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. 1
the complex numbers
complex numbers
z - z*
i^2
2. When two complex numbers are subtracted from one another.
Complex Subtraction
Complex Exponentiation
i^4
i^3
3. No i
Polar Coordinates - r
real
Polar Coordinates - Arg(z*)
i^4
4. 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
Real and Imaginary Parts
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z
5. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - r
Polar Coordinates - z?¹
can't get out of the complex numbers by adding (or subtracting) or multiplying two
The Complex Numbers
6. 3
i^3
Real and Imaginary Parts
v(-1)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
7. 2ib
cos z
Polar Coordinates - Arg(z*)
standard form of complex numbers
z - z*
8. The product of an imaginary number and its conjugate is
subtracting complex numbers
'i'
cos iy
a real number: (a + bi)(a - bi) = a² + b²
9. Divide moduli and subtract arguments
Polar Coordinates - Division
The Complex Numbers
complex
complex numbers
10. Has exactly n roots by the fundamental theorem of algebra
'i'
real
Any polynomial O(xn) - (n > 0)
multiply the numerator and the denominator by the complex conjugate of the denominator.
11. 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
i^3
point of inflection
We say that c+di and c-di are complex conjugates.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
12. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.
zz*
13. Imaginary number
Warning
: Invalid argument supplied for foreach() in
/var/www/html/basicversity.com/show_quiz.php
on line
183
14. I
i^0
v(-1)
|z| = mod(z)
cosh²y - sinh²y
15. (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
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to solve (2i+3)/(9-i)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Subtraction
16. (a + bi)(c + bi) =
Complex Subtraction
z + z*
Field
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
17. Any number not rational
Complex Number
Complex Multiplication
Roots of Unity
irrational
18. (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
How to multiply complex nubers(2+i)(2i-3)
The Complex Numbers
Polar Coordinates - Multiplication
Polar Coordinates - z?¹
19. x / r
cos z
Real Numbers
complex numbers
Polar Coordinates - cos?
20. 5th. Rule of Complex Arithmetic
z + z*
sin z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary number
21. 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.
Any polynomial O(xn) - (n > 0)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary number
How to find any Power
22. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Arg(z*)
e^(ln z)
How to solve (2i+3)/(9-i)
zz*
23. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
interchangeable
v(-1)
Roots of Unity
complex
24. A number that can be expressed as a fraction p/q where q is not equal to 0.
z1 / z2
cos z
subtracting complex numbers
Rational Number
25. 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
Subfield
radicals
-1
Complex Numbers: Add & subtract
26. ½(e^(iz) + e^(-iz))
cos z
sin iy
Euler's Formula
Complex Number
27. Given (4-2i) the complex conjugate would be (4+2i)
Argand diagram
Rational Number
|z-w|
Complex Conjugate
28. 1
i^0
irrational
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Addition
29. z1z2* / |z2|²
zz*
z1 / z2
z1 ^ (z2)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
30. Written as fractions - terminating + repeating decimals
|z| = mod(z)
Complex Numbers: Add & subtract
(cos? +isin?)n
rational
31. xpressions such as ``the complex number z'' - and ``the point z'' are now
integers
(cos? +isin?)n
Field
interchangeable
32. 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
cos iy
Complex Numbers: Multiply
real
Rules of Complex Arithmetic
33. 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
subtracting complex numbers
z1 ^ (z2)
Imaginary number
i²
34. 1
Polar Coordinates - z
How to find any Power
cosh²y - sinh²y
|z| = mod(z)
35. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate pairs
sin z
conjugate
Polar Coordinates - Multiplication
36. A plot of complex numbers as points.
conjugate
Square Root
the vector (a -b)
Argand diagram
37. 1
v(-1)
conjugate
Any polynomial O(xn) - (n > 0)
i^4
38. 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
39. Cos n? + i sin n? (for all n integers)
real
Complex Exponentiation
Roots of Unity
(cos? +isin?)n
40. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
cosh²y - sinh²y
point of inflection
How to add and subtract complex numbers (2-3i)-(4+6i)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
41. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Any polynomial O(xn) - (n > 0)
|z| = mod(z)
Absolute Value of a Complex Number
natural
42. Equivalent to an Imaginary Unit.
sin iy
Imaginary number
We say that c+di and c-di are complex conjugates.
a + bi for some real a and b.
43. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
0 if and only if a = b = 0
How to solve (2i+3)/(9-i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
44. I
How to solve (2i+3)/(9-i)
|z| = mod(z)
i^1
complex numbers
45. R^2 = x
i^4
standard form of complex numbers
Polar Coordinates - Arg(z*)
Square Root
46. The reals are just the
i^2 = -1
Affix
x-axis in the complex plane
For real a and b - a + bi = 0 if and only if a = b = 0
47. ? = -tan?
Imaginary Unit
sin z
Polar Coordinates - Division
Polar Coordinates - Arg(z*)
48. 1
i²
Complex Numbers: Multiply
Rules of Complex Arithmetic
De Moivre's Theorem
49. A subset within a field.
|z-w|
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0
Subfield
50. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
the distance from z to the origin in the complex plane
sin iy
interchangeable