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. 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.
Argand diagram
Complex Numbers: Multiply
Complex Addition
non-integers
2. Root negative - has letter i
Liouville's Theorem -
imaginary
rational
|z| = mod(z)
3. The field of all rational and irrational numbers.
x-axis in the complex plane
sin iy
Real Numbers
i^2 = -1
4. 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
Polar Coordinates - Multiplication
complex
Imaginary Unit
We say that c+di and c-di are complex conjugates.
5. Multiply moduli and add arguments
irrational
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Multiplication
Imaginary Unit
6. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
For real a and b - a + bi = 0 if and only if a = b = 0
The Complex Numbers
e^(ln z)
'i'
7. R?¹(cos? - isin?)
Imaginary number
complex
Polar Coordinates - z?¹
irrational
8. 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
i^3
complex
interchangeable
Rules of Complex Arithmetic
9. When two complex numbers are subtracted from one another.
Rational Number
i^2
z1 / z2
Complex Subtraction
10. y / r
Imaginary Numbers
Polar Coordinates - sin?
Complex Exponentiation
i^2 = -1
11. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
Liouville's Theorem -
the complex numbers
imaginary
|z| = mod(z)
12. The modulus of the complex number z= a + ib now can be interpreted as
0 if and only if a = b = 0
the distance from z to the origin in the complex plane
conjugate
Absolute Value of a Complex Number
13. (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 add and subtract complex numbers (2-3i)-(4+6i)
How to solve (2i+3)/(9-i)
radicals
natural
14. Starts at 1 - does not include 0
natural
i^2 = -1
four different numbers: i - -i - 1 - and -1.
i²
15. When two complex numbers are multipiled together.
De Moivre's Theorem
Complex Multiplication
transcendental
i^3
16. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
De Moivre's Theorem
integers
Imaginary number
17. A plot of complex numbers as points.
Complex numbers are points in the plane
Argand diagram
Complex Division
Complex Numbers: Add & subtract
18. Have radical
Absolute Value of a Complex Number
Affix
radicals
Imaginary number
19. 2ib
z - z*
Any polynomial O(xn) - (n > 0)
the vector (a -b)
Affix
20. (a + bi)(c + bi) =
the complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(cos? +isin?)n
sin iy
21. The product of an imaginary number and its conjugate is
Complex Addition
How to solve (2i+3)/(9-i)
zz*
a real number: (a + bi)(a - bi) = a² + b²
22. 3
ln z
Complex Number Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^3
23. 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
e^(ln z)
Complex Numbers: Add & subtract
cos iy
24. V(x² + y²) = |z|
transcendental
cosh²y - sinh²y
i^3
Polar Coordinates - r
25. (e^(iz) - e^(-iz)) / 2i
sin z
Rational Number
How to multiply complex nubers(2+i)(2i-3)
0 if and only if a = b = 0
26. xpressions such as ``the complex number z'' - and ``the point z'' are now
non-integers
How to multiply complex nubers(2+i)(2i-3)
Complex Exponentiation
interchangeable
27. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Exponentiation
Polar Coordinates - r
Absolute Value of a Complex Number
x-axis in the complex plane
28. 2nd. Rule of Complex Arithmetic
Warning
: Invalid argument supplied for foreach() in
/var/www/html/basicversity.com/show_quiz.php
on line
183
29. I^2 =
the distance from z to the origin in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
-1
Subfield
30. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
Warning
: Invalid argument supplied for foreach() in
/var/www/html/basicversity.com/show_quiz.php
on line
183
31. 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
Polar Coordinates - Division
the distance from z to the origin in the complex plane
Complex numbers are points in the plane
32. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Field
i^3
For real a and b - a + bi = 0 if and only if a = b = 0
33. 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
34. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the complex numbers
(a + c) + ( b + d)i
|z-w|
35. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
i²
Polar Coordinates - Division
The Complex Numbers
Real and Imaginary Parts
36. V(zz*) = v(a² + b²)
|z| = mod(z)
z1 ^ (z2)
How to find any Power
Complex Subtraction
37. 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
38. (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)
Euler's Formula
conjugate pairs
i^2
39. Not on the numberline
Absolute Value of a Complex Number
Complex Numbers: Add & subtract
non-integers
interchangeable
40. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
i^2 = -1
natural
multiplying complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
41. All the powers of i can be written as
Every complex number has the 'Standard Form': a + bi for some real a and b.
z + z*
ln z
four different numbers: i - -i - 1 - and -1.
42. 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
43. No i
Euler Formula
z - z*
real
i^2 = -1
44. A² + b² - real and non negative
zz*
a real number: (a + bi)(a - bi) = a² + b²
We say that c+di and c-di are complex conjugates.
i^2
45. The square root of -1.
Imaginary Unit
z1 ^ (z2)
Complex Numbers: Multiply
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
46. 1
i²
Complex Addition
adding complex numbers
Polar Coordinates - Multiplication by i
47. A + bi
standard form of complex numbers
integers
Euler Formula
i^0
48. I = imaginary unit - i² = -1 or i = v-1
Liouville's Theorem -
real
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary Numbers
49. When two complex numbers are divided.
a + bi for some real a and b.
non-integers
conjugate
Complex Division
50. 2a
Every complex number has the 'Standard Form': a + bi for some real a and b.
z + z*
Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i