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Test your basic knowledge |
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. Multiply moduli and add arguments
Polar Coordinates - Multiplication
For real a and b - a + bi = 0 if and only if a = b = 0
i^2
x-axis in the complex plane
2. E ^ (z2 ln z1)
Roots of Unity
z1 ^ (z2)
transcendental
Any polynomial O(xn) - (n > 0)
3. (a + bi) = (c + bi) =
Polar Coordinates - z?¹
(a + c) + ( b + d)i
subtracting complex numbers
Roots of Unity
4. Derives z = a+bi
Roots of Unity
Euler Formula
Complex Subtraction
Every complex number has the 'Standard Form': a + bi for some real a and b.
5. y / r
four different numbers: i - -i - 1 - and -1.
Field
Polar Coordinates - sin?
Subfield
6. When two complex numbers are multipiled together.
De Moivre's Theorem
Complex Multiplication
zz*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
7. The complex number z representing a+bi.
0 if and only if a = b = 0
cos z
Affix
Complex Division
8. In this amazing number field every algebraic equation in z with complex coefficients
integers
the vector (a -b)
has a solution.
standard form of complex numbers
9. 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
Imaginary Numbers
How to find any Power
the complex numbers
10. Root negative - has letter i
cosh²y - sinh²y
imaginary
Polar Coordinates - sin?
zz*
11. (e^(iz) - e^(-iz)) / 2i
four different numbers: i - -i - 1 - and -1.
point of inflection
Complex numbers are points in the plane
sin z
12. Any number not rational
natural
Imaginary Unit
subtracting complex numbers
irrational
13. 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
Affix
|z-w|
(a + c) + ( b + d)i
adding complex numbers
14. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Exponentiation
Integers
standard form of complex numbers
a + bi for some real a and b.
15. V(x² + y²) = |z|
Euler's Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r
i^2 = -1
16. x + iy = r(cos? + isin?) = re^(i?)
'i'
Imaginary number
Polar Coordinates - Multiplication by i
Polar Coordinates - z
17. (e^(-y) - e^(y)) / 2i = i sinh y
x-axis in the complex plane
conjugate pairs
sin iy
cosh²y - sinh²y
18. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
i^4
Euler's Formula
imaginary
19. R^2 = x
natural
Field
Square Root
Rules of Complex Arithmetic
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.
Polar Coordinates - cos?
How to find any Power
rational
radicals
21. 1
v(-1)
Imaginary number
the vector (a -b)
cosh²y - sinh²y
22. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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23. A subset within a field.
complex numbers
Subfield
Rules of Complex Arithmetic
(a + c) + ( b + d)i
24. (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
adding complex numbers
How to find any Power
How to multiply complex nubers(2+i)(2i-3)
a + bi for some real a and b.
25. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - z?¹
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Addition
Roots of Unity
26. Equivalent to an Imaginary Unit.
Imaginary number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin iy
27. The field of all rational and irrational numbers.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
zz*
Subfield
Real Numbers
28. A² + b² - real and non negative
zz*
(cos? +isin?)n
How to multiply complex nubers(2+i)(2i-3)
multiply the numerator and the denominator by the complex conjugate of the denominator.
29. z1z2* / |z2|²
z1 / z2
i^1
How to solve (2i+3)/(9-i)
-1
30. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Multiplication
The Complex Numbers
Complex Division
real
31. Have radical
z1 / z2
standard form of complex numbers
i^1
radicals
32. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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33. Rotates anticlockwise by p/2
zz*
Polar Coordinates - Multiplication by i
the vector (a -b)
Complex Numbers: Multiply
34. The reals are just the
0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number
x-axis in the complex plane
35. No i
Complex Addition
real
radicals
Liouville's Theorem -
36. The modulus of the complex number z= a + ib now can be interpreted as
z - z*
Square Root
cos iy
the distance from z to the origin in the complex plane
37. Not on the numberline
Irrational Number
non-integers
Complex Division
Euler's Formula
38. 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
i²
Complex Conjugate
Complex Numbers: Add & subtract
Any polynomial O(xn) - (n > 0)
39. Cos n? + i sin n? (for all n integers)
a + bi for some real a and b.
cosh²y - sinh²y
Complex Multiplication
(cos? +isin?)n
40. 1
cos z
transcendental
i^2
Imaginary Unit
41. Where the curvature of the graph changes
point of inflection
radicals
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex
42. 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.
Complex Numbers: Multiply
'i'
e^(ln z)
Polar Coordinates - cos?
43. 1
Liouville's Theorem -
-1
i²
i^1
44. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Real and Imaginary Parts
the distance from z to the origin in the complex plane
45. 2ib
z - z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Euler Formula
the complex numbers
46. ½(e^(iz) + e^(-iz))
Imaginary Numbers
sin z
Every complex number has the 'Standard Form': a + bi for some real a and b.
cos z
47. Numbers on a numberline
Rules of Complex Arithmetic
z - z*
integers
complex
48. 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
Complex Division
conjugate pairs
i^2
Rules of Complex Arithmetic
49. R?¹(cos? - isin?)
z1 / z2
Polar Coordinates - z?¹
For real a and b - a + bi = 0 if and only if a = b = 0
Rules of Complex Arithmetic
50. When two complex numbers are subtracted from one another.
Polar Coordinates - sin?
Complex Subtraction
interchangeable
multiplying complex numbers
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