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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
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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. We can also think of the point z= a+ ib as
Irrational Number
|z-w|
the vector (a -b)
Complex Numbers: Multiply
2. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Conjugate
non-integers
How to solve (2i+3)/(9-i)
Absolute Value of a Complex Number
3. ? = -tan?
cos iy
Polar Coordinates - z
Polar Coordinates - Arg(z*)
Affix
4. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Rules of Complex Arithmetic
De Moivre's Theorem
complex
5. All numbers
Complex Conjugate
the vector (a -b)
cos iy
complex
6. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
integers
the complex numbers
multiplying complex numbers
point of inflection
7. I
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z
has a solution.
i^1
8. When two complex numbers are subtracted from one another.
i^0
Complex Subtraction
conjugate
How to find any Power
9. 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.
point of inflection
|z-w|
Complex Numbers: Multiply
i^3
10. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
multiplying complex numbers
rational
transcendental
11. Every complex number has the 'Standard Form':
radicals
Polar Coordinates - r
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a + bi for some real a and b.
12. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
standard form of complex numbers
0 if and only if a = b = 0
Polar Coordinates - z?¹
13. I^2 =
Affix
-1
Complex Number
sin iy
14. The modulus of the complex number z= a + ib now can be interpreted as
natural
Polar Coordinates - r
irrational
the distance from z to the origin in the complex plane
15. Cos n? + i sin n? (for all n integers)
Real Numbers
a + bi for some real a and b.
(cos? +isin?)n
Field
16. For real a and b - a + bi =
Complex Subtraction
i²
complex
0 if and only if a = b = 0
17. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Numbers: Add & subtract
i^2
(cos? +isin?)n
18. E ^ (z2 ln z1)
z1 ^ (z2)
ln z
Any polynomial O(xn) - (n > 0)
e^(ln z)
19. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
conjugate pairs
Roots of Unity
(a + c) + ( b + d)i
'i'
20. I
Complex Numbers: Multiply
Absolute Value of a Complex Number
v(-1)
Polar Coordinates - sin?
21. (e^(iz) - e^(-iz)) / 2i
non-integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Numbers: Add & subtract
sin z
22. 1
i^0
a real number: (a + bi)(a - bi) = a² + b²
i^3
Affix
23. 3rd. Rule of Complex Arithmetic
Polar Coordinates - Arg(z*)
z + z*
cosh²y - sinh²y
For real a and b - a + bi = 0 if and only if a = b = 0
24. Rotates anticlockwise by p/2
(a + c) + ( b + d)i
Complex Conjugate
Subfield
Polar Coordinates - Multiplication by i
25. The field of all rational and irrational numbers.
Real Numbers
z1 ^ (z2)
point of inflection
Complex Numbers: Multiply
26. 2a
z + z*
has a solution.
conjugate pairs
Field
27. 1
Real and Imaginary Parts
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)
conjugate pairs
28. 1st. Rule of Complex Arithmetic
ln z
i^2 = -1
For real a and b - a + bi = 0 if and only if a = b = 0
standard form of complex numbers
29. When two complex numbers are added together.
Complex Addition
Imaginary number
'i'
interchangeable
30. xpressions such as ``the complex number z'' - and ``the point z'' are now
standard form of complex numbers
interchangeable
Complex Division
multiply the numerator and the denominator by the complex conjugate of the denominator.
31. Given (4-2i) the complex conjugate would be (4+2i)
natural
i^1
Complex Conjugate
(cos? +isin?)n
32. 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
rational
The Complex Numbers
Complex Number Formula
33. Numbers on a numberline
integers
subtracting complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
real
34. 1
Any polynomial O(xn) - (n > 0)
i^4
i^0
Every complex number has the 'Standard Form': a + bi for some real a and b.
35. A complex number and its conjugate
conjugate pairs
e^(ln z)
Real Numbers
subtracting complex numbers
36. 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 by i
Complex Number
Rules of Complex Arithmetic
We say that c+di and c-di are complex conjugates.
37. ½(e^(iz) + e^(-iz))
cos z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number
i^4
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
Subfield
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Add & subtract
For real a and b - a + bi = 0 if and only if a = b = 0
39. Real and imaginary numbers
complex numbers
z - z*
(cos? +isin?)n
Square Root
40. Imaginary number
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41. V(zz*) = v(a² + b²)
Irrational Number
Field
|z| = mod(z)
'i'
42. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Argand diagram
multiplying complex numbers
|z-w|
ln z
43. (a + bi) = (c + bi) =
Any polynomial O(xn) - (n > 0)
Euler Formula
(a + c) + ( b + d)i
Liouville's Theorem -
44. 1
interchangeable
i^2
i²
How to solve (2i+3)/(9-i)
45. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
multiply the numerator and the denominator by the complex conjugate of the denominator.
irrational
Rules of Complex Arithmetic
46. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Field
Complex Subtraction
Polar Coordinates - Multiplication by i
47. 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
interchangeable
Rules of Complex Arithmetic
Complex Number Formula
cos z
48. Multiply moduli and add arguments
Polar Coordinates - Multiplication
i^2 = -1
Polar Coordinates - r
conjugate pairs
49. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
How to multiply complex nubers(2+i)(2i-3)
Complex Conjugate
Field
-1
50. Derives z = a+bi
Euler Formula
i^3
imaginary
Complex Addition