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. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
Roots of Unity
the complex numbers
Complex Number
|z-w|


2. 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.
Rational Number
|z-w|
natural


3. To simplify a complex fraction
standard form of complex numbers
sin z
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field


4. (e^(iz) - e^(-iz)) / 2i
Argand diagram
sin z
Polar Coordinates - Arg(z*)
Imaginary Numbers


5. 2nd. Rule of Complex Arithmetic

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6. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
cos z
i^2 = -1
-1


7. 1
multiply the numerator and the denominator by the complex conjugate of the denominator.
rational
Roots of Unity
i^4


8. E ^ (z2 ln z1)
'i'
z1 ^ (z2)
Complex Subtraction
Every complex number has the 'Standard Form': a + bi for some real a and b.


9. Written as fractions - terminating + repeating decimals
rational
imaginary
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Unit


10. A complex number and its conjugate
a + bi for some real a and b.
ln z
real
conjugate pairs


11. I
i^0
v(-1)
subtracting complex numbers
Rules of Complex Arithmetic


12. ½(e^(-y) +e^(y)) = cosh y
cos iy
Polar Coordinates - z?¹
interchangeable
real


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

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14. 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
multiplying complex numbers
How to solve (2i+3)/(9-i)
Real and Imaginary Parts


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

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16. 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.
a real number: (a + bi)(a - bi) = a² + b²
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex numbers are points in the plane
Complex Conjugate


17. Divide moduli and subtract arguments
Polar Coordinates - Division
Polar Coordinates - Multiplication
Complex numbers are points in the plane
How to multiply complex nubers(2+i)(2i-3)


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

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19. (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)
(a + c) + ( b + d)i
adding complex numbers
Polar Coordinates - Multiplication by i


20. 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.
multiplying complex numbers
How to find any Power
Complex Number Formula
We say that c+di and c-di are complex conjugates.


21. 3
Complex Multiplication
Field
Complex Numbers: Add & subtract
i^3


22. 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
De Moivre's Theorem
Euler Formula
Complex Numbers: Add & subtract
the complex numbers


23. A+bi
Complex Conjugate
How to find any Power
z1 ^ (z2)
Complex Number Formula


24. E^(ln r) e^(i?) e^(2pin)
0 if and only if a = b = 0
We say that c+di and c-di are complex conjugates.
e^(ln z)
(cos? +isin?)n


25. I^2 =
real
-1
z1 ^ (z2)
Any polynomial O(xn) - (n > 0)


26. 1
Irrational Number
Complex Multiplication
i²
Rules of Complex Arithmetic


27. No i
real
i²
|z-w|
multiplying complex numbers


28. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Polar Coordinates - z?¹
Real and Imaginary Parts
i^0


29. Where the curvature of the graph changes
point of inflection
transcendental
imaginary
Absolute Value of a Complex Number


30. (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
How to multiply complex nubers(2+i)(2i-3)
Imaginary Unit
real
Rules of Complex Arithmetic


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
Euler Formula
Rules of Complex Arithmetic
Affix
Polar Coordinates - r


32. 2a
Polar Coordinates - Division
cos z
How to find any Power
z + z*


33. ? = -tan?
Square Root
e^(ln z)
conjugate pairs
Polar Coordinates - Arg(z*)


34. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - cos?
the distance from z to the origin in the complex plane
Complex Division
conjugate pairs


35. y / r
Polar Coordinates - z?¹
Polar Coordinates - sin?
point of inflection
How to add and subtract complex numbers (2-3i)-(4+6i)


36. 2ib
De Moivre's Theorem
x-axis in the complex plane
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*


37. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Polar Coordinates - r
Every complex number has the 'Standard Form': a + bi for some real a and b.
real


38. Imaginary number

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39. 3rd. Rule of Complex Arithmetic
cos z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary number


40. Multiply moduli and add arguments
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Multiplication
integers
imaginary


41. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - sin?
Complex Number
sin iy
integers


42. Numbers on a numberline
irrational
a real number: (a + bi)(a - bi) = a² + b²
zz*
integers


43. Root negative - has letter i
We say that c+di and c-di are complex conjugates.
interchangeable
imaginary
complex numbers


44. A² + b² - real and non negative
|z| = mod(z)
Polar Coordinates - Arg(z*)
How to find any Power
zz*


45. 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
cos iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers
Polar Coordinates - z?¹


46. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
complex
zz*


47. We can also think of the point z= a+ ib as
the vector (a -b)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number
rational


48. 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)
Polar Coordinates - Division
Irrational Number
-1


49. Cos n? + i sin n? (for all n integers)
z1 ^ (z2)
(cos? +isin?)n
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Division


50. A complex number may be taken to the power of another complex number.
multiplying complex numbers
Complex Exponentiation
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rules of Complex Arithmetic