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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. We see in this way that the distance between two points z and w in the complex plane is
complex numbers
|z-w|
How to solve (2i+3)/(9-i)
sin iy

2. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
rational
Polar Coordinates - sin?
complex

3. The modulus of the complex number z= a + ib now can be interpreted as
complex
Imaginary number
How to find any Power
the distance from z to the origin in the complex plane

4. Imaginary number

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5. 2ib
the vector (a -b)
interchangeable
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z - z*

6. (a + bi) = (c + bi) =
Euler's Formula
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + c) + ( b + d)i
adding complex numbers

7. The square root of -1.
Complex Exponentiation
Absolute Value of a Complex Number
Imaginary Unit
Square Root

8. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
(cos? +isin?)n
conjugate pairs
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

9. 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
the complex numbers
z - z*
Polar Coordinates - sin?
real

10. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Numbers: Multiply
conjugate
Polar Coordinates - r
(cos? +isin?)n

11. 2a
How to find any Power
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(cos? +isin?)n
z + z*

12. In this amazing number field every algebraic equation in z with complex coefficients
Field
Rules of Complex Arithmetic
has a solution.
Integers

13. 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.'
a real number: (a + bi)(a - bi) = a² + b²
Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z

14. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
|z-w|
multiplying complex numbers
sin z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

15. When two complex numbers are added together.
complex numbers
Complex Addition
Polar Coordinates - z
Liouville's Theorem -

16. Every complex number has the 'Standard Form':
Imaginary number
Rules of Complex Arithmetic
Field
a + bi for some real a and b.

17. 1st. Rule of Complex Arithmetic
-1
i^2 = -1
Polar Coordinates - r
a + bi for some real a and b.

18. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
cosh²y - sinh²y
How to add and subtract complex numbers (2-3i)-(4+6i)
point of inflection
Polar Coordinates - sin?

19. (a + bi)(c + bi) =
i^1
subtracting complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

20. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
z1 ^ (z2)
Complex Numbers: Add & subtract
zz*
Polar Coordinates - sin?

21. Rotates anticlockwise by p/2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication by i
(a + c) + ( b + d)i
|z-w|

22. Starts at 1 - does not include 0
Complex Number Formula
Complex Addition
Rational Number
natural

23. Not on the numberline
point of inflection
non-integers
cosh²y - sinh²y
has a solution.

24. Real and imaginary numbers
transcendental
Polar Coordinates - r
i^2 = -1
complex numbers

25. (e^(iz) - e^(-iz)) / 2i
Roots of Unity
Complex Addition
Complex Multiplication
sin z

26. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field
i^4
has a solution.

27. Derives z = a+bi
Euler Formula
i^2 = -1
(cos? +isin?)n
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

28. y / r
natural
Real Numbers
zz*
Polar Coordinates - sin?

29. Equivalent to an Imaginary Unit.
natural
i²
Imaginary number
Polar Coordinates - sin?

30. 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.
Roots of Unity
imaginary
Complex numbers are points in the plane
cos iy

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

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32. R^2 = x
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Unit
Square Root

33. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Addition
The Complex Numbers

34. Multiply moduli and add arguments
multiplying complex numbers
Polar Coordinates - Multiplication
i^3
cosh²y - sinh²y

35. I = imaginary unit - i² = -1 or i = v-1
|z| = mod(z)
How to find any Power
|z-w|
Imaginary Numbers

36. 2nd. Rule of Complex Arithmetic

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37. A number that can be expressed as a fraction p/q where q is not equal to 0.
0 if and only if a = b = 0
Real Numbers
Rational Number
has a solution.

38. 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
Euler's Formula
Rules of Complex Arithmetic
adding complex numbers
i^2 = -1

39. V(zz*) = v(a² + b²)
e^(ln z)
Complex Conjugate
|z| = mod(z)
cos z

40. Any number not rational
irrational
Imaginary Numbers
i²
Complex Subtraction

41. The reals are just the
Real Numbers
i^2
|z-w|
x-axis in the complex plane

42. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Polar Coordinates - Division
the vector (a -b)
Real Numbers

43. A complex number may be taken to the power of another complex number.
Argand diagram
zz*
Complex Exponentiation
conjugate pairs

44. 1
Absolute Value of a Complex Number
cosh²y - sinh²y
four different numbers: i - -i - 1 - and -1.
point of inflection

45. Numbers on a numberline
v(-1)
Affix
integers
Imaginary Unit

46. Have radical
-1
Every complex number has the 'Standard Form': a + bi for some real a and b.
radicals
Field

47. ? = -tan?
the distance from z to the origin in the complex plane
complex numbers
i^3
Polar Coordinates - Arg(z*)

48. A plot of complex numbers as points.
sin iy
Complex Number Formula
Complex Numbers: Multiply
Argand diagram

49. A complex number and its conjugate
conjugate pairs
Any polynomial O(xn) - (n > 0)
Complex Conjugate
interchangeable

50. (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
the distance from z to the origin in the complex plane
How to solve (2i+3)/(9-i)
Complex Numbers: Multiply
cos iy