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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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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. 1
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y
'i'
i^2
2. ? = -tan?
Absolute Value of a Complex Number
Field
Polar Coordinates - Arg(z*)
For real a and b - a + bi = 0 if and only if a = b = 0
3. Have radical
Polar Coordinates - z
-1
Integers
radicals
4. When two complex numbers are added together.
complex numbers
z + z*
Complex Addition
a real number: (a + bi)(a - bi) = a² + b²
5. 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
We say that c+di and c-di are complex conjugates.
real
Real and Imaginary Parts
i^1
6. y / r
rational
|z-w|
complex numbers
Polar Coordinates - sin?
7. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
a + bi for some real a and b.
z1 / z2
The Complex Numbers
Polar Coordinates - z?¹
8. Like pi
interchangeable
Complex numbers are points in the plane
Absolute Value of a Complex Number
transcendental
9. xpressions such as ``the complex number z'' - and ``the point z'' are now
cos z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
complex numbers
interchangeable
10. For real a and b - a + bi =
0 if and only if a = b = 0
(a + c) + ( b + d)i
z1 ^ (z2)
transcendental
11. The product of an imaginary number and its conjugate is
Euler's Formula
complex
adding complex numbers
a real number: (a + bi)(a - bi) = a² + b²
12. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
zz*
rational
i^4
13. I
v(-1)
|z-w|
How to add and subtract complex numbers (2-3i)-(4+6i)
cos z
14. A complex number and its conjugate
Imaginary number
conjugate pairs
the vector (a -b)
transcendental
15. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Field
imaginary
Polar Coordinates - cos?
16. 1
adding complex numbers
irrational
cosh²y - sinh²y
sin iy
17. Divide moduli and subtract arguments
interchangeable
Polar Coordinates - Division
complex numbers
Complex Number
18. Written as fractions - terminating + repeating decimals
a real number: (a + bi)(a - bi) = a² + b²
-1
Integers
rational
19. Any number not rational
irrational
Polar Coordinates - r
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rules of Complex Arithmetic
20. 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
i^4
rational
irrational
We say that c+di and c-di are complex conjugates.
21. R^2 = x
(cos? +isin?)n
Square Root
Polar Coordinates - r
four different numbers: i - -i - 1 - and -1.
22. E^(ln r) e^(i?) e^(2pin)
i^4
De Moivre's Theorem
e^(ln z)
a + bi for some real a and b.
23. x + iy = r(cos? + isin?) = re^(i?)
a + bi for some real a and b.
Polar Coordinates - Division
Complex Number Formula
Polar Coordinates - z
24. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - sin?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Any polynomial O(xn) - (n > 0)
adding complex numbers
25. 1
zz*
i^4
-1
Imaginary number
26. The modulus of the complex number z= a + ib now can be interpreted as
Complex numbers are points in the plane
rational
the distance from z to the origin in the complex plane
ln z
27. Imaginary number
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28. 1
i²
Subfield
irrational
i^1
29. Starts at 1 - does not include 0
cos z
Complex Division
natural
ln z
30. 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.'
Imaginary Numbers
e^(ln z)
Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
31. The complex number z representing a+bi.
the distance from z to the origin in the complex plane
complex
Affix
Field
32. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
We say that c+di and c-di are complex conjugates.
Complex Numbers: Multiply
How to find any Power
33. 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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34. 4th. Rule of Complex Arithmetic
Complex Exponentiation
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Euler's Formula
a + bi for some real a and b.
35. R?¹(cos? - isin?)
Polar Coordinates - z?¹
subtracting complex numbers
Euler Formula
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
36. x / r
De Moivre's Theorem
Polar Coordinates - cos?
z - z*
zz*
37. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
'i'
Complex Multiplication
i^1
38. A+bi
(a + c) + ( b + d)i
Any polynomial O(xn) - (n > 0)
Complex Number Formula
can't get out of the complex numbers by adding (or subtracting) or multiplying two
39. 2ib
conjugate
z - z*
Polar Coordinates - Arg(z*)
i^0
40. To simplify a complex fraction
Rational Number
conjugate pairs
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z?¹
41. All numbers
complex
conjugate pairs
Polar Coordinates - Multiplication by i
Irrational Number
42. V(zz*) = v(a² + b²)
Integers
transcendental
subtracting complex numbers
|z| = mod(z)
43. z1z2* / |z2|²
Complex Number Formula
z1 / z2
Rules of Complex Arithmetic
sin iy
44. 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
complex numbers
the complex numbers
Argand diagram
four different numbers: i - -i - 1 - and -1.
45. ½(e^(-y) +e^(y)) = cosh y
Complex Numbers: Add & subtract
complex
i^0
cos iy
46. 3
cosh²y - sinh²y
The Complex Numbers
i^3
a real number: (a + bi)(a - bi) = a² + b²
47. Numbers on a numberline
integers
Rules of Complex Arithmetic
sin iy
v(-1)
48. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Euler Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
rational
sin z
49. 2nd. Rule of Complex Arithmetic
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50. ½(e^(iz) + e^(-iz))
(cos? +isin?)n
Imaginary number
cos z
The Complex Numbers
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