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CLEP General Mathematics: Complex Numbers

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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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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. ½(e^(-y) +e^(y)) = cosh y
imaginary
Square Root
cos iy
0 if and only if a = b = 0

2. Not on the numberline
imaginary
has a solution.
multiply the numerator and the denominator by the complex conjugate of the denominator.
non-integers

3. 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.
How to find any Power
i²
Complex numbers are points in the plane
ln z

4. 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)
Absolute Value of a Complex Number
Complex Numbers: Multiply
i^1

5. 1
Complex Exponentiation
We say that c+di and c-di are complex conjugates.
transcendental
i^2

6. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
subtracting complex numbers
z + z*
Imaginary Numbers

7. 5th. Rule of Complex Arithmetic
Real Numbers
We say that c+di and c-di are complex conjugates.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
irrational

8. 3
sin iy
natural
i^3
cos iy

9. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Irrational Number
cos iy
Complex Number

10. ½(e^(iz) + e^(-iz))
multiplying complex numbers
z1 / z2
cos z
|z| = mod(z)

11. 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
transcendental
Imaginary number
Complex Number Formula
adding complex numbers

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

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13. A subset within a field.
x-axis in the complex plane
complex
Subfield
For real a and b - a + bi = 0 if and only if a = b = 0

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

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15. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Complex Division
Complex Addition
sin iy

16. All the powers of i can be written as
natural
irrational
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i

17. Like pi
|z| = mod(z)
Polar Coordinates - r
Polar Coordinates - Arg(z*)
transcendental

18. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
z - z*
complex
x-axis in the complex plane
Absolute Value of a Complex Number

19. xpressions such as ``the complex number z'' - and ``the point z'' are now
Real Numbers
Complex Division
interchangeable
multiplying complex numbers

20. 2ib
z - z*
conjugate pairs
Complex Number Formula
-1

21. A + bi
Polar Coordinates - Arg(z*)
standard form of complex numbers
cos z
Liouville's Theorem -

22. Divide moduli and subtract arguments
v(-1)
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Division
i^1

23. Written as fractions - terminating + repeating decimals
rational
'i'
Complex Number
z - z*

24. All numbers
z1 ^ (z2)
complex
Field
subtracting complex numbers

25. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
four different numbers: i - -i - 1 - and -1.
ln z
cosh²y - sinh²y
adding complex numbers

26. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
standard form of complex numbers
The Complex Numbers

27. 1
Complex Exponentiation
cosh²y - sinh²y
the complex numbers
four different numbers: i - -i - 1 - and -1.

28. R?¹(cos? - isin?)
z + z*
integers
Polar Coordinates - z?¹
Real and Imaginary Parts

29. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Imaginary number
Complex Subtraction
Complex Numbers: Multiply
Integers

30. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
multiplying complex numbers
Affix
i^1

31. Has exactly n roots by the fundamental theorem of algebra
cos iy
Polar Coordinates - cos?
Real Numbers
Any polynomial O(xn) - (n > 0)

32. (e^(iz) - e^(-iz)) / 2i
0 if and only if a = b = 0
sin z
adding complex numbers
Irrational Number

33. For real a and b - a + bi =
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - sin?
Liouville's Theorem -
0 if and only if a = b = 0

34. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)
zz*
Complex Addition

35. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Complex numbers are points in the plane
z + z*
i^2
sin z

36. 2nd. Rule of Complex Arithmetic

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37. When two complex numbers are subtracted from one another.
sin z
How to solve (2i+3)/(9-i)
Complex Subtraction
Polar Coordinates - z?¹

38. 1st. Rule of Complex Arithmetic
cos z
i^2 = -1
imaginary
Roots of Unity

39. 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
|z-w|
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Multiplication by i
conjugate

40. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Argand diagram
How to add and subtract complex numbers (2-3i)-(4+6i)
imaginary

41. 1
i^4
rational
|z-w|
Polar Coordinates - Multiplication by i

42. A plot of complex numbers as points.
conjugate pairs
Argand diagram
z - z*
Complex Exponentiation

43. I
zz*
Polar Coordinates - Multiplication by i
Complex Subtraction
i^1

44. We can also think of the point z= a+ ib as
Square Root
Polar Coordinates - cos?
irrational
the vector (a -b)

45. 2a
z + z*
For real a and b - a + bi = 0 if and only if a = b = 0
z1 / z2
Complex Division

46. Every complex number has the 'Standard Form':
transcendental
Roots of Unity
a + bi for some real a and b.
How to multiply complex nubers(2+i)(2i-3)

47. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Complex Subtraction
We say that c+di and c-di are complex conjugates.
Liouville's Theorem -

48. (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
integers
How to multiply complex nubers(2+i)(2i-3)
Real Numbers
the complex numbers

49. We see in this way that the distance between two points z and w in the complex plane is
How to multiply complex nubers(2+i)(2i-3)
irrational
|z-w|
can't get out of the complex numbers by adding (or subtracting) or multiplying two

50. Given (4-2i) the complex conjugate would be (4+2i)
Rational Number
The Complex Numbers
i²
Complex Conjugate

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CLEP General Mathematics: Complex Numbers

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