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CLEP General Mathematics: Complex Numbers
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Subjects
:
clep
,
math
Instructions:
Answer 50 questions in 15 minutes.
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.
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. (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)
cos iy
has a solution.
the vector (a -b)
2. z1z2* / |z2|²
Absolute Value of a Complex Number
i^3
z1 / z2
Polar Coordinates - Division
3. R^2 = x
Affix
sin z
'i'
Square Root
4. A subset within a field.
Subfield
We say that c+di and c-di are complex conjugates.
Imaginary Numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
5. A complex number and its conjugate
'i'
conjugate pairs
irrational
non-integers
6. 5th. Rule of Complex Arithmetic
conjugate
Euler's Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
7. 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
transcendental
How to multiply complex nubers(2+i)(2i-3)
the complex numbers
i^0
8. 3
Complex Number Formula
Polar Coordinates - z
i^3
z1 / z2
9. 1
Complex Number Formula
cosh²y - sinh²y
integers
0 if and only if a = b = 0
10. Where the curvature of the graph changes
point of inflection
sin z
ln z
Euler Formula
11. We can also think of the point z= a+ ib as
Complex Number
cos iy
the vector (a -b)
|z| = mod(z)
12. 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
Polar Coordinates - sin?
Complex Multiplication
How to find any Power
Rules of Complex Arithmetic
13. A² + b² - real and non negative
z1 / z2
cos z
zz*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
14. 2a
Complex Numbers: Multiply
Subfield
Square Root
z + z*
15. 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.
How to find any Power
point of inflection
Field
Any polynomial O(xn) - (n > 0)
16. Real and imaginary numbers
i^1
(cos? +isin?)n
Polar Coordinates - Multiplication
complex numbers
17. ? = -tan?
Rational Number
Polar Coordinates - Arg(z*)
a + bi for some real a and b.
i²
18. The field of all rational and irrational numbers.
z - z*
Real Numbers
non-integers
Absolute Value of a Complex Number
19. When two complex numbers are multipiled together.
Complex Addition
Complex Multiplication
has a solution.
z + z*
20. R?¹(cos? - isin?)
|z| = mod(z)
Polar Coordinates - z?¹
Complex Exponentiation
irrational
21. A plot of complex numbers as points.
(a + c) + ( b + d)i
interchangeable
Argand diagram
zz*
22. When two complex numbers are divided.
x-axis in the complex plane
non-integers
Complex Division
subtracting complex numbers
23. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
e^(ln z)
complex
Field
Every complex number has the 'Standard Form': a + bi for some real a and b.
24. When two complex numbers are subtracted from one another.
Complex Subtraction
Complex Multiplication
Polar Coordinates - cos?
Square Root
25. I = imaginary unit - i² = -1 or i = v-1
Every complex number has the 'Standard Form': a + bi for some real a and b.
z1 ^ (z2)
Imaginary Numbers
transcendental
26. Has exactly n roots by the fundamental theorem of algebra
How to find any Power
Any polynomial O(xn) - (n > 0)
Complex Multiplication
a real number: (a + bi)(a - bi) = a² + b²
27. The reals are just the
natural
z - z*
x-axis in the complex plane
We say that c+di and c-di are complex conjugates.
28. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
real
Every complex number has the 'Standard Form': a + bi for some real a and b.
Any polynomial O(xn) - (n > 0)
29. A number that can be expressed as a fraction p/q where q is not equal to 0.
transcendental
Rational Number
cos iy
four different numbers: i - -i - 1 - and -1.
30. To simplify a complex fraction
Polar Coordinates - cos?
multiply the numerator and the denominator by the complex conjugate of the denominator.
irrational
Affix
31. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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32. E^(ln r) e^(i?) e^(2pin)
(a + c) + ( b + d)i
v(-1)
'i'
e^(ln z)
33. 4th. Rule of Complex Arithmetic
Complex Addition
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)
Euler Formula
34. Like pi
Complex Conjugate
subtracting complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
transcendental
35. I^2 =
Argand diagram
Irrational Number
cos z
-1
36. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
cos iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers
37. Have radical
Complex Conjugate
radicals
Rational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
38. All numbers
The Complex Numbers
complex
Polar Coordinates - r
i^2 = -1
39. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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40. 1st. Rule of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^2 = -1
the vector (a -b)
the distance from z to the origin in the complex plane
41. (e^(iz) - e^(-iz)) / 2i
sin z
Field
Complex numbers are points in the plane
Complex Addition
42. To simplify the square root of a negative number
multiplying complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Conjugate
transcendental
43. A complex number may be taken to the power of another complex number.
transcendental
Polar Coordinates - Multiplication by i
conjugate
Complex Exponentiation
44. Derives z = a+bi
i^4
Field
Euler Formula
'i'
45. 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(a + c) + ( b + d)i
a + bi for some real a and b.
i²
46. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
x-axis in the complex plane
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication by i
multiplying complex numbers
47. Multiply moduli and add arguments
|z-w|
Polar Coordinates - Multiplication
Complex Numbers: Add & subtract
Complex Addition
48. 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
sin z
natural
x-axis in the complex plane
49. ½(e^(iz) + e^(-iz))
How to find any Power
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
50. For real a and b - a + bi =
0 if and only if a = b = 0
Complex Numbers: Add & subtract
can't get out of the complex numbers by adding (or subtracting) or multiplying two
e^(ln z)
Sorry!:) No result found.
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