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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.
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. The field of all rational and irrational numbers.
Real Numbers
integers
i^1
z1 ^ (z2)
2. (e^(iz) - e^(-iz)) / 2i
complex numbers
How to solve (2i+3)/(9-i)
sin z
four different numbers: i - -i - 1 - and -1.
3. 1
-1
i^2
irrational
a + bi for some real a and b.
4. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - sin?
Euler Formula
Any polynomial O(xn) - (n > 0)
Affix
5. 1
i²
i^0
Euler's Formula
radicals
6. Multiply moduli and add arguments
zz*
i^2 = -1
Polar Coordinates - Multiplication
Imaginary Numbers
7. Have radical
cos iy
z1 / z2
radicals
Subfield
8. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
cosh²y - sinh²y
has a solution.
De Moivre's Theorem
9. The reals are just the
x-axis in the complex plane
Polar Coordinates - r
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)
10. Derives z = a+bi
Polar Coordinates - Multiplication by i
cos z
Euler Formula
Rules of Complex Arithmetic
11. 2ib
z - z*
Polar Coordinates - Division
multiplying complex numbers
Complex Number
12. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
v(-1)
multiplying complex numbers
Absolute Value of a Complex Number
standard form of complex numbers
13. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Argand diagram
i²
ln z
a + bi for some real a and b.
14. ? = -tan?
ln z
zz*
Polar Coordinates - Arg(z*)
i²
15. I
i^1
z1 ^ (z2)
i²
Polar Coordinates - z?¹
16. 5th. Rule of Complex Arithmetic
|z| = mod(z)
natural
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
The Complex Numbers
17. A plot of complex numbers as points.
Complex numbers are points in the plane
transcendental
Imaginary Numbers
Argand diagram
18. Numbers on a numberline
Rational Number
sin iy
integers
conjugate
19. All numbers
i^1
complex
point of inflection
Absolute Value of a Complex Number
20. 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
Polar Coordinates - r
Rules of Complex Arithmetic
De Moivre's Theorem
0 if and only if a = b = 0
21. Root negative - has letter i
imaginary
Complex Conjugate
the vector (a -b)
0 if and only if a = b = 0
22. We see in this way that the distance between two points z and w in the complex plane is
subtracting complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
|z-w|
Polar Coordinates - z?¹
23. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
multiply the numerator and the denominator by the complex conjugate of the denominator.
radicals
i^2
conjugate
24. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
standard form of complex numbers
Integers
Complex Number Formula
Complex Numbers: Multiply
25. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
'i'
real
Any polynomial O(xn) - (n > 0)
26. A subset within a field.
0 if and only if a = b = 0
Complex Conjugate
Subfield
z - z*
27. Where the curvature of the graph changes
point of inflection
sin z
Complex Addition
Every complex number has the 'Standard Form': a + bi for some real a and b.
28. Any number not rational
irrational
the vector (a -b)
cos iy
Polar Coordinates - sin?
29. x / r
point of inflection
Polar Coordinates - cos?
Any polynomial O(xn) - (n > 0)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
30. 1st. Rule of Complex Arithmetic
i^2 = -1
integers
the vector (a -b)
Polar Coordinates - sin?
31. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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32. 2a
z + z*
a + bi for some real a and b.
Polar Coordinates - Multiplication by i
Real and Imaginary Parts
33. 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
Complex Conjugate
Imaginary Unit
Polar Coordinates - sin?
Complex Numbers: Add & subtract
34. A number that can be expressed as a fraction p/q where q is not equal to 0.
subtracting complex numbers
Polar Coordinates - cos?
sin z
Rational Number
35. In this amazing number field every algebraic equation in z with complex coefficients
rational
adding complex numbers
i^0
has a solution.
36. 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.
How to find any Power
z + z*
Complex Numbers: Multiply
cos iy
37. 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.
Rules of Complex Arithmetic
e^(ln z)
Imaginary number
Complex Numbers: Multiply
38. To simplify a complex fraction
integers
|z-w|
multiply the numerator and the denominator by the complex conjugate of the denominator.
Affix
39. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
Real and Imaginary Parts
i^2
the complex numbers
Complex Addition
40. I = imaginary unit - i² = -1 or i = v-1
We say that c+di and c-di are complex conjugates.
natural
sin iy
Imaginary Numbers
41. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
For real a and b - a + bi = 0 if and only if a = b = 0
Real and Imaginary Parts
Roots of Unity
i^1
42. A number that cannot be expressed as a fraction for any integer.
How to add and subtract complex numbers (2-3i)-(4+6i)
Irrational Number
cos z
Complex Numbers: Add & subtract
43. R^2 = x
Complex Multiplication
natural
Square Root
Complex Addition
44. ½(e^(iz) + e^(-iz))
the distance from z to the origin in the complex plane
cos z
adding complex numbers
complex
45. No i
De Moivre's Theorem
Roots of Unity
ln z
real
46. For real a and b - a + bi =
|z| = mod(z)
Euler Formula
0 if and only if a = b = 0
can't get out of the complex numbers by adding (or subtracting) or multiplying two
47. When two complex numbers are multipiled together.
0 if and only if a = b = 0
Polar Coordinates - Multiplication
Real and Imaginary Parts
Complex Multiplication
48. Equivalent to an Imaginary Unit.
Imaginary number
Rules of Complex Arithmetic
Complex numbers are points in the plane
Complex Number
49. The square root of -1.
Imaginary Unit
the distance from z to the origin in the complex plane
Square Root
the complex numbers
50. (a + bi)(c + bi) =
i^1
Polar Coordinates - cos?
Complex Subtraction
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Can you answer 50 questions in 15 minutes?
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