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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
Start Test
Study First
Subjects
:
clep
,
math
Instructions:
Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can
study here
.
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. ½(e^(-y) +e^(y)) = cosh y
complex
cos z
0 if and only if a = b = 0
cos iy
2. When two complex numbers are subtracted from one another.
x-axis in the complex plane
z - z*
Complex Subtraction
sin iy
3. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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4. All numbers
|z| = mod(z)
Polar Coordinates - Multiplication by i
zz*
complex
5. A subset within a field.
z - z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication
Subfield
6. A + bi
standard form of complex numbers
Complex Multiplication
cos iy
0 if and only if a = b = 0
7. Root negative - has letter i
imaginary
a real number: (a + bi)(a - bi) = a² + b²
non-integers
Polar Coordinates - sin?
8. When two complex numbers are multipiled together.
For real a and b - a + bi = 0 if and only if a = b = 0
natural
Irrational Number
Complex Multiplication
9. 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)
natural
sin z
z + z*
10. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Division
multiplying complex numbers
Argand diagram
Complex Addition
11. 2nd. Rule of Complex Arithmetic
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12. E ^ (z2 ln z1)
z1 / z2
sin z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z1 ^ (z2)
13. (e^(iz) - e^(-iz)) / 2i
a + bi for some real a and b.
z - z*
Complex Addition
sin z
14. I
integers
How to add and subtract complex numbers (2-3i)-(4+6i)
How to solve (2i+3)/(9-i)
i^1
15. Written as fractions - terminating + repeating decimals
Complex Numbers: Multiply
rational
Complex Subtraction
multiplying complex numbers
16. 3
We say that c+di and c-di are complex conjugates.
i^3
Complex Division
Polar Coordinates - Arg(z*)
17. I^2 =
standard form of complex numbers
-1
Polar Coordinates - z?¹
Rules of Complex Arithmetic
18. Has exactly n roots by the fundamental theorem of algebra
can't get out of the complex numbers by adding (or subtracting) or multiplying two
four different numbers: i - -i - 1 - and -1.
adding complex numbers
Any polynomial O(xn) - (n > 0)
19. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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20. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
e^(ln z)
conjugate
(a + c) + ( b + d)i
irrational
21. A² + b² - real and non negative
a + bi for some real a and b.
i^0
zz*
imaginary
22. For real a and b - a + bi =
0 if and only if a = b = 0
Irrational Number
i^1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
23. (a + bi) = (c + bi) =
|z-w|
a real number: (a + bi)(a - bi) = a² + b²
(a + c) + ( b + d)i
Complex Exponentiation
24. ½(e^(iz) + e^(-iz))
Rational Number
imaginary
cos z
Integers
25. The complex number z representing a+bi.
adding complex numbers
Polar Coordinates - Multiplication
Affix
How to add and subtract complex numbers (2-3i)-(4+6i)
26. Where the curvature of the graph changes
real
point of inflection
Complex Conjugate
For real a and b - a + bi = 0 if and only if a = b = 0
27. 1
Affix
point of inflection
cosh²y - sinh²y
Complex Number
28. No i
sin iy
integers
Complex numbers are points in the plane
real
29. 1
i^2
conjugate
zz*
How to find any Power
30. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
How to find any Power
Roots of Unity
Field
'i'
31. 1
Euler's Formula
0 if and only if a = b = 0
radicals
i²
32. Rotates anticlockwise by p/2
(a + c) + ( b + d)i
Euler's Formula
Polar Coordinates - Multiplication by i
Field
33. y / r
Polar Coordinates - Multiplication
Polar Coordinates - sin?
Roots of Unity
complex
34. 1st. Rule of Complex Arithmetic
i^2 = -1
non-integers
Complex Number Formula
ln z
35. Cos n? + i sin n? (for all n integers)
irrational
(cos? +isin?)n
Polar Coordinates - Multiplication
i²
36. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Complex Addition
multiplying complex numbers
Polar Coordinates - z?¹
37. Divide moduli and subtract arguments
cos z
Polar Coordinates - Division
Euler's Formula
integers
38. z1z2* / |z2|²
|z-w|
Polar Coordinates - z?¹
Any polynomial O(xn) - (n > 0)
z1 / z2
39. When two complex numbers are divided.
Complex Division
Imaginary number
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Unit
40. In this amazing number field every algebraic equation in z with complex coefficients
imaginary
has a solution.
Complex Numbers: Add & subtract
Complex Division
41. E^(ln r) e^(i?) e^(2pin)
has a solution.
e^(ln z)
Imaginary number
i^2 = -1
42. V(x² + y²) = |z|
We say that c+di and c-di are complex conjugates.
Polar Coordinates - r
Real and Imaginary Parts
subtracting complex numbers
43. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
Polar Coordinates - cos?
natural
44. V(zz*) = v(a² + b²)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers
Roots of Unity
|z| = mod(z)
45. (a + bi)(c + bi) =
Affix
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers
Polar Coordinates - r
46. 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
Polar Coordinates - Multiplication by i
subtracting complex numbers
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
47. xpressions such as ``the complex number z'' - and ``the point z'' are now
conjugate pairs
interchangeable
Complex Multiplication
Polar Coordinates - Division
48. Have radical
radicals
Every complex number has the 'Standard Form': a + bi for some real a and b.
Affix
For real a and b - a + bi = 0 if and only if a = b = 0
49. Starts at 1 - does not include 0
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
natural
Roots of Unity
50. All the powers of i can be written as
De Moivre's Theorem
sin z
four different numbers: i - -i - 1 - and -1.
Rational Number