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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. A subset within a field.
Liouville's Theorem -
Subfield
zz*
Complex Number

2. The field of all rational and irrational numbers.
standard form of complex numbers
Affix
How to find any Power
Real Numbers

3. 1
How to solve (2i+3)/(9-i)
irrational
Polar Coordinates - Multiplication by i
i^4

4. 3rd. Rule of Complex Arithmetic
z + z*
Polar Coordinates - r
For real a and b - a + bi = 0 if and only if a = b = 0
the vector (a -b)

5. Starts at 1 - does not include 0
-1
i^4
natural
x-axis in the complex plane

6. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Affix
a real number: (a + bi)(a - bi) = a² + b²
Integers
v(-1)

7. V(zz*) = v(a² + b²)
For real a and b - a + bi = 0 if and only if a = b = 0
natural
z1 / z2
|z| = mod(z)

8. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Imaginary Unit
Any polynomial O(xn) - (n > 0)
Euler's Formula

9. Derives z = a+bi
z1 ^ (z2)
cos z
For real a and b - a + bi = 0 if and only if a = b = 0
Euler Formula

10. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
natural
multiplying complex numbers
Imaginary Numbers
z1 ^ (z2)

11. All numbers
complex
non-integers
interchangeable
The Complex Numbers

12. I
How to solve (2i+3)/(9-i)
point of inflection
v(-1)
Any polynomial O(xn) - (n > 0)

13. 1st. Rule of Complex Arithmetic
i^2 = -1
How to add and subtract complex numbers (2-3i)-(4+6i)
sin iy
non-integers

14. 1
|z| = mod(z)
Polar Coordinates - cos?
Polar Coordinates - Multiplication
cosh²y - sinh²y

15. 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.'
Roots of Unity
e^(ln z)
Complex Number
point of inflection

16. E ^ (z2 ln z1)
z1 ^ (z2)
How to find any Power
Complex numbers are points in the plane
sin iy

17. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Integers
(cos? +isin?)n
(a + c) + ( b + d)i
Absolute Value of a Complex Number

18. xpressions such as ``the complex number z'' - and ``the point z'' are now
a real number: (a + bi)(a - bi) = a² + b²
Euler's Formula
cos z
interchangeable

19. 2a
the vector (a -b)
the distance from z to the origin in the complex plane
z + z*
z1 ^ (z2)

20. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
natural
e^(ln z)
How to add and subtract complex numbers (2-3i)-(4+6i)
i²

21. (a + bi) = (c + bi) =
non-integers
(a + c) + ( b + d)i
Imaginary Unit
How to multiply complex nubers(2+i)(2i-3)

22. Any number not rational
De Moivre's Theorem
How to solve (2i+3)/(9-i)
(a + c) + ( b + d)i
irrational

23. (a + bi)(c + bi) =
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition
Complex Numbers: Add & subtract

24. 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
real
Complex Numbers: Add & subtract
rational
Rational Number

25. All the powers of i can be written as
Complex Conjugate
(a + c) + ( b + d)i
the vector (a -b)
four different numbers: i - -i - 1 - and -1.

26. A number that cannot be expressed as a fraction for any integer.
v(-1)
i^4
How to multiply complex nubers(2+i)(2i-3)
Irrational Number

27. For real a and b - a + bi =
Complex Conjugate
Polar Coordinates - z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
0 if and only if a = b = 0

28. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Polar Coordinates - z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to solve (2i+3)/(9-i)

29. Has exactly n roots by the fundamental theorem of algebra
z1 ^ (z2)
the complex numbers
Any polynomial O(xn) - (n > 0)
Complex Addition

30. The product of an imaginary number and its conjugate is
Polar Coordinates - z
Complex Conjugate
i^2
a real number: (a + bi)(a - bi) = a² + b²

31. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
sin iy
complex
Rules of Complex Arithmetic

32. When two complex numbers are added together.
Polar Coordinates - Multiplication
ln z
Complex Addition
i^3

33. 5th. Rule of Complex Arithmetic
i^4
i²
|z| = mod(z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

34. When two complex numbers are multipiled together.
complex numbers
How to multiply complex nubers(2+i)(2i-3)
the vector (a -b)
Complex Multiplication

35. Imaginary number

36. ½(e^(iz) + e^(-iz))
Complex Addition
How to multiply complex nubers(2+i)(2i-3)
cos z
Complex Division

37. The square root of -1.
Rational Number
Imaginary Unit
Subfield
adding complex numbers

38. I
i^1
Polar Coordinates - Division
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Roots of Unity

39. Root negative - has letter i
irrational
imaginary
x-axis in the complex plane
z - z*

40. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
imaginary
Imaginary Unit
0 if and only if a = b = 0

41. 1
e^(ln z)
i^0
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

42. (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
has a solution.
How to solve (2i+3)/(9-i)
Affix
imaginary

43. Have radical
How to multiply complex nubers(2+i)(2i-3)
real
We say that c+di and c-di are complex conjugates.
radicals

44. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Complex Subtraction
Any polynomial O(xn) - (n > 0)
Rules of Complex Arithmetic

45. 3
i^3
i²
Complex Number
How to multiply complex nubers(2+i)(2i-3)

46. R?¹(cos? - isin?)
De Moivre's Theorem
Polar Coordinates - z?¹
radicals
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

47. 2nd. Rule of Complex Arithmetic

48. We can also think of the point z= a+ ib as
The Complex Numbers
the vector (a -b)
conjugate
Polar Coordinates - Arg(z*)

49. V(x² + y²) = |z|
Polar Coordinates - r
Polar Coordinates - z
Complex Number Formula
|z-w|

50. 1
i²
cos iy
Roots of Unity
z + z*