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CLEP General Mathematics: Complex Numbers
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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. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
multiply the numerator and the denominator by the complex conjugate of the denominator.
four different numbers: i - -i - 1 - and -1.
Roots of Unity
(cos? +isin?)n
2. ? = -tan?
Imaginary number
Polar Coordinates - Arg(z*)
|z-w|
-1
3. (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
z + z*
How to multiply complex nubers(2+i)(2i-3)
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to solve (2i+3)/(9-i)
4. 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
Rational Number
i^1
integers
Rules of Complex Arithmetic
5. The square root of -1.
i^1
i^0
sin iy
Imaginary Unit
6. 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
Polar Coordinates - z
i²
Polar Coordinates - Arg(z*)
7. 1st. Rule of Complex Arithmetic
point of inflection
Euler Formula
i^2 = -1
Complex Division
8. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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9. 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 - Multiplication by i
adding complex numbers
e^(ln z)
Imaginary Numbers
10. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
sin z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
conjugate
11. Starts at 1 - does not include 0
a + bi for some real a and b.
Complex Multiplication
natural
Complex numbers are points in the plane
12. I = imaginary unit - i² = -1 or i = v-1
How to multiply complex nubers(2+i)(2i-3)
Imaginary Numbers
Subfield
Euler Formula
13. Equivalent to an Imaginary Unit.
Complex Addition
the vector (a -b)
Imaginary number
Complex Division
14. 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
Square Root
Field
the complex numbers
i^0
15. A number that can be expressed as a fraction p/q where q is not equal to 0.
i²
interchangeable
-1
Rational Number
16. Any number not rational
Polar Coordinates - sin?
0 if and only if a = b = 0
irrational
Imaginary number
17. E^(ln r) e^(i?) e^(2pin)
Roots of Unity
adding complex numbers
z + z*
e^(ln z)
18. Written as fractions - terminating + repeating decimals
'i'
rational
Complex Numbers: Multiply
i^0
19. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
i^0
cos iy
i^4
20. 2nd. Rule of Complex Arithmetic
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21. Numbers on a numberline
non-integers
integers
Polar Coordinates - r
cos iy
22. ½(e^(iz) + e^(-iz))
imaginary
(a + c) + ( b + d)i
cos z
real
23. The field of all rational and irrational numbers.
Real Numbers
z + z*
i^3
Complex Number
24. 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
Argand diagram
Square Root
Complex Numbers: Add & subtract
0 if and only if a = b = 0
25. Like pi
We say that c+di and c-di are complex conjugates.
the complex numbers
Imaginary Unit
transcendental
26. 5th. Rule of Complex Arithmetic
the vector (a -b)
rational
sin iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
27. x / r
Polar Coordinates - sin?
Polar Coordinates - cos?
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + c) + ( b + d)i
28. ½(e^(-y) +e^(y)) = cosh y
Any polynomial O(xn) - (n > 0)
cos iy
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Division
29. Have radical
standard form of complex numbers
radicals
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Addition
30. 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
Roots of Unity
Imaginary Unit
subtracting complex numbers
i^3
31. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
radicals
The Complex Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^0
32. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
i^4
ln z
Complex numbers are points in the plane
|z-w|
33. 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
Liouville's Theorem -
Absolute Value of a Complex Number
34. Derives z = a+bi
-1
conjugate pairs
Euler Formula
De Moivre's Theorem
35. For real a and b - a + bi =
Polar Coordinates - z
subtracting complex numbers
Complex Numbers: Multiply
0 if and only if a = b = 0
36. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Multiplication
cos z
37. 1
ln z
i^4
Complex Numbers: Add & subtract
adding complex numbers
38. Not on the numberline
v(-1)
How to solve (2i+3)/(9-i)
non-integers
Polar Coordinates - cos?
39. Every complex number has the 'Standard Form':
Complex Division
the vector (a -b)
a + bi for some real a and b.
Field
40. 2a
Polar Coordinates - Division
Polar Coordinates - z?¹
z1 ^ (z2)
z + z*
41. A number that cannot be expressed as a fraction for any integer.
Polar Coordinates - Multiplication by i
Irrational Number
interchangeable
v(-1)
42. Has exactly n roots by the fundamental theorem of algebra
the complex numbers
Any polynomial O(xn) - (n > 0)
cos z
adding complex numbers
43. Cos n? + i sin n? (for all n integers)
cos z
(cos? +isin?)n
Complex Number
Real Numbers
44. 2ib
conjugate
complex numbers
Affix
z - z*
45. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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46. V(x² + y²) = |z|
Polar Coordinates - z?¹
z1 / z2
z - z*
Polar Coordinates - r
47. Root negative - has letter i
i^0
imaginary
Subfield
complex numbers
48. A plot of complex numbers as points.
sin iy
Argand diagram
z + z*
Complex Division
49. (a + bi)(c + bi) =
-1
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4
Argand diagram
50. 3rd. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
cos iy
i^2 = -1
For real a and b - a + bi = 0 if and only if a = b = 0
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