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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. Cos n? + i sin n? (for all n integers)
a + bi for some real a and b.
the distance from z to the origin in the complex plane
(cos? +isin?)n
Euler's Formula

2. A subset within a field.
(a + c) + ( b + d)i
Subfield
Euler Formula
Field

3. Where the curvature of the graph changes
point of inflection
real
complex numbers
i^0

4. Imaginary number

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5. A complex number and its conjugate
Complex Division
conjugate pairs
(a + c) + ( b + d)i
a real number: (a + bi)(a - bi) = a² + b²

6. Equivalent to an Imaginary Unit.
complex numbers
Imaginary number
adding complex numbers
point of inflection

7. To simplify the square root of a negative number
zz*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number
How to add and subtract complex numbers (2-3i)-(4+6i)

8. 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.
z - z*
Complex numbers are points in the plane
(cos? +isin?)n
natural

9. 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
subtracting complex numbers
adding complex numbers
standard form of complex numbers
(a + c) + ( b + d)i

10. E^(ln r) e^(i?) e^(2pin)
cosh²y - sinh²y
e^(ln z)
four different numbers: i - -i - 1 - and -1.
conjugate

11. 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.
Complex Addition
cos z
non-integers

12. x + iy = r(cos? + isin?) = re^(i?)
a + bi for some real a and b.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z
Field

13. I^2 =
interchangeable
cosh²y - sinh²y
Complex Conjugate
-1

14. The product of an imaginary number and its conjugate is
complex
De Moivre's Theorem
cos z
a real number: (a + bi)(a - bi) = a² + b²

15. A number that cannot be expressed as a fraction for any integer.
Polar Coordinates - cos?
Rules of Complex Arithmetic
Complex Exponentiation
Irrational Number

16. A+bi
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula
complex
i^3

17. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
integers
Complex Exponentiation
Rational Number

18. (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
Complex numbers are points in the plane
How to solve (2i+3)/(9-i)
rational
Complex Numbers: Add & subtract

19. x / r
Complex Number Formula
'i'
Polar Coordinates - cos?
Polar Coordinates - z?¹

20. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
i^2
conjugate
Absolute Value of a Complex Number
integers

21. When two complex numbers are subtracted from one another.
a real number: (a + bi)(a - bi) = a² + b²
Complex Subtraction
Square Root
Rules of Complex Arithmetic

22. 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.'
We say that c+di and c-di are complex conjugates.
Complex Number
natural
0 if and only if a = b = 0

23. ½(e^(iz) + e^(-iz))
cos z
conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Addition

24. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Polar Coordinates - r
Integers
z1 / z2
How to multiply complex nubers(2+i)(2i-3)

25. When two complex numbers are multipiled together.
four different numbers: i - -i - 1 - and -1.
Complex Multiplication
point of inflection
Integers

26. 3
Any polynomial O(xn) - (n > 0)
i^3
standard form of complex numbers
Complex Exponentiation

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

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28. (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
How to multiply complex nubers(2+i)(2i-3)
Complex Division
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Arg(z*)

29. A plot of complex numbers as points.
Irrational Number
Argand diagram
Polar Coordinates - z?¹
The Complex Numbers

30. 1
point of inflection
We say that c+di and c-di are complex conjugates.
i^0
Imaginary number

31. V(zz*) = v(a² + b²)
|z| = mod(z)
complex
z1 ^ (z2)
Every complex number has the 'Standard Form': a + bi for some real a and b.

32. Rotates anticlockwise by p/2
conjugate
Polar Coordinates - Multiplication by i
Rational Number
i^2 = -1

33. Root negative - has letter i
e^(ln z)
Polar Coordinates - z?¹
z - z*
imaginary

34. y / r
rational
cos z
Polar Coordinates - sin?
multiplying complex numbers

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

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36. A + bi
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - sin?
standard form of complex numbers
rational

37. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
e^(ln z)
Argand diagram
i^4

38. I
interchangeable
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)
i^1

39. Written as fractions - terminating + repeating decimals
-1
Roots of Unity
the complex numbers
rational

40. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
We say that c+di and c-di are complex conjugates.
Polar Coordinates - cos?
Absolute Value of a Complex Number
Integers

41. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
0 if and only if a = b = 0
Polar Coordinates - r
Euler Formula

42. 2nd. Rule of Complex Arithmetic

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43. R^2 = x
multiplying complex numbers
radicals
Square Root
i^1

44. 2ib
cos iy
z - z*
Polar Coordinates - Multiplication by i
sin z

45. 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
irrational
Polar Coordinates - z?¹
Polar Coordinates - Multiplication

46. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - Multiplication by i
Liouville's Theorem -
Field
sin iy

47. No i
real
integers
z + z*
Argand diagram

48. 1
How to find any Power
irrational
i^4
subtracting complex numbers

49. The modulus of the complex number z= a + ib now can be interpreted as
Complex Conjugate
the distance from z to the origin in the complex plane
z - z*
Subfield

50. 1
adding complex numbers
Polar Coordinates - Arg(z*)
cos z
cosh²y - sinh²y