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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. 2a
'i'
De Moivre's Theorem
z + z*
|z-w|

2. 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
For real a and b - a + bi = 0 if and only if a = b = 0
Absolute Value of a Complex Number
Real and Imaginary Parts
(a + bi) = (c + bi) = (a + c) + ( b + d)i

3. I
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^1
the distance from z to the origin in the complex plane
Square Root

4. The field of all rational and irrational numbers.
Affix
Complex Number Formula
Real Numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.

5. x + iy = r(cos? + isin?) = re^(i?)
v(-1)
Polar Coordinates - r
i^2 = -1
Polar Coordinates - z

6. 1
Real Numbers
adding complex numbers
i^2
Roots of Unity

7. Equivalent to an Imaginary Unit.
a real number: (a + bi)(a - bi) = a² + b²
ln z
Imaginary number
Any polynomial O(xn) - (n > 0)

8. Rotates anticlockwise by p/2
cosh²y - sinh²y
How to find any Power
Polar Coordinates - Multiplication by i
Integers

9. A complex number may be taken to the power of another complex number.
z1 ^ (z2)
four different numbers: i - -i - 1 - and -1.
Complex Exponentiation
Complex Subtraction

10. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
v(-1)
Affix
complex
Absolute Value of a Complex Number

11. In this amazing number field every algebraic equation in z with complex coefficients
non-integers
How to multiply complex nubers(2+i)(2i-3)
a real number: (a + bi)(a - bi) = a² + b²
has a solution.

12. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Multiplication
natural
sin z
cos z

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

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14. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
sin iy
z + z*
Integers
interchangeable

15. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos z
cos iy
Real and Imaginary Parts

16. y / r
'i'
Affix
-1
Polar Coordinates - sin?

17. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Polar Coordinates - r
Subfield
Complex Exponentiation

18. For real a and b - a + bi =
Any polynomial O(xn) - (n > 0)
0 if and only if a = b = 0
cos z
Liouville's Theorem -

19. 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
i^2
Polar Coordinates - Multiplication
can't get out of the complex numbers by adding (or subtracting) or multiplying two
subtracting complex numbers

20. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
can't get out of the complex numbers by adding (or subtracting) or multiplying two
the distance from z to the origin in the complex plane
cosh²y - sinh²y

21. V(zz*) = v(a² + b²)
i^2 = -1
|z| = mod(z)
cos iy
i^3

22. ½(e^(iz) + e^(-iz))
i^4
cos z
Complex Addition
ln z

23. Like pi
Polar Coordinates - sin?
transcendental
Irrational Number
(cos? +isin?)n

24. 3
v(-1)
i^4
natural
i^3

25. A complex number and its conjugate
conjugate pairs
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.
the vector (a -b)

26. When two complex numbers are added together.
z - z*
Complex Addition
Subfield
zz*

27. I^2 =
Field
ln z
the distance from z to the origin in the complex plane
-1

28. Cos n? + i sin n? (for all n integers)
sin z
sin iy
interchangeable
(cos? +isin?)n

29. Starts at 1 - does not include 0
Imaginary number
i^0
natural
complex

30. Have radical
ln z
Complex Number
Real and Imaginary Parts
radicals

31. x / r
(cos? +isin?)n
sin iy
Complex Number
Polar Coordinates - cos?

32. We can also think of the point z= a+ ib as
z + z*
ln z
the vector (a -b)
Polar Coordinates - Arg(z*)

33. The reals are just the
x-axis in the complex plane
Liouville's Theorem -
interchangeable
conjugate

34. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
The Complex Numbers
Polar Coordinates - sin?
Imaginary number
How to add and subtract complex numbers (2-3i)-(4+6i)

35. Real and imaginary numbers
Complex Division
complex numbers
Affix
four different numbers: i - -i - 1 - and -1.

36. The complex number z representing a+bi.
Polar Coordinates - cos?
sin z
Affix
e^(ln z)

37. A² + b² - real and non negative
zz*
x-axis in the complex plane
|z| = mod(z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

38. 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.
Polar Coordinates - Multiplication
Complex numbers are points in the plane
the distance from z to the origin in the complex plane
We say that c+di and c-di are complex conjugates.

39. The square root of -1.
Polar Coordinates - Multiplication
Imaginary Unit
Square Root
Absolute Value of a Complex Number

40. Written as fractions - terminating + repeating decimals
can't get out of the complex numbers by adding (or subtracting) or multiplying two
interchangeable
rational
Complex Numbers: Multiply

41. Any number not rational
Euler Formula
i^3
irrational
rational

42. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
natural
Complex numbers are points in the plane

43. 2ib
Affix
Complex Number Formula
Polar Coordinates - cos?
z - z*

44. 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
Rules of Complex Arithmetic
Euler's Formula
Polar Coordinates - z?¹
How to add and subtract complex numbers (2-3i)-(4+6i)

45. 2nd. Rule of Complex Arithmetic

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46. V(x² + y²) = |z|
Complex Numbers: Multiply
Polar Coordinates - r
Field
De Moivre's Theorem

47. 5th. Rule of Complex Arithmetic
cosh²y - sinh²y
rational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate pairs

48. 3rd. Rule of Complex Arithmetic
radicals
Complex Number
Polar Coordinates - z
For real a and b - a + bi = 0 if and only if a = b = 0

49. 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
adding complex numbers
real
zz*
Integers

50. Imaginary number

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