the numerator and the denominator by the complex conjugate of the denominator.

35. For real a and b - a + bi =
(a + c) + ( b + d)i
0 if and only if a = b = 0
z1 / z2
imaginary

36. When two complex numbers are subtracted from one another.
ln z
a real number: (a + bi)(a - bi) = a² + b²
Complex Subtraction
real

37. Like pi
Imaginary Numbers
transcendental
Affix
Complex Numbers: Add & subtract

38. Starts at 1 - does not include 0
-1
z + z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
natural

39. Not on the numberline
i^3
non-integers
Complex Exponentiation
How to solve (2i+3)/(9-i)

40. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Number Formula
Complex Conjugate
transcendental
multiplying complex numbers

41. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Square Root
|z-w|
Complex Conjugate

42. 1
Field
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos z
i^0

43. A² + b² - real and non negative
a real number: (a + bi)(a - bi) = a² + b²
complex
sin z
zz*

44. The field of all rational and irrational numbers.
i²
Real Numbers
real
sin z

45. I
adding complex numbers
Complex Multiplication
irrational
i^1

46. (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
Square Root
Affix
How to multiply complex nubers(2+i)(2i-3)
sin iy

47. I
Polar Coordinates - cos?
ln z
v(-1)
For real a and b - a + bi = 0 if and only if a = b = 0

48. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i^1
How to find any Power
Integers
Polar Coordinates - Division

49. ? = -tan?
Polar Coordinates - Arg(z*)
complex numbers
i^1
Rules of Complex Arithmetic

50. 2a
z + z*
Imaginary Unit
Roots of Unity
Polar Coordinates - r

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CLEP General Mathematics: Complex Numbers

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