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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

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. 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.
Integers
i^3
De Moivre's Theorem

2. When two complex numbers are multipiled together.
real
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Multiplication
0 if and only if a = b = 0

3. When two complex numbers are subtracted from one another.
z1 / z2
adding complex numbers
Complex Subtraction
z - z*

4. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
conjugate
i^1
Field
Roots of Unity

5. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
has a solution.
Polar Coordinates - Division
The Complex Numbers
rational

6. ½(e^(-y) +e^(y)) = cosh y
cos iy
irrational
complex numbers
subtracting complex numbers

7. I^2 =
For real a and b - a + bi = 0 if and only if a = b = 0
conjugate pairs
-1
e^(ln z)

8. In this amazing number field every algebraic equation in z with complex coefficients
Imaginary Unit
conjugate pairs
the vector (a -b)
has a solution.

9. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Complex Subtraction
Complex Numbers: Add & subtract
cos iy

10. Starts at 1 - does not include 0
cos z
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i
ln z

11. (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
'i'
sin z
e^(ln z)
How to solve (2i+3)/(9-i)

12. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Number Formula
Real and Imaginary Parts
0 if and only if a = b = 0
Rational Number

13. 2a
How to find any Power
(cos? +isin?)n
Rational Number
z + z*

14. 2ib
z - z*
z + z*
complex numbers
the complex numbers

15. (e^(-y) - e^(y)) / 2i = i sinh y
has a solution.
sin iy
Complex Number
multiplying complex numbers

16. V(zz*) = v(a² + b²)
|z| = mod(z)
i^1
Complex numbers are points in the plane
How to solve (2i+3)/(9-i)

17. 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.
i^4
e^(ln z)
point of inflection
Complex numbers are points in the plane

18. Derives z = a+bi
Euler Formula
imaginary
four different numbers: i - -i - 1 - and -1.
Real Numbers

19. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
De Moivre's Theorem
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number
Liouville's Theorem -

20. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
irrational
Polar Coordinates - Division
i^3
conjugate

21. A² + b² - real and non negative
Polar Coordinates - Arg(z*)
the distance from z to the origin in the complex plane
Real Numbers
zz*

22. 5th. Rule of Complex Arithmetic
How to solve (2i+3)/(9-i)
complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2

23. y / r
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
a + bi for some real a and b.
Absolute Value of a Complex Number

24. z1z2* / |z2|²
Complex Division
(a + c) + ( b + d)i
Complex Multiplication
z1 / z2

25. Have radical
Rules of Complex Arithmetic
multiply the numerator and the denominator by the complex conjugate of the denominator.
0 if and only if a = b = 0
radicals

26. To simplify the square root of a negative number
sin z
point of inflection
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
irrational

27. 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.
i^1
Polar Coordinates - Multiplication by i
How to solve (2i+3)/(9-i)
How to find any Power

28. (a + bi) = (c + bi) =
Imaginary Numbers
Real Numbers
Argand diagram
(a + c) + ( b + d)i

29. Rotates anticlockwise by p/2
Polar Coordinates - cos?
|z-w|
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication by i

30. 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
z1 ^ (z2)
rational
Rules of Complex Arithmetic
Polar Coordinates - sin?

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

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32. ½(e^(iz) + e^(-iz))
sin iy
How to find any Power
cos z
Rules of Complex Arithmetic

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

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34. The field of all rational and irrational numbers.
Every complex number has the 'Standard Form': a + bi for some real a and b.
ln z
Real Numbers
radicals

35. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to find any Power
i^1
De Moivre's Theorem

36. The square root of -1.
(cos? +isin?)n
Affix
Complex Division
Imaginary Unit

37. Any number not rational
imaginary
Polar Coordinates - z?¹
(a + bi) = (c + bi) = (a + c) + ( b + d)i
irrational

38. 4th. Rule of Complex Arithmetic
Complex Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
radicals
(cos? +isin?)n

39. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Irrational Number
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Numbers
Polar Coordinates - Multiplication by i

40. When two complex numbers are added together.
Liouville's Theorem -
(a + c) + ( b + d)i
Complex Addition
Rules of Complex Arithmetic

41. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
sin iy
i^1
De Moivre's Theorem

42. (e^(iz) - e^(-iz)) / 2i
Complex Conjugate
i^4
z1 ^ (z2)
sin z

43. E^(ln r) e^(i?) e^(2pin)
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Number Formula
e^(ln z)
Complex Conjugate

44. 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
the distance from z to the origin in the complex plane
i^4
i^2
Real and Imaginary Parts

45. Imaginary number

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46. Cos n? + i sin n? (for all n integers)
Polar Coordinates - Arg(z*)
real
(cos? +isin?)n
Absolute Value of a Complex Number

47. No i
the complex numbers
real
De Moivre's Theorem
Argand diagram

48. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Absolute Value of a Complex Number
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Unit

49. 1
Liouville's Theorem -
i^4
|z-w|
Rational Number

50. For real a and b - a + bi =
sin z
four different numbers: i - -i - 1 - and -1.
Complex Conjugate
0 if and only if a = b = 0