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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. Divide moduli and subtract arguments
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Division
the complex numbers
i^3
2. (a + bi)(c + bi) =
Imaginary number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number
Polar Coordinates - z
3. Derives z = a+bi
imaginary
Complex Numbers: Add & subtract
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler Formula
4. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
-1
(a + c) + ( b + d)i
z + z*
5. The modulus of the complex number z= a + ib now can be interpreted as
Liouville's Theorem -
the distance from z to the origin in the complex plane
Polar Coordinates - Multiplication by i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
6. ½(e^(iz) + e^(-iz))
Complex Multiplication
four different numbers: i - -i - 1 - and -1.
cos z
(a + c) + ( b + d)i
7. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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8. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - r
Polar Coordinates - Multiplication
Complex numbers are points in the plane
9. A complex number may be taken to the power of another complex number.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary Unit
Complex Exponentiation
'i'
10. When two complex numbers are subtracted from one another.
Complex Subtraction
zz*
Complex Division
i^0
11. Where the curvature of the graph changes
point of inflection
How to multiply complex nubers(2+i)(2i-3)
Complex Number Formula
Euler's Formula
12. R?¹(cos? - isin?)
Complex Number
Polar Coordinates - z?¹
i^0
can't get out of the complex numbers by adding (or subtracting) or multiplying two
13. We can also think of the point z= a+ ib as
interchangeable
Argand diagram
the vector (a -b)
How to multiply complex nubers(2+i)(2i-3)
14. To simplify the square root of a negative number
standard form of complex numbers
zz*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
conjugate
15. When two complex numbers are divided.
Polar Coordinates - Division
four different numbers: i - -i - 1 - and -1.
Complex Division
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
16. All numbers
i^2
We say that c+di and c-di are complex conjugates.
real
complex
17. The complex number z representing a+bi.
Affix
How to solve (2i+3)/(9-i)
Complex Numbers: Add & subtract
i^3
18. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
the distance from z to the origin in the complex plane
conjugate
Field
e^(ln z)
19. z1z2* / |z2|²
Complex Addition
cos z
Liouville's Theorem -
z1 / z2
20. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Conjugate
z1 ^ (z2)
21. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
How to solve (2i+3)/(9-i)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
the vector (a -b)
complex numbers
22. A + bi
i^4
Polar Coordinates - cos?
natural
standard form of complex numbers
23. All the powers of i can be written as
a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.
four different numbers: i - -i - 1 - and -1.
ln z
24. 1
adding complex numbers
i^2
|z| = mod(z)
ln z
25. 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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26. R^2 = x
Square Root
i^1
radicals
Field
27. A subset within a field.
the vector (a -b)
i^2 = -1
Field
Subfield
28. A² + b² - real and non negative
zz*
sin iy
Complex Numbers: Add & subtract
Imaginary Numbers
29. I
How to add and subtract complex numbers (2-3i)-(4+6i)
i^1
Complex Multiplication
Polar Coordinates - Arg(z*)
30. We see in this way that the distance between two points z and w in the complex plane is
Affix
Any polynomial O(xn) - (n > 0)
i²
|z-w|
31. I
v(-1)
has a solution.
a real number: (a + bi)(a - bi) = a² + b²
i^2
32. 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
Complex Number Formula
subtracting complex numbers
cosh²y - sinh²y
Complex Numbers: Multiply
33. Written as fractions - terminating + repeating decimals
Imaginary number
Polar Coordinates - sin?
multiplying complex numbers
rational
34. ? = -tan?
x-axis in the complex plane
Irrational Number
Polar Coordinates - Arg(z*)
z1 / z2
35. Multiply moduli and add arguments
i²
Polar Coordinates - Multiplication
Imaginary Unit
How to multiply complex nubers(2+i)(2i-3)
36. 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
How to find any Power
0 if and only if a = b = 0
adding complex numbers
a + bi for some real a and b.
37. Numbers on a numberline
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Affix
adding complex numbers
integers
38. 2a
conjugate
Polar Coordinates - Division
|z-w|
z + z*
39. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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40. ½(e^(-y) +e^(y)) = cosh y
natural
cos iy
Irrational Number
Affix
41. A plot of complex numbers as points.
Argand diagram
(a + c) + ( b + d)i
Real Numbers
real
42. I^2 =
|z| = mod(z)
-1
adding complex numbers
Real Numbers
43. No i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to solve (2i+3)/(9-i)
conjugate pairs
real
44. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Polar Coordinates - Division
Polar Coordinates - z
Polar Coordinates - Multiplication by i
45. To simplify a complex fraction
subtracting complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
multiplying complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
46. (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
Affix
Polar Coordinates - Arg(z*)
How to multiply complex nubers(2+i)(2i-3)
Square Root
47. 1
i^0
Imaginary number
a real number: (a + bi)(a - bi) = a² + b²
Absolute Value of a Complex Number
48. V(zz*) = v(a² + b²)
Complex Number
Affix
Field
|z| = mod(z)
49. Any number not rational
Complex Number
irrational
v(-1)
standard form of complex numbers
50. 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
|z| = mod(z)
e^(ln z)
Rules of Complex Arithmetic