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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. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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2. When two complex numbers are subtracted from one another.
|z| = mod(z)
Field
Complex Subtraction
Integers
3. ½(e^(-y) +e^(y)) = cosh y
Subfield
complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos iy
4. For real a and b - a + bi =
Complex Division
complex numbers
0 if and only if a = b = 0
z - z*
5. When two complex numbers are divided.
Polar Coordinates - z
Complex Division
Subfield
integers
6. Where the curvature of the graph changes
i^1
point of inflection
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Multiplication
7. 5th. Rule of Complex Arithmetic
Complex Subtraction
real
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex
8. 1
i^3
i^0
Complex Multiplication
e^(ln z)
9. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Conjugate
'i'
v(-1)
conjugate
10. Every complex number has the 'Standard Form':
Complex Multiplication
can't get out of the complex numbers by adding (or subtracting) or multiplying two
a + bi for some real a and b.
Argand diagram
11. V(x² + y²) = |z|
Complex Multiplication
natural
Polar Coordinates - r
ln z
12. Any number not rational
i^3
cos iy
irrational
Polar Coordinates - r
13. V(zz*) = v(a² + b²)
Polar Coordinates - z
|z| = mod(z)
z - z*
imaginary
14. 2ib
Polar Coordinates - Division
i^2
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
15. No i
real
Polar Coordinates - Multiplication by i
Complex Numbers: Add & subtract
Irrational Number
16. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
point of inflection
Polar Coordinates - z?¹
Field
Absolute Value of a Complex Number
17. x + iy = r(cos? + isin?) = re^(i?)
Imaginary Numbers
Polar Coordinates - Multiplication
ln z
Polar Coordinates - z
18. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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19. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication
Polar Coordinates - sin?
Polar Coordinates - Multiplication by i
Polar Coordinates - Arg(z*)
20. All the powers of i can be written as
conjugate pairs
Polar Coordinates - Multiplication by i
natural
four different numbers: i - -i - 1 - and -1.
21. In this amazing number field every algebraic equation in z with complex coefficients
Absolute Value of a Complex Number
Complex Number
has a solution.
Complex Numbers: Add & subtract
22. The square root of -1.
z + z*
Irrational Number
Imaginary Unit
conjugate
23. Real and imaginary numbers
complex numbers
Any polynomial O(xn) - (n > 0)
i^0
Polar Coordinates - Arg(z*)
24. We see in this way that the distance between two points z and w in the complex plane is
Affix
v(-1)
|z-w|
Polar Coordinates - z?¹
25. To simplify the square root of a negative number
Argand diagram
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number Formula
Complex Exponentiation
26. A² + b² - real and non negative
zz*
transcendental
Complex numbers are points in the plane
complex
27. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
real
We say that c+di and c-di are complex conjugates.
multiplying complex numbers
Any polynomial O(xn) - (n > 0)
28. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
'i'
|z-w|
Complex Division
29. 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 Numbers: Add & subtract
the vector (a -b)
x-axis in the complex plane
Integers
30. 1
complex numbers
Polar Coordinates - Multiplication
has a solution.
i²
31. Have radical
The Complex Numbers
Complex Exponentiation
radicals
Complex numbers are points in the plane
32. ½(e^(iz) + e^(-iz))
Absolute Value of a Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
the complex numbers
33. (e^(-y) - e^(y)) / 2i = i sinh y
the complex numbers
Complex Addition
Polar Coordinates - z
sin iy
34. I
ln z
v(-1)
(cos? +isin?)n
multiply the numerator and the denominator by the complex conjugate of the denominator.
35. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Complex Numbers: Multiply
How to add and subtract complex numbers (2-3i)-(4+6i)
How to find any Power
Complex Number
36. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
How to find any Power
The Complex Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field
37. Given (4-2i) the complex conjugate would be (4+2i)
Real and Imaginary Parts
Complex Conjugate
subtracting complex numbers
Absolute Value of a Complex Number
38. 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
adding complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
complex
i^0
39. 1
cosh²y - sinh²y
Polar Coordinates - Arg(z*)
Complex Division
|z-w|
40. 2nd. Rule of Complex Arithmetic
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41. A + bi
Polar Coordinates - z
standard form of complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication
42. 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.
cosh²y - sinh²y
How to find any Power
Complex Numbers: Add & subtract
irrational
43. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
real
Roots of Unity
Irrational Number
subtracting complex numbers
44. R^2 = x
Square Root
Imaginary Unit
x-axis in the complex plane
Polar Coordinates - z
45. 1
i^2
Any polynomial O(xn) - (n > 0)
We say that c+di and c-di are complex conjugates.
Irrational Number
46. Imaginary number
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47. When two complex numbers are multipiled together.
Complex Multiplication
transcendental
Polar Coordinates - z?¹
Affix
48. Derives z = a+bi
Euler Formula
z1 ^ (z2)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
49. 1
multiplying complex numbers
i^4
irrational
conjugate pairs
50. 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
Complex Multiplication
z1 ^ (z2)
We say that c+di and c-di are complex conjugates.
(cos? +isin?)n
Can you answer 50 questions in 15 minutes?
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