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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. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
z + z*
i^0
We say that c+di and c-di are complex conjugates.

2. 2nd. Rule of Complex Arithmetic

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3. 1
cosh²y - sinh²y
conjugate
e^(ln z)
Liouville's Theorem -

4. ½(e^(-y) +e^(y)) = cosh y
Polar Coordinates - r
cos iy
Absolute Value of a Complex Number
Liouville's Theorem -

5. Given (4-2i) the complex conjugate would be (4+2i)
standard form of complex numbers
Complex Numbers: Multiply
Polar Coordinates - cos?
Complex Conjugate

6. No i
real
Polar Coordinates - Arg(z*)
Rules of Complex Arithmetic
sin z

7. (a + bi) = (c + bi) =
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + c) + ( b + d)i
sin z
complex numbers

8. 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.
i^1
Complex Number Formula
cos z

9. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Imaginary Unit
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to add and subtract complex numbers (2-3i)-(4+6i)
Square Root

10. Where the curvature of the graph changes
Complex Numbers: Multiply
cos z
complex
point of inflection

11. 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
a + bi for some real a and b.
adding complex numbers
Rules of Complex Arithmetic
v(-1)

12. x + iy = r(cos? + isin?) = re^(i?)
Irrational Number
Field
Polar Coordinates - z
irrational

13. All the powers of i can be written as
complex numbers
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication

14. (e^(iz) - e^(-iz)) / 2i
Any polynomial O(xn) - (n > 0)
Euler's Formula
sin z
Imaginary Unit

15. z1z2* / |z2|²
Affix
e^(ln z)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z1 / z2

16. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
x-axis in the complex plane
adding complex numbers
Argand diagram
conjugate

17. We can also think of the point z= a+ ib as
standard form of complex numbers
transcendental
the vector (a -b)
(a + c) + ( b + d)i

18. Real and imaginary numbers
(cos? +isin?)n
Polar Coordinates - sin?
complex numbers
Euler Formula

19. Root negative - has letter i
Complex Addition
imaginary
How to solve (2i+3)/(9-i)
has a solution.

20. When two complex numbers are subtracted from one another.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)
Complex Subtraction
Irrational Number

21. 1
i^0
a real number: (a + bi)(a - bi) = a² + b²
i^1
Polar Coordinates - r

22. 1
|z-w|
Complex Addition
x-axis in the complex plane
i^2

23. I^2 =
conjugate
i^0
-1
How to solve (2i+3)/(9-i)

24. 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
subtracting complex numbers
Polar Coordinates - z?¹
e^(ln z)
Complex Numbers: Add & subtract

25. Divide moduli and subtract arguments
Irrational Number
complex
0 if and only if a = b = 0
Polar Coordinates - Division

26. A number that cannot be expressed as a fraction for any integer.
Irrational Number
interchangeable
(a + c) + ( b + d)i
zz*

27. Equivalent to an Imaginary Unit.
cos z
De Moivre's Theorem
Any polynomial O(xn) - (n > 0)
Imaginary number

28. When two complex numbers are multipiled together.
natural
Complex Multiplication
How to multiply complex nubers(2+i)(2i-3)
sin iy

29. All numbers
i²
four different numbers: i - -i - 1 - and -1.
complex numbers
complex

30. 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
imaginary
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication by i
Real and Imaginary Parts

31. V(zz*) = v(a² + b²)
Polar Coordinates - z?¹
non-integers
|z| = mod(z)
the vector (a -b)

32. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
complex
ln z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiply the numerator and the denominator by the complex conjugate of the denominator.

33. 1st. Rule of Complex Arithmetic
i^2 = -1
Polar Coordinates - Multiplication
radicals
i^4

34. I
zz*
Complex Number
|z| = mod(z)
v(-1)

35. (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
z + z*
How to multiply complex nubers(2+i)(2i-3)
Field
interchangeable

36. 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.
Field
Square Root
multiplying complex numbers
Complex Numbers: Multiply

37. xpressions such as ``the complex number z'' - and ``the point z'' are now
subtracting complex numbers
Rules of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
interchangeable

38. 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.
i²
How to find any Power
Polar Coordinates - sin?
Any polynomial O(xn) - (n > 0)

39. 3rd. Rule of Complex Arithmetic
the distance from z to the origin in the complex plane
radicals
Real and Imaginary Parts
For real a and b - a + bi = 0 if and only if a = b = 0

40. Have radical
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Addition
radicals
sin iy

41. I
i^1
irrational
integers
rational

42. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
i^1
cos iy
Absolute Value of a Complex Number
Complex Numbers: Multiply

43. 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.'
Polar Coordinates - sin?
the distance from z to the origin in the complex plane
Complex Number
a real number: (a + bi)(a - bi) = a² + b²

44. The modulus of the complex number z= a + ib now can be interpreted as
zz*
the distance from z to the origin in the complex plane
Integers
cos iy

45. I = imaginary unit - i² = -1 or i = v-1
Integers
Imaginary Numbers
the complex numbers
cosh²y - sinh²y

46. To simplify the square root of a negative number
Real Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
|z-w|
point of inflection

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

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48. Starts at 1 - does not include 0
the vector (a -b)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
natural
z1 / z2

49. Has exactly n roots by the fundamental theorem of algebra
i^0
the distance from z to the origin in the complex plane
Complex Multiplication
Any polynomial O(xn) - (n > 0)

50. A + bi
a + bi for some real a and b.
standard form of complex numbers
Complex Subtraction
i^0