CLEP General Mathematics: Complex Numbers

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• 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 modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
Imaginary number
Roots of Unity
cosh²y - sinh²y


2. 2ib
z - z*
imaginary
four different numbers: i - -i - 1 - and -1.
Complex Division


3. (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
How to solve (2i+3)/(9-i)
Complex Numbers: Add & subtract
natural
z + z*


4. Every complex number has the 'Standard Form':
a + bi for some real a and b.
x-axis in the complex plane
Complex Exponentiation
Euler Formula


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

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6. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
How to find any Power
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number


7. R?¹(cos? - isin?)
e^(ln z)
x-axis in the complex plane
integers
Polar Coordinates - z?¹


8. 1
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Unit
Irrational Number
i^0


9. R^2 = x
Irrational Number
Square Root
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
real


10. The reals are just the
x-axis in the complex plane
Rules of Complex Arithmetic
cosh²y - sinh²y
Complex Number


11. A complex number and its conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 ^ (z2)
conjugate pairs
How to solve (2i+3)/(9-i)


12. 5th. Rule of Complex Arithmetic
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
-1


13. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
a + bi for some real a and b.
Complex Conjugate
The Complex Numbers
Real Numbers


14. Imaginary number

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15. z1z2* / |z2|²
has a solution.
z1 / z2
i^4
Polar Coordinates - sin?


16. Derives z = a+bi
imaginary
Polar Coordinates - Multiplication
Euler Formula
non-integers


17. 1
-1
Polar Coordinates - z
i^4
Complex Number


18. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
Real and Imaginary Parts
v(-1)
How to find any Power
i^1


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

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20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Imaginary Numbers
The Complex Numbers
adding complex numbers
Absolute Value of a Complex Number


21. Numbers on a numberline
integers
Complex Multiplication
Complex Number Formula
Polar Coordinates - cos?


22. I
i^1
interchangeable
integers
adding complex numbers


23. The square root of -1.
Imaginary Unit
Subfield
ln z
Irrational Number


24. Multiply moduli and add arguments
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - z
Complex Multiplication
Polar Coordinates - Multiplication


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.
Imaginary number
i²
rational
Complex numbers are points in the plane


26. 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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27. (e^(-y) - e^(y)) / 2i = i sinh y
(a + c) + ( b + d)i
cosh²y - sinh²y
sin iy
i^1


28. 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
How to multiply complex nubers(2+i)(2i-3)
has a solution.
Polar Coordinates - Arg(z*)


29. Equivalent to an Imaginary Unit.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary number
subtracting complex numbers
Affix


30. 3rd. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
i^1
Complex numbers are points in the plane
For real a and b - a + bi = 0 if and only if a = b = 0


31. 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
subtracting complex numbers
zz*
v(-1)
For real a and b - a + bi = 0 if and only if a = b = 0


32. 3
conjugate
real
i^3
Complex Subtraction


33. y / r
Polar Coordinates - sin?
Euler's Formula
Complex Addition
zz*


34. Starts at 1 - does not include 0
Complex Subtraction
a real number: (a + bi)(a - bi) = a² + b²
natural
real


35. 2nd. Rule of Complex Arithmetic

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36. The field of all rational and irrational numbers.
Real Numbers
Complex Number
(a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0


37. Real and imaginary numbers
(cos? +isin?)n
multiplying complex numbers
radicals
complex numbers


38. 2a
Field
z + z*
Polar Coordinates - sin?
i^2


39. 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.
Real and Imaginary Parts
e^(ln z)
non-integers
Complex Numbers: Multiply


40. 4th. Rule of Complex Arithmetic
Complex Conjugate
standard form of complex numbers
Imaginary number
(a + bi) = (c + bi) = (a + c) + ( b + d)i


41. 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
interchangeable
adding complex numbers
Polar Coordinates - Arg(z*)
Polar Coordinates - cos?


42. When two complex numbers are multipiled together.
Polar Coordinates - Multiplication
Complex Multiplication
sin iy
conjugate


43. Root negative - has letter i
Rules of Complex Arithmetic
Complex Number Formula
imaginary
complex numbers


44. ? = -tan?
v(-1)
Polar Coordinates - Arg(z*)
cosh²y - sinh²y
Affix


45. We see in this way that the distance between two points z and w in the complex plane is
(a + c) + ( b + d)i
Complex Addition
Complex Conjugate
|z-w|


46. (a + bi)(c + bi) =
v(-1)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
Field


47. A² + b² - real and non negative
i^4
cos z
We say that c+di and c-di are complex conjugates.
zz*


48. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Roots of Unity
zz*
complex


49. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
z1 / z2
conjugate
We say that c+di and c-di are complex conjugates.
Rules of Complex Arithmetic


50. Not on the numberline
-1
non-integers
adding complex numbers
Real Numbers