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. xpressions such as ``the complex number z'' - and ``the point z'' are now
subtracting complex numbers
interchangeable
Subfield
Complex Division
2. The field of all rational and irrational numbers.
Complex Number
conjugate
imaginary
Real Numbers
3. Not on the numberline
multiplying complex numbers
integers
non-integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
4. Root negative - has letter i
imaginary
Rules of Complex Arithmetic
Complex Numbers: Multiply
natural
5. V(x² + y²) = |z|
-1
Polar Coordinates - z
Subfield
Polar Coordinates - r
6. Multiply moduli and add arguments
Absolute Value of a Complex Number
Field
Polar Coordinates - Multiplication
Polar Coordinates - z
7. y / r
Polar Coordinates - sin?
Complex numbers are points in the plane
a real number: (a + bi)(a - bi) = a² + b²
the distance from z to the origin in the complex plane
8. A complex number and its conjugate
i^4
a + bi for some real a and b.
conjugate pairs
natural
9. I^2 =
-1
complex numbers
i^1
i^3
10. x + iy = r(cos? + isin?) = re^(i?)
x-axis in the complex plane
|z-w|
Polar Coordinates - z
Integers
11. Equivalent to an Imaginary Unit.
Imaginary number
Roots of Unity
interchangeable
the distance from z to the origin in the complex plane
12. Given (4-2i) the complex conjugate would be (4+2i)
Complex Numbers: Add & subtract
How to solve (2i+3)/(9-i)
a + bi for some real a and b.
Complex Conjugate
13. We see in this way that the distance between two points z and w in the complex plane is
Euler Formula
|z-w|
Rules of Complex Arithmetic
standard form of complex numbers
14. Every complex number has the 'Standard Form':
Integers
Polar Coordinates - cos?
a + bi for some real a and b.
i^0
15. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
the distance from z to the origin in the complex plane
adding complex numbers
Complex Exponentiation
16. When two complex numbers are subtracted from one another.
transcendental
zz*
complex
Complex Subtraction
17. Written as fractions - terminating + repeating decimals
-1
rational
Real Numbers
Polar Coordinates - Multiplication by i
18. A + bi
zz*
Imaginary number
standard form of complex numbers
multiplying complex numbers
19. E^(ln r) e^(i?) e^(2pin)
integers
Complex Division
Imaginary Numbers
e^(ln z)
20. 3
a real number: (a + bi)(a - bi) = a² + b²
Euler Formula
i^3
point of inflection
21. 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.'
Liouville's Theorem -
Complex Number
z - z*
complex numbers
22. (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
integers
Complex Numbers: Add & subtract
z1 / z2
How to multiply complex nubers(2+i)(2i-3)
23. A+bi
sin z
Argand diagram
Complex Subtraction
Complex Number Formula
24. In this amazing number field every algebraic equation in z with complex coefficients
(a + c) + ( b + d)i
has a solution.
Complex Addition
complex numbers
25. ½(e^(-y) +e^(y)) = cosh y
i^1
Polar Coordinates - cos?
the vector (a -b)
cos iy
26. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
sin iy
Euler Formula
multiplying complex numbers
The Complex Numbers
27. 1st. Rule of Complex Arithmetic
Rules of Complex Arithmetic
How to solve (2i+3)/(9-i)
the complex numbers
i^2 = -1
28. A plot of complex numbers as points.
Subfield
Argand diagram
How to solve (2i+3)/(9-i)
Affix
29. Any number not rational
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Number
irrational
(cos? +isin?)n
30. ? = -tan?
cos iy
Rational Number
Polar Coordinates - Arg(z*)
Real Numbers
31. Derives z = a+bi
Argand diagram
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler Formula
Polar Coordinates - z
32. 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.
How to solve (2i+3)/(9-i)
ln z
Affix
Complex Numbers: Multiply
33. 2ib
How to find any Power
the distance from z to the origin in the complex plane
z - z*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
34. Where the curvature of the graph changes
z - z*
We say that c+di and c-di are complex conjugates.
Complex Exponentiation
point of inflection
35. 4th. Rule of Complex Arithmetic
Affix
interchangeable
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
36. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Field
Roots of Unity
Complex Number
z + z*
37. 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.
e^(ln z)
Polar Coordinates - Multiplication
Polar Coordinates - Multiplication by i
38. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos iy
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^4
39. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Complex Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Field
subtracting complex numbers
40. A² + b² - real and non negative
zz*
Polar Coordinates - z?¹
Complex Conjugate
Polar Coordinates - sin?
41. 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
How to solve (2i+3)/(9-i)
complex numbers
Complex Exponentiation
42. When two complex numbers are added together.
Complex Addition
Polar Coordinates - Multiplication
Polar Coordinates - sin?
Complex Subtraction
43. The modulus of the complex number z= a + ib now can be interpreted as
radicals
the distance from z to the origin in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.
has a solution.
44. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
Warning
: Invalid argument supplied for foreach() in
/var/www/html/basicversity.com/show_quiz.php
on line
183
45. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Complex Addition
For real a and b - a + bi = 0 if and only if a = b = 0
Euler Formula
46. For real a and b - a + bi =
Complex Addition
Complex Conjugate
0 if and only if a = b = 0
natural
47. Divide moduli and subtract arguments
Polar Coordinates - Division
point of inflection
i^4
complex numbers
48. When two complex numbers are divided.
non-integers
Complex Number
Real and Imaginary Parts
Complex Division
49. x / r
Polar Coordinates - cos?
Subfield
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a + bi for some real a and b.
50. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
i^1
Polar Coordinates - Multiplication
z + z*