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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
Start Test
Study First
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. I^2 =
i²
the distance from z to the origin in the complex plane
Polar Coordinates - r
-1
2. 1
standard form of complex numbers
Polar Coordinates - cos?
cosh²y - sinh²y
How to find any Power
3. 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
subtracting complex numbers
The Complex Numbers
Roots of Unity
i^2
4. The product of an imaginary number and its conjugate is
transcendental
irrational
Affix
a real number: (a + bi)(a - bi) = a² + b²
5. ½(e^(-y) +e^(y)) = cosh y
|z| = mod(z)
Imaginary Numbers
cos iy
Square Root
6. Divide moduli and subtract arguments
Polar Coordinates - Division
conjugate pairs
complex numbers
Roots of Unity
7. Numbers on a numberline
Argand diagram
integers
Polar Coordinates - Multiplication by i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
8. 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.
Square Root
0 if and only if a = b = 0
Imaginary Unit
Complex numbers are points in the plane
9. The reals are just the
the vector (a -b)
We say that c+di and c-di are complex conjugates.
x-axis in the complex plane
De Moivre's Theorem
10. Real and imaginary numbers
transcendental
complex numbers
i^0
Argand diagram
11. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Roots of Unity
natural
Polar Coordinates - r
How to add and subtract complex numbers (2-3i)-(4+6i)
12. (a + bi)(c + bi) =
Every complex number has the 'Standard Form': a + bi for some real a and b.
-1
We say that c+di and c-di are complex conjugates.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
13. x / r
interchangeable
Polar Coordinates - cos?
Rules of Complex Arithmetic
v(-1)
14. 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
Real and Imaginary Parts
Imaginary Unit
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Numbers: Add & subtract
15. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
How to solve (2i+3)/(9-i)
Integers
Polar Coordinates - Multiplication
Subfield
16. I
a + bi for some real a and b.
v(-1)
Liouville's Theorem -
ln z
17. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Complex Addition
four different numbers: i - -i - 1 - and -1.
i^2 = -1
18. A complex number and its conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
i^3
conjugate pairs
19. ? = -tan?
z + z*
Field
Polar Coordinates - Arg(z*)
Complex Number
20. When two complex numbers are subtracted from one another.
multiplying complex numbers
Complex Subtraction
(a + c) + ( b + d)i
four different numbers: i - -i - 1 - and -1.
21. 1
i^0
irrational
Polar Coordinates - Multiplication by i
Complex Subtraction
22. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Polar Coordinates - Division
imaginary
Argand diagram
23. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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24. y / r
Absolute Value of a Complex Number
|z-w|
How to solve (2i+3)/(9-i)
Polar Coordinates - sin?
25. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Euler's Formula
a + bi for some real a and b.
Roots of Unity
How to multiply complex nubers(2+i)(2i-3)
26. 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.'
Argand diagram
Complex Number
the distance from z to the origin in the complex plane
Complex Conjugate
27. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
(cos? +isin?)n
non-integers
How to multiply complex nubers(2+i)(2i-3)
28. R?¹(cos? - isin?)
Complex Division
Polar Coordinates - z?¹
Polar Coordinates - cos?
De Moivre's Theorem
29. (e^(-y) - e^(y)) / 2i = i sinh y
i^3
sin iy
Roots of Unity
Complex numbers are points in the plane
30. 2ib
'i'
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z - z*
How to add and subtract complex numbers (2-3i)-(4+6i)
31. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.
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32. We can also think of the point z= a+ ib as
the vector (a -b)
Subfield
(cos? +isin?)n
cos iy
33. Given (4-2i) the complex conjugate would be (4+2i)
Complex Numbers: Add & subtract
Complex Conjugate
(a + c) + ( b + d)i
Absolute Value of a Complex Number
34. A plot of complex numbers as points.
Complex Subtraction
Polar Coordinates - z?¹
|z-w|
Argand diagram
35. 4th. Rule of Complex Arithmetic
Any polynomial O(xn) - (n > 0)
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i
irrational
36. Derives z = a+bi
Euler Formula
0 if and only if a = b = 0
Imaginary Numbers
Complex numbers are points in the plane
37. 2a
Complex Number
the complex numbers
a + bi for some real a and b.
z + z*
38. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
i^1
Complex Number
cos z
39. The modulus of the complex number z= a + ib now can be interpreted as
Any polynomial O(xn) - (n > 0)
Real Numbers
Complex Exponentiation
the distance from z to the origin in the complex plane
40. (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
a + bi for some real a and b.
Polar Coordinates - Division
i^2 = -1
How to solve (2i+3)/(9-i)
41. Cos n? + i sin n? (for all n integers)
non-integers
Imaginary Numbers
Polar Coordinates - Arg(z*)
(cos? +isin?)n
42. Rotates anticlockwise by p/2
Absolute Value of a Complex Number
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication by i
complex numbers
43. 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
|z| = mod(z)
z - z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
44. No i
the complex numbers
standard form of complex numbers
rational
real
45. A subset within a field.
Subfield
interchangeable
cosh²y - sinh²y
point of inflection
46. We see in this way that the distance between two points z and w in the complex plane is
Complex Subtraction
Euler Formula
|z-w|
z + z*
47. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
non-integers
|z| = mod(z)
complex
48. A² + b² - real and non negative
zz*
complex
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number
49. ½(e^(iz) + e^(-iz))
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to find any Power
a + bi for some real a and b.
cos z
50. Not on the numberline
Complex Division
non-integers
subtracting complex numbers
i^3