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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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Match each statement with the correct term.
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1. Any number not rational
How to add and subtract complex numbers (2-3i)-(4+6i)
complex numbers
z - z*
irrational

2. 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.
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to solve (2i+3)/(9-i)
How to find any Power
z - z*

3. z1z2* / |z2|²
z1 / z2
adding complex numbers
Imaginary Unit
complex numbers

4. A complex number may be taken to the power of another complex number.
Complex Exponentiation
the complex numbers
cosh²y - sinh²y
Polar Coordinates - Multiplication by i

5. The field of all rational and irrational numbers.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)
irrational
Real Numbers

6. Real and imaginary numbers
complex numbers
interchangeable
z1 ^ (z2)
Irrational Number

7. 1st. Rule of Complex Arithmetic
i^2 = -1
The Complex Numbers
zz*
non-integers

8. 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.
Real Numbers
z - z*
(cos? +isin?)n

9. Divide moduli and subtract arguments
Polar Coordinates - z
Polar Coordinates - Division
Integers
non-integers

10. For real a and b - a + bi =
Polar Coordinates - Multiplication
natural
i^1
0 if and only if a = b = 0

11. A number that can be expressed as a fraction p/q where q is not equal to 0.
zz*
i^2 = -1
Polar Coordinates - Multiplication
Rational Number

12. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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13. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
How to find any Power
Absolute Value of a Complex Number
i^3
cos iy

14. We see in this way that the distance between two points z and w in the complex plane is
Imaginary Numbers
cosh²y - sinh²y
|z-w|
Polar Coordinates - Arg(z*)

15. 1
Polar Coordinates - Division
adding complex numbers
z1 ^ (z2)
i^2

16. ½(e^(iz) + e^(-iz))
integers
x-axis in the complex plane
cos z
|z| = mod(z)

17. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Subtraction
Complex Conjugate
Polar Coordinates - z?¹

18. Written as fractions - terminating + repeating decimals
|z| = mod(z)
Imaginary Unit
rational
How to multiply complex nubers(2+i)(2i-3)

19. 5th. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
cos z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
radicals

20. Has exactly n roots by the fundamental theorem of algebra
Complex Addition
Any polynomial O(xn) - (n > 0)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Integers

21. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - Multiplication
complex
interchangeable
z1 / z2

22. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
cosh²y - sinh²y
Square Root
Polar Coordinates - cos?

23. Every complex number has the 'Standard Form':
Euler's Formula
v(-1)
a + bi for some real a and b.
Complex Exponentiation

24. A complex number and its conjugate
conjugate pairs
cos iy
Polar Coordinates - Division
Polar Coordinates - Multiplication

25. 2ib
complex
conjugate
z - z*
cosh²y - sinh²y

26. The reals are just the
'i'
x-axis in the complex plane
Complex Conjugate
Real Numbers

27. V(zz*) = v(a² + b²)
Any polynomial O(xn) - (n > 0)
Complex Multiplication
|z| = mod(z)
Polar Coordinates - sin?

28. Equivalent to an Imaginary Unit.
integers
adding complex numbers
Imaginary number
sin z

29. Imaginary number

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30. No i
(a + c) + ( b + d)i
real
Complex Number
How to multiply complex nubers(2+i)(2i-3)

31. ? = -tan?
Polar Coordinates - Arg(z*)
Polar Coordinates - sin?
x-axis in the complex plane
Euler Formula

32. E^(ln r) e^(i?) e^(2pin)
integers
How to solve (2i+3)/(9-i)
e^(ln z)
Rational Number

33. The product of an imaginary number and its conjugate is
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Numbers

34. Where the curvature of the graph changes
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
Square Root
point of inflection

35. 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
conjugate
Polar Coordinates - Arg(z*)
Complex Number

36. 2nd. Rule of Complex Arithmetic

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37. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
i^1
z1 ^ (z2)
We say that c+di and c-di are complex conjugates.
Roots of Unity

38. (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
Any polynomial O(xn) - (n > 0)
adding complex numbers
Real and Imaginary Parts
How to multiply complex nubers(2+i)(2i-3)

39. I^2 =
Complex Numbers: Add & subtract
Complex Number Formula
-1
natural

40. R^2 = x
i^4
Real Numbers
has a solution.
Square Root

41. ½(e^(-y) +e^(y)) = cosh y
Imaginary Numbers
ln z
Subfield
cos iy

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
Rules of Complex Arithmetic
radicals
The Complex Numbers
z + z*

43. 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.
i^1
Complex Numbers: Multiply
rational
How to add and subtract complex numbers (2-3i)-(4+6i)

44. 2a
Argand diagram
a real number: (a + bi)(a - bi) = a² + b²
Rules of Complex Arithmetic
z + z*

45. V(x² + y²) = |z|
Polar Coordinates - r
0 if and only if a = b = 0
sin z
|z| = mod(z)

46. 1
|z-w|
the vector (a -b)
i^4
complex numbers

47. 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
adding complex numbers
Polar Coordinates - sin?
Complex Multiplication
Complex Subtraction

48. All numbers
complex
For real a and b - a + bi = 0 if and only if a = b = 0
z1 / z2
imaginary

49. Given (4-2i) the complex conjugate would be (4+2i)
rational
|z| = mod(z)
Complex Conjugate
the vector (a -b)

50. y / r
Imaginary Unit
Polar Coordinates - sin?
Roots of Unity
i^2

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CLEP General Mathematics: Complex Numbers

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