SUBJECTS
|
BROWSE
|
CAREER CENTER
|
POPULAR
|
JOIN
|
LOGIN
Business Skills
|
Soft Skills
|
Basic Literacy
|
Certifications
About
|
Help
|
Privacy
|
Terms
|
Email
Search
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. When two complex numbers are divided.
the complex numbers
Rules of Complex Arithmetic
Complex Division
Polar Coordinates - Division
2. A² + b² - real and non negative
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
zz*
(a + bi) = (c + bi) = (a + c) + ( b + d)i
3. 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
Rules of Complex Arithmetic
a + bi for some real a and b.
transcendental
Polar Coordinates - Arg(z*)
4. I^2 =
Complex Multiplication
has a solution.
Complex Conjugate
-1
5. A + bi
Complex Subtraction
Polar Coordinates - Division
Complex numbers are points in the plane
standard form of complex numbers
6. Starts at 1 - does not include 0
natural
v(-1)
i^2 = -1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
7. ½(e^(-y) +e^(y)) = cosh y
Complex Subtraction
cos iy
The Complex Numbers
Complex Numbers: Multiply
8. A complex number may be taken to the power of another complex number.
complex
v(-1)
Euler's Formula
Complex Exponentiation
9. (e^(-y) - e^(y)) / 2i = i sinh y
conjugate pairs
Liouville's Theorem -
sin iy
0 if and only if a = b = 0
10. Written as fractions - terminating + repeating decimals
rational
sin z
'i'
Square Root
11. When two complex numbers are added together.
Complex Addition
i^4
Complex Multiplication
conjugate
12. ? = -tan?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the complex numbers
Polar Coordinates - Arg(z*)
For real a and b - a + bi = 0 if and only if a = b = 0
13. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
Warning
: Invalid argument supplied for foreach() in
/var/www/html/basicversity.com/show_quiz.php
on line
183
14. 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
Complex Number Formula
Field
multiplying complex numbers
adding complex numbers
15. V(zz*) = v(a² + b²)
|z| = mod(z)
Polar Coordinates - Division
standard form of complex numbers
'i'
16. E ^ (z2 ln z1)
e^(ln z)
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 ^ (z2)
De Moivre's Theorem
17. Imaginary number
Warning
: Invalid argument supplied for foreach() in
/var/www/html/basicversity.com/show_quiz.php
on line
183
18. 1
Polar Coordinates - z
For real a and b - a + bi = 0 if and only if a = b = 0
cosh²y - sinh²y
Rules of Complex Arithmetic
19. 1
i^4
a + bi for some real a and b.
Imaginary Unit
transcendental
20. z1z2* / |z2|²
Subfield
Polar Coordinates - z
The Complex Numbers
z1 / z2
21. No i
e^(ln z)
real
x-axis in the complex plane
Complex Numbers: Add & subtract
22. y / r
Complex Subtraction
Polar Coordinates - cos?
Polar Coordinates - sin?
point of inflection
23. In this amazing number field every algebraic equation in z with complex coefficients
How to find any Power
Polar Coordinates - Multiplication by i
the complex numbers
has a solution.
24. 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
z + z*
'i'
subtracting complex numbers
We say that c+di and c-di are complex conjugates.
25. V(x² + y²) = |z|
i²
Polar Coordinates - r
i^2
Polar Coordinates - Multiplication
26. 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
Euler's Formula
Real Numbers
Complex Exponentiation
27. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
zz*
z1 ^ (z2)
adding complex numbers
28. The reals are just the
Polar Coordinates - Arg(z*)
x-axis in the complex plane
imaginary
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
29. I
v(-1)
i²
Integers
standard form of complex numbers
30. Derives z = a+bi
i^4
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers
Euler Formula
31. Numbers on a numberline
rational
complex numbers
integers
i^1
32. Divide moduli and subtract arguments
complex
sin iy
Polar Coordinates - Division
De Moivre's Theorem
33. I = imaginary unit - i² = -1 or i = v-1
multiply the numerator and the denominator by the complex conjugate of the denominator.
Imaginary Numbers
The Complex Numbers
non-integers
34. Not on the numberline
non-integers
Complex Addition
multiply the numerator and the denominator by the complex conjugate of the denominator.
z1 ^ (z2)
35. 1st. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
Integers
i^2 = -1
Real Numbers
36. A plot of complex numbers as points.
How to multiply complex nubers(2+i)(2i-3)
Argand diagram
Imaginary Numbers
natural
37. x + iy = r(cos? + isin?) = re^(i?)
zz*
Field
Subfield
Polar Coordinates - z
38. Real and imaginary numbers
the complex numbers
has a solution.
complex numbers
Polar Coordinates - Arg(z*)
39. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
rational
How to multiply complex nubers(2+i)(2i-3)
For real a and b - a + bi = 0 if and only if a = b = 0
40. Where the curvature of the graph changes
Complex Division
cos iy
point of inflection
Imaginary Numbers
41. R?¹(cos? - isin?)
Polar Coordinates - z?¹
z1 / z2
(cos? +isin?)n
De Moivre's Theorem
42. The product of an imaginary number and its conjugate is
Complex Number Formula
ln z
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Multiplication by i
43. 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.
Warning
: Invalid argument supplied for foreach() in
/var/www/html/basicversity.com/show_quiz.php
on line
183
44. 3
real
De Moivre's Theorem
i^3
Affix
45. A number that cannot be expressed as a fraction for any integer.
(a + c) + ( b + d)i
Irrational Number
Complex Division
How to solve (2i+3)/(9-i)
46. We can also think of the point z= a+ ib as
Complex Addition
imaginary
Complex Conjugate
the vector (a -b)
47. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
transcendental
How to add and subtract complex numbers (2-3i)-(4+6i)
48. (a + bi) = (c + bi) =
i²
(a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)
conjugate pairs
49. 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.
Real and Imaginary Parts
Argand diagram
Complex Numbers: Multiply
conjugate
50. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Absolute Value of a Complex Number
i^1
conjugate
the distance from z to the origin in the complex plane