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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. The reals are just the
Rules of Complex Arithmetic
x-axis in the complex plane
Euler's Formula
Liouville's Theorem -

2. 5th. Rule of Complex Arithmetic
a + bi for some real a and b.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
De Moivre's Theorem

3. Equivalent to an Imaginary Unit.
has a solution.
complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number

4. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Number Formula
complex
interchangeable
Complex Numbers: Add & subtract

5. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin iy
z1 ^ (z2)
ln z

6. (e^(iz) - e^(-iz)) / 2i
Rules of Complex Arithmetic
|z-w|
conjugate
sin z

7. ? = -tan?
Argand diagram
Complex Numbers: Multiply
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Arg(z*)

8. 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
How to find any Power
Polar Coordinates - z?¹
z - z*

9. 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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10. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Affix
i^2 = -1
i^0

11. 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
Subfield
Rules of Complex Arithmetic
Real Numbers
Real and Imaginary Parts

12. y / r
Euler's Formula
(cos? +isin?)n
Polar Coordinates - sin?
Imaginary Numbers

13. Have radical
i^0
radicals
Polar Coordinates - r
sin z

14. The modulus of the complex number z= a + ib now can be interpreted as
Real and Imaginary Parts
the distance from z to the origin in the complex plane
(cos? +isin?)n
four different numbers: i - -i - 1 - and -1.

15. Multiply moduli and add arguments
|z-w|
-1
non-integers
Polar Coordinates - Multiplication

16. ½(e^(iz) + e^(-iz))
cosh²y - sinh²y
cos z
How to find any Power
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. Starts at 1 - does not include 0
a + bi for some real a and b.
transcendental
How to add and subtract complex numbers (2-3i)-(4+6i)
natural

18. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Polar Coordinates - z?¹
z + z*
Complex Exponentiation
multiplying complex numbers

19. Where the curvature of the graph changes
point of inflection
Complex Addition
i^3
cos z

20. (a + bi)(c + bi) =
Polar Coordinates - z?¹
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
e^(ln z)
the distance from z to the origin in the complex plane

21. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*
the vector (a -b)

22. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
irrational
conjugate
Real Numbers
(cos? +isin?)n

23. 1
(cos? +isin?)n
standard form of complex numbers
i^4
complex

24. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
complex
Roots of Unity
Complex numbers are points in the plane
standard form of complex numbers

25. Divide moduli and subtract arguments
non-integers
Polar Coordinates - z
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division

26. All numbers
standard form of complex numbers
complex
has a solution.
conjugate pairs

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

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28. When two complex numbers are added together.
i^1
How to multiply complex nubers(2+i)(2i-3)
cos iy
Complex Addition

29. R?¹(cos? - isin?)
interchangeable
Polar Coordinates - z?¹
sin iy
Every complex number has the 'Standard Form': a + bi for some real a and b.

30. The field of all rational and irrational numbers.
Complex Number Formula
Absolute Value of a Complex Number
Real Numbers
Imaginary number

31. I
ln z
z1 / z2
-1
i^1

32. x / r
Absolute Value of a Complex Number
radicals
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - cos?

33. Given (4-2i) the complex conjugate would be (4+2i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z
Complex Conjugate
non-integers

34. A plot of complex numbers as points.
Complex Exponentiation
Polar Coordinates - sin?
z + z*
Argand diagram

35. Not on the numberline
non-integers
Complex Numbers: Multiply
We say that c+di and c-di are complex conjugates.
imaginary

36. 3
Euler's Formula
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^2 = -1
i^3

37. Root negative - has letter i
imaginary
Square Root
e^(ln z)
For real a and b - a + bi = 0 if and only if a = b = 0

38. A + bi
standard form of complex numbers
Complex numbers are points in the plane
Euler Formula
Affix

39. 1
cosh²y - sinh²y
Euler's Formula
i^0
cos z

40. Cos n? + i sin n? (for all n integers)
Every complex number has the 'Standard Form': a + bi for some real a and b.
For real a and b - a + bi = 0 if and only if a = b = 0
i^3
(cos? +isin?)n

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

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42. Real and imaginary numbers
The Complex Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos iy
complex numbers

43. E ^ (z2 ln z1)
Complex Conjugate
z1 ^ (z2)
How to solve (2i+3)/(9-i)
Polar Coordinates - Arg(z*)

44. (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 - Multiplication
imaginary
Subfield
How to multiply complex nubers(2+i)(2i-3)

45. A subset within a field.
Complex Multiplication
Subfield
For real a and b - a + bi = 0 if and only if a = b = 0
real

46. V(zz*) = v(a² + b²)
complex numbers
x-axis in the complex plane
|z| = mod(z)
0 if and only if a = b = 0

47. A² + b² - real and non negative
Complex Subtraction
conjugate pairs
interchangeable
zz*

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

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49. 4th. Rule of Complex Arithmetic
z - z*
transcendental
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate

50. 1
has a solution.
i^2
complex
Square Root