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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.
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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. 2nd. Rule of Complex Arithmetic
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2. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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3. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Complex Exponentiation
cosh²y - sinh²y
point of inflection
4. 1
i^2
Complex Addition
transcendental
z1 ^ (z2)
5. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
0 if and only if a = b = 0
rational
Absolute Value of a Complex Number
point of inflection
6. A complex number and its conjugate
Complex Number
conjugate pairs
How to add and subtract complex numbers (2-3i)-(4+6i)
Square Root
7. No i
|z| = mod(z)
We say that c+di and c-di are complex conjugates.
Complex Numbers: Add & subtract
real
8. Starts at 1 - does not include 0
natural
i^1
Polar Coordinates - Multiplication
sin z
9. Have radical
radicals
the complex numbers
Real and Imaginary Parts
ln z
10. A plot of complex numbers as points.
(cos? +isin?)n
zz*
irrational
Argand diagram
11. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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12. I
Polar Coordinates - r
z1 / z2
Euler's Formula
i^1
13. V(x² + y²) = |z|
Complex Numbers: Multiply
the vector (a -b)
Polar Coordinates - r
irrational
14. 2a
z + z*
The Complex Numbers
Argand diagram
Complex Addition
15. E ^ (z2 ln z1)
z1 ^ (z2)
(a + c) + ( b + d)i
Rules of Complex Arithmetic
Complex Number
16. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
How to find any Power
Euler's Formula
Complex numbers are points in the plane
the vector (a -b)
17. (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)
Polar Coordinates - z?¹
Field
Polar Coordinates - Multiplication
18. ½(e^(-y) +e^(y)) = cosh y
integers
natural
cos iy
(cos? +isin?)n
19. 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
Polar Coordinates - Multiplication by i
Complex Numbers: Multiply
interchangeable
20. Rotates anticlockwise by p/2
Imaginary Unit
Absolute Value of a Complex Number
Polar Coordinates - Multiplication by i
z + z*
21. To simplify the square root of a negative number
rational
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - z
0 if and only if a = b = 0
22. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - sin?
Any polynomial O(xn) - (n > 0)
Rational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
23. 3rd. Rule of Complex Arithmetic
cos iy
For real a and b - a + bi = 0 if and only if a = b = 0
complex
point of inflection
24. Cos n? + i sin n? (for all n integers)
Affix
i^4
the complex numbers
(cos? +isin?)n
25. 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
How to solve (2i+3)/(9-i)
Complex Multiplication
the complex numbers
ln z
26. A complex number may be taken to the power of another complex number.
Complex Number Formula
Complex Exponentiation
Irrational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
27. (a + bi) = (c + bi) =
standard form of complex numbers
Complex Addition
(a + c) + ( b + d)i
Rational Number
28. (a + bi)(c + bi) =
Polar Coordinates - r
Complex Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
29. Root negative - has letter i
How to find any Power
imaginary
sin iy
multiplying complex numbers
30. Derives z = a+bi
the vector (a -b)
Euler Formula
i^2
standard form of complex numbers
31. Divide moduli and subtract arguments
|z-w|
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(cos? +isin?)n
Polar Coordinates - Division
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
Integers
Polar Coordinates - sin?
For real a and b - a + bi = 0 if and only if a = b = 0
Rules of Complex Arithmetic
33. 1
z1 / z2
i²
How to solve (2i+3)/(9-i)
i^3
34. All the powers of i can be written as
Irrational Number
four different numbers: i - -i - 1 - and -1.
Complex Division
radicals
35. 4th. Rule of Complex Arithmetic
Field
|z| = mod(z)
Complex Exponentiation
(a + bi) = (c + bi) = (a + c) + ( b + d)i
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
We say that c+di and c-di are complex conjugates.
complex
natural
Polar Coordinates - Multiplication
37. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers
Complex numbers are points in the plane
38. A + bi
standard form of complex numbers
Imaginary Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
v(-1)
39. R^2 = x
Square Root
radicals
Real and Imaginary Parts
can't get out of the complex numbers by adding (or subtracting) or multiplying two
40. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
We say that c+di and c-di are complex conjugates.
How to find any Power
ln z
the vector (a -b)
41. 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
Complex Addition
Imaginary Unit
Complex Numbers: Add & subtract
Real Numbers
42. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
rational
Irrational Number
(a + c) + ( b + d)i
43. When two complex numbers are multipiled together.
Imaginary number
Complex Multiplication
Real and Imaginary Parts
Complex Number
44. V(zz*) = v(a² + b²)
multiplying complex numbers
transcendental
x-axis in the complex plane
|z| = mod(z)
45. x / r
Integers
Polar Coordinates - cos?
real
Euler Formula
46. 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
Complex Subtraction
subtracting complex numbers
Complex Division
multiply the numerator and the denominator by the complex conjugate of the denominator.
47. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
radicals
Rules of Complex Arithmetic
i^4
48. x + iy = r(cos? + isin?) = re^(i?)
i^3
Polar Coordinates - z
Complex Number
Complex Multiplication
49. The field of all rational and irrational numbers.
x-axis in the complex plane
the distance from z to the origin in the complex plane
Real Numbers
i^3
50. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Number Formula
complex numbers
x-axis in the complex plane
multiplying complex numbers
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