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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. 2ib
Polar Coordinates - Multiplication
i^3
Real Numbers
z - z*

2. Given (4-2i) the complex conjugate would be (4+2i)
natural
Complex Multiplication
Rules of Complex Arithmetic
Complex Conjugate

3. 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.
Complex Numbers: Multiply
Imaginary Unit
De Moivre's Theorem
ln z

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

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5. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - Multiplication
a + bi for some real a and b.
Imaginary Numbers
integers

6. Like pi
can't get out of the complex numbers by adding (or subtracting) or multiplying two
transcendental
Polar Coordinates - Arg(z*)
cos iy

7. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
|z| = mod(z)
conjugate pairs
How to multiply complex nubers(2+i)(2i-3)

8. 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.'
adding complex numbers
Polar Coordinates - z
How to find any Power
Complex Number

9. 5th. Rule of Complex Arithmetic
Argand diagram
Complex numbers are points in the plane
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Roots of Unity

10. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Imaginary Numbers
real
Absolute Value of a Complex Number
multiplying complex numbers

11. 1
i^1
i^2
Polar Coordinates - Multiplication
integers

12. Divide moduli and subtract arguments
sin iy
Rational Number
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Division

13. 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
Polar Coordinates - z
Euler Formula
subtracting complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two

14. Cos n? + i sin n? (for all n integers)
Square Root
(cos? +isin?)n
i²
v(-1)

15. (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
Polar Coordinates - sin?
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to solve (2i+3)/(9-i)
cosh²y - sinh²y

16. A number that cannot be expressed as a fraction for any integer.
Irrational Number
point of inflection
(cos? +isin?)n
i^4

17. No i
Imaginary number
Euler Formula
Complex Number
real

18. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Subfield
conjugate
Polar Coordinates - z?¹
Irrational Number

19. 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
Any polynomial O(xn) - (n > 0)
adding complex numbers
Field
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

20. The field of all rational and irrational numbers.
Real Numbers
adding complex numbers
a + bi for some real a and b.
The Complex Numbers

21. We see in this way that the distance between two points z and w in the complex plane is
De Moivre's Theorem
Polar Coordinates - Division
How to solve (2i+3)/(9-i)
|z-w|

22. When two complex numbers are multipiled together.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication
Imaginary Unit
Polar Coordinates - cos?

23. Every complex number has the 'Standard Form':
i^2 = -1
The Complex Numbers
a + bi for some real a and b.
transcendental

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

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25. 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
i^0
complex
Complex numbers are points in the plane

26. E ^ (z2 ln z1)
sin z
Polar Coordinates - z?¹
z1 ^ (z2)
has a solution.

27. x / r
Argand diagram
the distance from z to the origin in the complex plane
v(-1)
Polar Coordinates - cos?

28. 2nd. Rule of Complex Arithmetic

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29. To simplify a complex fraction
x-axis in the complex plane
point of inflection
multiply the numerator and the denominator by the complex conjugate of the denominator.
can't get out of the complex numbers by adding (or subtracting) or multiplying two

30. 1
i^4
|z| = mod(z)
point of inflection
Imaginary Unit

31. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Real and Imaginary Parts
Integers
e^(ln z)
Imaginary number

32. When two complex numbers are divided.
Complex Division
real
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - r

33. R?¹(cos? - isin?)
Polar Coordinates - z?¹
non-integers
z1 ^ (z2)
(cos? +isin?)n

34. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex numbers are points in the plane
Absolute Value of a Complex Number
Irrational Number
the complex numbers

35. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
complex numbers
Rational Number
has a solution.

36. Any number not rational
irrational
Complex Subtraction
e^(ln z)
-1

37. I^2 =
-1
How to add and subtract complex numbers (2-3i)-(4+6i)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Rational Number

38. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex Subtraction
We say that c+di and c-di are complex conjugates.
Imaginary Unit

39. I
v(-1)
the vector (a -b)
the complex numbers
(cos? +isin?)n

40. (a + bi) = (c + bi) =
Polar Coordinates - r
the vector (a -b)
(a + c) + ( b + d)i
Real and Imaginary Parts

41. V(x² + y²) = |z|
Complex Number
Polar Coordinates - r
-1
Complex Subtraction

42. (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
Polar Coordinates - cos?
How to multiply complex nubers(2+i)(2i-3)
Complex Addition
radicals

43. 4th. Rule of Complex Arithmetic
v(-1)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to find any Power
the complex numbers

44. R^2 = x
z1 / z2
conjugate
v(-1)
Square Root

45. 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
standard form of complex numbers
(a + c) + ( b + d)i
Rules of Complex Arithmetic
cos z

46. We can also think of the point z= a+ ib as
|z| = mod(z)
subtracting complex numbers
Complex numbers are points in the plane
the vector (a -b)

47. 2a
complex numbers
z + z*
radicals
Complex Subtraction

48. Real and imaginary numbers
conjugate
natural
complex numbers
cos z

49. A complex number and its conjugate
z1 / z2
conjugate pairs
point of inflection
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

50. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)
interchangeable
cosh²y - sinh²y