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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. 3
complex numbers
Polar Coordinates - cos?
i^3
Complex Subtraction
2. ? = -tan?
transcendental
Polar Coordinates - Arg(z*)
v(-1)
z + z*
3. Have radical
interchangeable
conjugate
radicals
cos z
4. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
multiplying complex numbers
i^0
i^4
5. Imaginary number
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6. xpressions such as ``the complex number z'' - and ``the point z'' are now
z - z*
interchangeable
Square Root
Imaginary Unit
7. Has exactly n roots by the fundamental theorem of algebra
Complex Numbers: Add & subtract
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)
Complex Number
8. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.
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9. Where the curvature of the graph changes
For real a and b - a + bi = 0 if and only if a = b = 0
i^2 = -1
Liouville's Theorem -
point of inflection
10. When two complex numbers are divided.
|z| = mod(z)
Complex Division
|z-w|
complex
11. When two complex numbers are subtracted from one another.
irrational
e^(ln z)
Complex Subtraction
Polar Coordinates - sin?
12. x + iy = r(cos? + isin?) = re^(i?)
rational
Complex Multiplication
Polar Coordinates - z
For real a and b - a + bi = 0 if and only if a = b = 0
13. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Argand diagram
cosh²y - sinh²y
|z| = mod(z)
14. Divide moduli and subtract arguments
Field
Polar Coordinates - cos?
De Moivre's Theorem
Polar Coordinates - Division
15. z1z2* / |z2|²
i^3
Integers
z1 / z2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
16. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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17. Every complex number has the 'Standard Form':
irrational
Complex Addition
the distance from z to the origin in the complex plane
a + bi for some real a and b.
18. (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)
Subfield
imaginary
19. y / r
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - sin?
How to add and subtract complex numbers (2-3i)-(4+6i)
adding complex numbers
20. Numbers on a numberline
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos z
integers
the complex numbers
21. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
e^(ln z)
a + bi for some real a and b.
Polar Coordinates - z
22. V(x² + y²) = |z|
cosh²y - sinh²y
sin iy
Polar Coordinates - r
zz*
23. 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.
Irrational Number
Roots of Unity
Real Numbers
24. V(zz*) = v(a² + b²)
i^1
interchangeable
|z| = mod(z)
multiplying complex numbers
25. ½(e^(-y) +e^(y)) = cosh y
cos iy
point of inflection
|z| = mod(z)
i^0
26. A number that cannot be expressed as a fraction for any integer.
How to multiply complex nubers(2+i)(2i-3)
(a + c) + ( b + d)i
Irrational Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
27. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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28. 1
i^2 = -1
Rational Number
Complex Conjugate
i²
29. 2a
Complex Division
How to multiply complex nubers(2+i)(2i-3)
z + z*
Imaginary Unit
30. When two complex numbers are multipiled together.
z - z*
x-axis in the complex plane
e^(ln z)
Complex Multiplication
31. For real a and b - a + bi =
0 if and only if a = b = 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
sin z
natural
32. To simplify a complex fraction
interchangeable
has a solution.
How to find any Power
multiply the numerator and the denominator by the complex conjugate of the denominator.
33. Starts at 1 - does not include 0
Polar Coordinates - Arg(z*)
natural
complex numbers
Square Root
34. Multiply moduli and add arguments
Polar Coordinates - Multiplication
Complex Exponentiation
Field
subtracting complex numbers
35. Written as fractions - terminating + repeating decimals
rational
multiply the numerator and the denominator by the complex conjugate of the denominator.
has a solution.
conjugate pairs
36. We can also think of the point z= a+ ib as
the vector (a -b)
Liouville's Theorem -
Complex Division
cosh²y - sinh²y
37. (e^(iz) - e^(-iz)) / 2i
sin z
rational
zz*
Complex Division
38. The field of all rational and irrational numbers.
Real Numbers
irrational
Every complex number has the 'Standard Form': a + bi for some real a and b.
transcendental
39. A plot of complex numbers as points.
Polar Coordinates - r
-1
Imaginary Numbers
Argand diagram
40. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
We say that c+di and c-di are complex conjugates.
Complex Multiplication
radicals
the complex numbers
41. No i
Polar Coordinates - Multiplication by i
real
Real and Imaginary Parts
Integers
42. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Complex Number
cosh²y - sinh²y
'i'
ln z
43. When two complex numbers are added together.
Complex Addition
Polar Coordinates - Multiplication
Subfield
z + z*
44. Any number not rational
natural
irrational
zz*
Liouville's Theorem -
45. Given (4-2i) the complex conjugate would be (4+2i)
a + bi for some real a and b.
the distance from z to the origin in the complex plane
i^2
Complex Conjugate
46. 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
adding complex numbers
Complex Addition
conjugate pairs
How to multiply complex nubers(2+i)(2i-3)
47. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
How to add and subtract complex numbers (2-3i)-(4+6i)
non-integers
conjugate
|z-w|
48. E^(ln r) e^(i?) e^(2pin)
cos iy
conjugate pairs
e^(ln z)
x-axis in the complex plane
49. To simplify the square root of a negative number
complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Roots of Unity
natural
50. 1
z - z*
i^4
Complex Numbers: Multiply
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)