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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. Numbers on a numberline
integers
conjugate pairs
i^2
e^(ln z)

2. 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.'
i^4
Complex Subtraction
imaginary
Complex Number

3. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
|z-w|
the vector (a -b)
Roots of Unity
a real number: (a + bi)(a - bi) = a² + b²

4. E ^ (z2 ln z1)
z1 ^ (z2)
0 if and only if a = b = 0
imaginary
Subfield

5. ½(e^(iz) + e^(-iz))
Rules of Complex Arithmetic
cos z
cos iy
the complex numbers

6. We see in this way that the distance between two points z and w in the complex plane is
Field
|z-w|
Square Root
z + z*

7. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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8. 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.
Polar Coordinates - Arg(z*)
Imaginary Numbers
Any polynomial O(xn) - (n > 0)
How to find any Power

9. 1
Any polynomial O(xn) - (n > 0)
integers
i^2
rational

10. V(x² + y²) = |z|
a + bi for some real a and b.
conjugate
Polar Coordinates - z?¹
Polar Coordinates - r

11. (e^(iz) - e^(-iz)) / 2i
z + z*
sin z
real
Complex Conjugate

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

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13. Every complex number has the 'Standard Form':
'i'
a + bi for some real a and b.
complex
The Complex Numbers

14. V(zz*) = v(a² + b²)
(cos? +isin?)n
|z| = mod(z)
the complex numbers
Euler's Formula

15. 2a
Euler's Formula
the distance from z to the origin in the complex plane
Imaginary Numbers
z + z*

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

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17. 1
0 if and only if a = b = 0
Subfield
i^4
How to multiply complex nubers(2+i)(2i-3)

18. I
the complex numbers
(cos? +isin?)n
v(-1)
Polar Coordinates - Division

19. x / r
i^2
imaginary
Polar Coordinates - cos?
z1 / z2

20. 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?
i^2 = -1
rational

21. The field of all rational and irrational numbers.
Real Numbers
point of inflection
Complex Conjugate
Polar Coordinates - sin?

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
How to solve (2i+3)/(9-i)
How to find any Power
Irrational Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

23. Divide moduli and subtract arguments
Polar Coordinates - Division
complex numbers
Rational Number
Complex Conjugate

24. I^2 =
Complex Subtraction
-1
Polar Coordinates - z?¹
How to add and subtract complex numbers (2-3i)-(4+6i)

25. I
cos z
Complex Subtraction
four different numbers: i - -i - 1 - and -1.
i^1

26. 2ib
(cos? +isin?)n
z - z*
Rules of Complex Arithmetic
We say that c+di and c-di are complex conjugates.

27. Derives z = a+bi
Euler Formula
How to find any Power
How to multiply complex nubers(2+i)(2i-3)
0 if and only if a = b = 0

28. Have radical
the vector (a -b)
radicals
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers

29. 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
Polar Coordinates - z?¹
cosh²y - sinh²y
How to multiply complex nubers(2+i)(2i-3)
the complex numbers

30. In this amazing number field every algebraic equation in z with complex coefficients
i^2 = -1
z1 ^ (z2)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
has a solution.

31. 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.
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Numbers: Multiply
'i'
point of inflection

32. When two complex numbers are subtracted from one another.
i^2
-1
Complex Subtraction
How to multiply complex nubers(2+i)(2i-3)

33. Has exactly n roots by the fundamental theorem of algebra
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Numbers: Multiply
Any polynomial O(xn) - (n > 0)
conjugate pairs

34. To simplify the square root of a negative number
Complex Numbers: Add & subtract
x-axis in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the complex numbers

35. Rotates anticlockwise by p/2
Square Root
sin z
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Multiplication by i

36. ½(e^(-y) +e^(y)) = cosh y
cos iy
i²
Polar Coordinates - sin?
z1 ^ (z2)

37. x + iy = r(cos? + isin?) = re^(i?)
a real number: (a + bi)(a - bi) = a² + b²
Euler's Formula
Polar Coordinates - z
irrational

38. The complex number z representing a+bi.
-1
Complex Conjugate
Affix
Square Root

39. No i
Polar Coordinates - Division
real
Polar Coordinates - cos?
Affix

40. 1st. Rule of Complex Arithmetic
i^2 = -1
How to add and subtract complex numbers (2-3i)-(4+6i)
z1 / z2
i^2

41. A number that can be expressed as a fraction p/q where q is not equal to 0.
Square Root
Rational Number
i^3
How to find any Power

42. 4th. Rule of Complex Arithmetic
Complex Subtraction
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Euler's Formula
Integers

43. A plot of complex numbers as points.
Square Root
Absolute Value of a Complex Number
Argand diagram
i^3

44. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Complex Numbers: Add & subtract
z1 / z2
i^4

45. 3rd. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
multiplying complex numbers
adding complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0

46. Equivalent to an Imaginary Unit.
rational
Complex Subtraction
Complex Numbers: Add & subtract
Imaginary number

47. 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
i^0
point of inflection
Polar Coordinates - z
We say that c+di and c-di are complex conjugates.

48. R^2 = x
Square Root
the distance from z to the origin in the complex plane
x-axis in the complex plane
Polar Coordinates - Multiplication by i

49. Multiply moduli and add arguments
'i'
Polar Coordinates - Multiplication
Complex Subtraction
four different numbers: i - -i - 1 - and -1.

50. Not on the numberline
transcendental
Euler's Formula
non-integers
integers