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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. In this amazing number field every algebraic equation in z with complex coefficients
Affix
Complex Subtraction
z - z*
has a solution.
2. Real and imaginary numbers
Euler's Formula
complex numbers
subtracting complex numbers
Real and Imaginary Parts
3. Given (4-2i) the complex conjugate would be (4+2i)
Real Numbers
multiplying complex numbers
Complex Conjugate
Imaginary Unit
4. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
-1
cos z
Absolute Value of a Complex Number
conjugate
5. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
v(-1)
imaginary
6. We can also think of the point z= a+ ib as
the vector (a -b)
(a + c) + ( b + d)i
Polar Coordinates - Multiplication by i
the complex numbers
7. The field of all rational and irrational numbers.
Real Numbers
'i'
non-integers
i^4
8. 2a
Polar Coordinates - cos?
Polar Coordinates - Arg(z*)
transcendental
z + z*
9. To simplify a complex fraction
the distance from z to the origin in the complex plane
complex
multiply the numerator and the denominator by the complex conjugate of the denominator.
(cos? +isin?)n
10. 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
i²
Imaginary Numbers
Polar Coordinates - r
11. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Polar Coordinates - Arg(z*)
cos z
Integers
v(-1)
12. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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13. ½(e^(-y) +e^(y)) = cosh y
-1
cos iy
radicals
Polar Coordinates - Arg(z*)
14. (e^(iz) - e^(-iz)) / 2i
sin z
i^1
Complex Numbers: Add & subtract
How to add and subtract complex numbers (2-3i)-(4+6i)
15. 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
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Rules of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
16. I^2 =
For real a and b - a + bi = 0 if and only if a = b = 0
-1
integers
How to solve (2i+3)/(9-i)
17. ? = -tan?
De Moivre's Theorem
Polar Coordinates - Arg(z*)
a real number: (a + bi)(a - bi) = a² + b²
Rules of Complex Arithmetic
18. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
(a + c) + ( b + d)i
Absolute Value of a Complex Number
a + bi for some real a and b.
19. Multiply moduli and add arguments
e^(ln z)
conjugate
Rational Number
Polar Coordinates - Multiplication
20. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Field
We say that c+di and c-di are complex conjugates.
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
21. 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^1
We say that c+di and c-di are complex conjugates.
Any polynomial O(xn) - (n > 0)
De Moivre's Theorem
22. Rotates anticlockwise by p/2
imaginary
radicals
Complex Numbers: Multiply
Polar Coordinates - Multiplication by i
23. When two complex numbers are multipiled together.
Any polynomial O(xn) - (n > 0)
Complex Multiplication
Imaginary number
Complex Number
24. 1st. Rule of Complex Arithmetic
e^(ln z)
cos iy
Polar Coordinates - r
i^2 = -1
25. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
standard form of complex numbers
Any polynomial O(xn) - (n > 0)
The Complex Numbers
Real and Imaginary Parts
26. 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
i^2 = -1
Roots of Unity
standard form of complex numbers
adding complex numbers
27. Every complex number has the 'Standard Form':
Absolute Value of a Complex Number
cosh²y - sinh²y
a + bi for some real a and b.
i^3
28. Has exactly n roots by the fundamental theorem of algebra
multiplying complex numbers
Any polynomial O(xn) - (n > 0)
i²
sin z
29. A number that cannot be expressed as a fraction for any integer.
Polar Coordinates - Multiplication
Real Numbers
has a solution.
Irrational Number
30. 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.
i²
How to find any Power
Imaginary Unit
Subfield
31. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Euler Formula
conjugate
rational
32. A subset within a field.
Subfield
irrational
How to solve (2i+3)/(9-i)
conjugate
33. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Real Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
ln z
Complex numbers are points in the plane
34. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Polar Coordinates - Multiplication
Euler's Formula
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
35. The square root of -1.
Imaginary Unit
Complex Numbers: Add & subtract
Complex Multiplication
z1 ^ (z2)
36. x + iy = r(cos? + isin?) = re^(i?)
v(-1)
i^1
Polar Coordinates - cos?
Polar Coordinates - z
37. (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
Irrational Number
How to solve (2i+3)/(9-i)
integers
Rational Number
38. When two complex numbers are subtracted from one another.
Polar Coordinates - Arg(z*)
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Subtraction
Real and Imaginary Parts
39. Not on the numberline
Complex Numbers: Multiply
Polar Coordinates - z
non-integers
adding complex numbers
40. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
the distance from z to the origin in the complex plane
Complex Conjugate
Field
Complex Division
41. 1
integers
cosh²y - sinh²y
transcendental
the distance from z to the origin in the complex plane
42. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
i^1
How to solve (2i+3)/(9-i)
zz*
43. All numbers
complex
adding complex numbers
Field
How to multiply complex nubers(2+i)(2i-3)
44. Numbers on a numberline
i^2 = -1
Imaginary number
integers
i^0
45. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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46. Divide moduli and subtract arguments
Polar Coordinates - cos?
Polar Coordinates - Division
real
Euler Formula
47. 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.
subtracting complex numbers
Complex Numbers: Multiply
Polar Coordinates - Multiplication
non-integers
48. V(x² + y²) = |z|
zz*
i^4
radicals
Polar Coordinates - r
49. A + bi
standard form of complex numbers
v(-1)
the complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
50. 1
i^2
irrational
the distance from z to the origin in the complex plane
i^2 = -1
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