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

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2. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Multiplication by i
e^(ln z)
Polar Coordinates - Multiplication
The Complex Numbers

3. The complex number z representing a+bi.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix
conjugate pairs
Complex Subtraction

4. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Complex Subtraction
interchangeable
Polar Coordinates - sin?

5. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
i²
complex numbers
We say that c+di and c-di are complex conjugates.

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
complex numbers
Imaginary Numbers
subtracting complex numbers
v(-1)

7. The reals are just the
x-axis in the complex plane
e^(ln z)
|z| = mod(z)
integers

8. 2ib
Complex Subtraction
integers
Irrational Number
z - z*

9. z1z2* / |z2|²
Complex Conjugate
z1 / z2
Polar Coordinates - Arg(z*)
four different numbers: i - -i - 1 - and -1.

10. 2a
z + z*
sin iy
the distance from z to the origin in the complex plane
Polar Coordinates - Multiplication by i

11. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
radicals
Euler's Formula
(a + c) + ( b + d)i

12. 4th. Rule of Complex Arithmetic
transcendental
z1 ^ (z2)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Conjugate

13. R?¹(cos? - isin?)
v(-1)
How to solve (2i+3)/(9-i)
Polar Coordinates - z?¹
(a + bi) = (c + bi) = (a + c) + ( b + d)i

14. Where the curvature of the graph changes
irrational
multiply the numerator and the denominator by the complex conjugate of the denominator.
point of inflection
Imaginary Numbers

15. When two complex numbers are divided.
Polar Coordinates - Division
Subfield
Complex Division
v(-1)

16. 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.'
rational
transcendental
Complex Number
i^2

17. V(zz*) = v(a² + b²)
Complex Conjugate
Real and Imaginary Parts
|z| = mod(z)
four different numbers: i - -i - 1 - and -1.

18. 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
i^0
Real and Imaginary Parts
Euler Formula
Absolute Value of a Complex Number

19. 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.
conjugate
integers
Complex numbers are points in the plane
i^1

20. Starts at 1 - does not include 0
z1 ^ (z2)
sin z
natural
How to multiply complex nubers(2+i)(2i-3)

21. A plot of complex numbers as points.
standard form of complex numbers
Argand diagram
integers
z - z*

22. (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
standard form of complex numbers
Imaginary Numbers
Imaginary number
How to solve (2i+3)/(9-i)

23. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
i^1
How to add and subtract complex numbers (2-3i)-(4+6i)
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z

24. A + bi
Complex Numbers: Multiply
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs
standard form of complex numbers

25. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
zz*
complex numbers
Polar Coordinates - Arg(z*)

26. 1
i^0
has a solution.
Real and Imaginary Parts
The Complex Numbers

27. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Argand diagram
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field
|z-w|

28. I
Polar Coordinates - z?¹
Rational Number
i^1
a real number: (a + bi)(a - bi) = a² + b²

29. Has exactly n roots by the fundamental theorem of algebra
(cos? +isin?)n
natural
Any polynomial O(xn) - (n > 0)
i^1

30. ½(e^(-y) +e^(y)) = cosh y
Imaginary number
cos iy
We say that c+di and c-di are complex conjugates.
Subfield

31. x / r
Imaginary Numbers
Polar Coordinates - Arg(z*)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - cos?

32. Like pi
Argand diagram
z1 ^ (z2)
transcendental
the vector (a -b)

33. I^2 =
-1
ln z
complex
Polar Coordinates - Multiplication

34. A subset within a field.
Imaginary Numbers
Subfield
real
(a + bi) = (c + bi) = (a + c) + ( b + d)i

35. Derives z = a+bi
Imaginary number
Complex Conjugate
How to multiply complex nubers(2+i)(2i-3)
Euler Formula

36. (a + bi) = (c + bi) =
i^3
natural
'i'
(a + c) + ( b + d)i

37. V(x² + y²) = |z|
|z| = mod(z)
e^(ln z)
Polar Coordinates - r
four different numbers: i - -i - 1 - and -1.

38. We see in this way that the distance between two points z and w in the complex plane is
The Complex Numbers
Complex Addition
i^3
|z-w|

39. The modulus of the complex number z= a + ib now can be interpreted as
real
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the distance from z to the origin in the complex plane
Complex Conjugate

40. x + iy = r(cos? + isin?) = re^(i?)
the complex numbers
irrational
the distance from z to the origin in the complex plane
Polar Coordinates - z

41. For real a and b - a + bi =
0 if and only if a = b = 0
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Conjugate
i²

42. 3rd. Rule of Complex Arithmetic
a + bi for some real a and b.
Polar Coordinates - Multiplication
|z-w|
For real a and b - a + bi = 0 if and only if a = b = 0

43. Multiply moduli and add arguments
the distance from z to the origin in the complex plane
Polar Coordinates - Multiplication
Polar Coordinates - Division
Polar Coordinates - sin?

44. When two complex numbers are multipiled together.
multiplying complex numbers
cos z
Complex Number Formula
Complex Multiplication

45. Given (4-2i) the complex conjugate would be (4+2i)
conjugate
Complex Conjugate
De Moivre's Theorem
cos iy

46. ? = -tan?
Liouville's Theorem -
Polar Coordinates - Arg(z*)
Absolute Value of a Complex Number
We say that c+di and c-di are complex conjugates.

47. y / r
natural
0 if and only if a = b = 0
has a solution.
Polar Coordinates - sin?

48. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
conjugate pairs
Complex Numbers: Add & subtract

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

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50. Numbers on a numberline
sin z
a real number: (a + bi)(a - bi) = a² + b²
|z-w|
integers