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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. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Argand diagram
non-integers
i^4

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

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3. When two complex numbers are divided.
Complex Division
i^1
Imaginary Numbers
ln z

4. Multiply moduli and add arguments
|z-w|
Integers
zz*
Polar Coordinates - Multiplication

5. A complex number may be taken to the power of another complex number.
Complex Exponentiation
'i'
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Every complex number has the 'Standard Form': a + bi for some real a and b.

6. When two complex numbers are multipiled together.
z + z*
complex
Complex Multiplication
Integers

7. V(x² + y²) = |z|
multiplying complex numbers
Polar Coordinates - Division
a + bi for some real a and b.
Polar Coordinates - r

8. 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
Polar Coordinates - cos?
Any polynomial O(xn) - (n > 0)
adding complex numbers
i^2 = -1

9. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
i^4
The Complex Numbers
the vector (a -b)
Integers

10. When two complex numbers are subtracted from one another.
irrational
Argand diagram
rational
Complex Subtraction

11. A+bi
Complex Number Formula
a + bi for some real a and b.
i^3
Polar Coordinates - z

12. The complex number z representing a+bi.
Affix
radicals
integers
Liouville's Theorem -

13. 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
integers
conjugate
Complex Addition
Rules of Complex Arithmetic

14. Derives z = a+bi
Subfield
|z| = mod(z)
i^3
Euler Formula

15. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - Arg(z*)
Any polynomial O(xn) - (n > 0)
the complex numbers
conjugate pairs

16. E ^ (z2 ln z1)
Field
z1 ^ (z2)
0 if and only if a = b = 0
the complex numbers

17. A number that cannot be expressed as a fraction for any integer.
i^1
irrational
Irrational Number
Complex Number Formula

18. For real a and b - a + bi =
irrational
0 if and only if a = b = 0
Affix
multiply the numerator and the denominator by the complex conjugate of the denominator.

19. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
cosh²y - sinh²y
'i'
the complex numbers

20. 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.
Complex numbers are points in the plane
|z| = mod(z)
the distance from z to the origin in the complex plane
complex

21. (a + bi)(c + bi) =
Real Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to find any Power
Complex Conjugate

22. ½(e^(iz) + e^(-iz))
How to multiply complex nubers(2+i)(2i-3)
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Numbers: Add & subtract
cos z

23. 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
i^2 = -1
interchangeable
Real and Imaginary Parts
i^2

24. Rotates anticlockwise by p/2
Absolute Value of a Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication by i
multiply the numerator and the denominator by the complex conjugate of the denominator.

25. Have radical
Any polynomial O(xn) - (n > 0)
Complex numbers are points in the plane
radicals
cos z

26. 1
-1
Euler Formula
i²
Polar Coordinates - z

27. The product of an imaginary number and its conjugate is
Integers
the complex numbers
Subfield
a real number: (a + bi)(a - bi) = a² + b²

28. 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.'
(a + c) + ( b + d)i
Complex Number
v(-1)
adding complex numbers

29. 1
cosh²y - sinh²y
Imaginary Unit
Liouville's Theorem -
cos iy

30. 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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31. We see in this way that the distance between two points z and w in the complex plane is
a + bi for some real a and b.
Imaginary number
cosh²y - sinh²y
|z-w|

32. We can also think of the point z= a+ ib as
Polar Coordinates - Arg(z*)
real
four different numbers: i - -i - 1 - and -1.
the vector (a -b)

33. In this amazing number field every algebraic equation in z with complex coefficients
Euler's Formula
Polar Coordinates - cos?
has a solution.
(a + bi) = (c + bi) = (a + c) + ( b + d)i

34. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
the vector (a -b)
the complex numbers
How to solve (2i+3)/(9-i)

35. xpressions such as ``the complex number z'' - and ``the point z'' are now
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
Rational Number
interchangeable

36. A subset within a field.
four different numbers: i - -i - 1 - and -1.
Subfield
Affix
z + z*

37. y / r
Complex Numbers: Multiply
Argand diagram
Polar Coordinates - sin?
transcendental

38. (e^(iz) - e^(-iz)) / 2i
sin z
Polar Coordinates - cos?
Field
can't get out of the complex numbers by adding (or subtracting) or multiplying two

39. Equivalent to an Imaginary Unit.
Irrational Number
Imaginary number
cos z
i^1

40. x / r
0 if and only if a = b = 0
Polar Coordinates - cos?
multiply the numerator and the denominator by the complex conjugate of the denominator.
real

41. Cos n? + i sin n? (for all n integers)
Subfield
Polar Coordinates - Multiplication
(cos? +isin?)n
cos z

42. Root negative - has letter i
z1 ^ (z2)
Field
Rules of Complex Arithmetic
imaginary

43. I
i^1
i^2
Complex Number
complex

44. Real and imaginary numbers
Complex Number
complex numbers
Real and Imaginary Parts
conjugate pairs

45. Imaginary number

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46. 1
i^2
transcendental
the complex numbers
sin z

47. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Irrational Number
cos iy
zz*
conjugate

48. A complex number and its conjugate
conjugate pairs
sin z
Affix
ln z

49. To simplify a complex fraction
sin iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i
irrational
multiply the numerator and the denominator by the complex conjugate of the denominator.

50. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Complex Numbers: Multiply
Polar Coordinates - Arg(z*)
(cos? +isin?)n