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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.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

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. Root negative - has letter i
Polar Coordinates - z
imaginary
Rational Number
De Moivre's Theorem

2. We see in this way that the distance between two points z and w in the complex plane is
irrational
|z-w|
Irrational Number
real

3. (e^(iz) - e^(-iz)) / 2i
imaginary
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)
sin z

4. E^(ln r) e^(i?) e^(2pin)
cosh²y - sinh²y
Roots of Unity
Polar Coordinates - sin?
e^(ln z)

5. A number that cannot be expressed as a fraction for any integer.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Multiply
Irrational Number
Integers

6. A plot of complex numbers as points.
Argand diagram
radicals
z - z*
Polar Coordinates - cos?

7. ½(e^(iz) + e^(-iz))
cos z
|z| = mod(z)
0 if and only if a = b = 0
How to solve (2i+3)/(9-i)

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

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9. A² + b² - real and non negative
a + bi for some real a and b.
interchangeable
zz*
i^2

10. 1
Real and Imaginary Parts
How to multiply complex nubers(2+i)(2i-3)
i^0
i^2

11. To simplify a complex fraction
cosh²y - sinh²y
Irrational Number
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.

12. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
four different numbers: i - -i - 1 - and -1.
Roots of Unity
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

13. The complex number z representing a+bi.
point of inflection
Affix
Real and Imaginary Parts
Real Numbers

14. The square root of -1.
conjugate pairs
Imaginary Unit
(cos? +isin?)n
subtracting complex numbers

15. The reals are just the
x-axis in the complex plane
(a + c) + ( b + d)i
a + bi for some real a and b.
e^(ln z)

16. Any number not rational
Imaginary Unit
Polar Coordinates - cos?
irrational
interchangeable

17. V(zz*) = v(a² + b²)
How to add and subtract complex numbers (2-3i)-(4+6i)
|z| = mod(z)
Complex Multiplication
Absolute Value of a Complex Number

18. V(x² + y²) = |z|
Polar Coordinates - r
Complex Addition
i^2 = -1
z1 ^ (z2)

19. To simplify the square root of a negative number
Square Root
interchangeable
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)

20. y / r
Polar Coordinates - sin?
Polar Coordinates - z?¹
Complex Subtraction
The Complex Numbers

21. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4
Complex Subtraction
Polar Coordinates - Multiplication by i

22. In this amazing number field every algebraic equation in z with complex coefficients
x-axis in the complex plane
has a solution.
the complex numbers
z1 / z2

23. 1
sin z
i²
i^3
conjugate

24. 3
i^3
Real and Imaginary Parts
How to find any Power
v(-1)

25. Like pi
Euler Formula
i^3
Polar Coordinates - z
transcendental

26. 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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27. Given (4-2i) the complex conjugate would be (4+2i)
Complex Division
z + z*
Complex Conjugate
De Moivre's Theorem

28. Not on the numberline
non-integers
Rational Number
Rules of Complex Arithmetic
i²

29. 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
Imaginary Numbers
Complex Numbers: Add & subtract
transcendental
e^(ln z)

30. Divide moduli and subtract arguments
Field
(cos? +isin?)n
Polar Coordinates - Division
Complex numbers are points in the plane

31. Where the curvature of the graph changes
point of inflection
Roots of Unity
Polar Coordinates - Division
i^4

32. Equivalent to an Imaginary Unit.
Imaginary number
i^1
How to add and subtract complex numbers (2-3i)-(4+6i)
irrational

33. A subset within a field.
non-integers
Polar Coordinates - z
Subfield
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

34. We can also think of the point z= a+ ib as
adding complex numbers
z + z*
the vector (a -b)
complex numbers

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

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36. 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
0 if and only if a = b = 0
conjugate pairs
the complex numbers
Imaginary number

37. When two complex numbers are divided.
complex numbers
Complex Division
multiplying complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

38. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
irrational
Subfield
i^2
How to add and subtract complex numbers (2-3i)-(4+6i)

39. For real a and b - a + bi =
z + z*
complex
0 if and only if a = b = 0
Polar Coordinates - r

40. Written as fractions - terminating + repeating decimals
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
rational
For real a and b - a + bi = 0 if and only if a = b = 0

41. Multiply moduli and add arguments
Polar Coordinates - Multiplication
rational
i^0
Complex Subtraction

42. All numbers
complex
point of inflection
Complex Conjugate
non-integers

43. I
real
Roots of Unity
i^1
the vector (a -b)

44. Derives z = a+bi
subtracting complex numbers
Polar Coordinates - Multiplication by i
z + z*
Euler Formula

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

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46. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Roots of Unity
imaginary
non-integers

47. (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)
Polar Coordinates - z?¹
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z1 / z2

48. A+bi
Complex Number Formula
Imaginary Unit
Integers
Real Numbers

49. R?¹(cos? - isin?)
Polar Coordinates - z?¹
a real number: (a + bi)(a - bi) = a² + b²
Imaginary number
real

50. I
imaginary
Polar Coordinates - sin?
v(-1)
cosh²y - sinh²y