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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. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - Arg(z*)
Irrational Number
The Complex Numbers
multiplying complex numbers
2. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
standard form of complex numbers
multiplying complex numbers
Complex Exponentiation
3. 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
Subfield
multiplying complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
4. We can also think of the point z= a+ ib as
the vector (a -b)
i^2
Complex Numbers: Multiply
Complex Division
5. z1z2* / |z2|²
z1 / z2
x-axis in the complex plane
We say that c+di and c-di are complex conjugates.
integers
6. y / r
Imaginary Numbers
Complex numbers are points in the plane
Irrational Number
Polar Coordinates - sin?
7. 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 - Multiplication
complex numbers
Rules of Complex Arithmetic
Polar Coordinates - z?¹
8. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
a + bi for some real a and b.
Absolute Value of a Complex Number
Any polynomial O(xn) - (n > 0)
multiplying complex numbers
9. When two complex numbers are multipiled together.
z - z*
Polar Coordinates - Multiplication by i
Polar Coordinates - sin?
Complex Multiplication
10. V(zz*) = v(a² + b²)
|z| = mod(z)
Complex Numbers: Add & subtract
Complex numbers are points in the plane
Complex Exponentiation
11. 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
The Complex Numbers
Complex Numbers: Add & subtract
cos iy
has a solution.
12. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^2 = -1
'i'
13. 1
x-axis in the complex plane
Field
i^2
Complex Number Formula
14. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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15. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0
Imaginary Unit
adding complex numbers
16. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a + bi for some real a and b.
Real Numbers
We say that c+di and c-di are complex conjugates.
17. All numbers
How to multiply complex nubers(2+i)(2i-3)
complex
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z?¹
18. E ^ (z2 ln z1)
z1 ^ (z2)
We say that c+di and c-di are complex conjugates.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0
19. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Polar Coordinates - Division
Every complex number has the 'Standard Form': a + bi for some real a and b.
Real and Imaginary Parts
20. A plot of complex numbers as points.
Argand diagram
cosh²y - sinh²y
Rules of Complex Arithmetic
How to multiply complex nubers(2+i)(2i-3)
21. Not on the numberline
Polar Coordinates - z
non-integers
Polar Coordinates - cos?
Affix
22. V(x² + y²) = |z|
v(-1)
z1 / z2
Polar Coordinates - r
multiply the numerator and the denominator by the complex conjugate of the denominator.
23. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Roots of Unity
sin z
Complex Subtraction
24. 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.
real
radicals
z + z*
How to find any Power
25. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Absolute Value of a Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
26. (e^(iz) - e^(-iz)) / 2i
sin z
|z-w|
four different numbers: i - -i - 1 - and -1.
adding complex numbers
27. A subset within a field.
Argand diagram
Polar Coordinates - cos?
i^4
Subfield
28. R^2 = x
z1 ^ (z2)
adding complex numbers
i^2 = -1
Square Root
29. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
e^(ln z)
Real and Imaginary Parts
natural
complex
30. 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.
Rules of Complex Arithmetic
Complex Numbers: Multiply
i^3
a + bi for some real a and b.
31. 1
'i'
Polar Coordinates - r
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^4
32. Where the curvature of the graph changes
z - z*
complex numbers
point of inflection
Complex Number Formula
33. Root negative - has letter i
complex
Polar Coordinates - Multiplication
Complex Numbers: Add & subtract
imaginary
34. When two complex numbers are added together.
Complex Division
Square Root
Complex Addition
For real a and b - a + bi = 0 if and only if a = b = 0
35. Rotates anticlockwise by p/2
Complex numbers are points in the plane
Polar Coordinates - Multiplication by i
Liouville's Theorem -
-1
36. 3
i^3
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication
can't get out of the complex numbers by adding (or subtracting) or multiplying two
37. Every complex number has the 'Standard Form':
complex numbers
a + bi for some real a and b.
Real and Imaginary Parts
Irrational Number
38. 2ib
imaginary
z - z*
Complex Number Formula
i^1
39. A complex number and its conjugate
conjugate pairs
x-axis in the complex plane
non-integers
Complex numbers are points in the plane
40. 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
zz*
complex
How to multiply complex nubers(2+i)(2i-3)
the complex numbers
41. (a + bi) = (c + bi) =
z + z*
How to find any Power
Complex Numbers: Multiply
(a + c) + ( b + d)i
42. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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43. 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.'
z + z*
Complex Number
Polar Coordinates - cos?
Absolute Value of a Complex Number
44. Real and imaginary numbers
Polar Coordinates - sin?
complex numbers
Polar Coordinates - Multiplication by i
Imaginary Unit
45. x + iy = r(cos? + isin?) = re^(i?)
Complex Addition
irrational
Complex Numbers: Add & subtract
Polar Coordinates - z
46. Multiply moduli and add arguments
standard form of complex numbers
Polar Coordinates - Multiplication
point of inflection
Complex Addition
47. In this amazing number field every algebraic equation in z with complex coefficients
cosh²y - sinh²y
has a solution.
transcendental
Complex Numbers: Multiply
48. A+bi
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Number Formula
Complex Addition
'i'
49. A + bi
Roots of Unity
Complex Numbers: Multiply
standard form of complex numbers
ln z
50. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
How to find any Power
The Complex Numbers
ln z
How to multiply complex nubers(2+i)(2i-3)