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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. No i
real
zz*
De Moivre's Theorem
i²
2. Where the curvature of the graph changes
ln z
Liouville's Theorem -
point of inflection
Polar Coordinates - r
3. x / r
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 / z2
Polar Coordinates - cos?
Field
4. Divide moduli and subtract arguments
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Division
the complex numbers
Complex Division
5. All numbers
Euler Formula
Affix
subtracting complex numbers
complex
6. Derives z = a+bi
Any polynomial O(xn) - (n > 0)
i^1
natural
Euler Formula
7. 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.'
Polar Coordinates - r
z - z*
Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
8. Like pi
Real and Imaginary Parts
transcendental
Complex Number
x-axis in the complex plane
9. 1
cosh²y - sinh²y
Square Root
ln z
z - z*
10. For real a and b - a + bi =
0 if and only if a = b = 0
'i'
Rational Number
four different numbers: i - -i - 1 - and -1.
11. z1z2* / |z2|²
Subfield
z1 / z2
Roots of Unity
Polar Coordinates - r
12. 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)
Irrational Number
Polar Coordinates - z
non-integers
13. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Multiplication by i
Imaginary number
i^1
14. ½(e^(-y) +e^(y)) = cosh y
cos iy
subtracting complex numbers
rational
i²
15. I
irrational
i^1
We say that c+di and c-di are complex conjugates.
the complex numbers
16. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Real Numbers
Complex Number
Polar Coordinates - cos?
17. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Multiplication
radicals
How to add and subtract complex numbers (2-3i)-(4+6i)
Integers
18. Given (4-2i) the complex conjugate would be (4+2i)
Square Root
Complex Conjugate
conjugate pairs
imaginary
19. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin z
z - z*
Real and Imaginary Parts
20. 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.
Complex Numbers: Multiply
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)
complex numbers
21. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Conjugate
The Complex Numbers
Subfield
Absolute Value of a Complex Number
22. (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
How to solve (2i+3)/(9-i)
z - z*
i^3
non-integers
23. Multiply moduli and add arguments
subtracting complex numbers
Polar Coordinates - z?¹
i^2 = -1
Polar Coordinates - Multiplication
24. A² + b² - real and non negative
Integers
zz*
z - z*
Every complex number has the 'Standard Form': a + bi for some real a and b.
25. Equivalent to an Imaginary Unit.
the vector (a -b)
Imaginary number
-1
Roots of Unity
26. When two complex numbers are multipiled together.
natural
Complex Multiplication
Subfield
standard form of complex numbers
27. A complex number and its conjugate
Polar Coordinates - Division
conjugate pairs
How to find any Power
conjugate
28. Have radical
De Moivre's Theorem
radicals
standard form of complex numbers
a + bi for some real a and b.
29. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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30. (e^(-y) - e^(y)) / 2i = i sinh y
e^(ln z)
sin iy
'i'
Complex Number
31. 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
transcendental
Polar Coordinates - cos?
adding complex numbers
We say that c+di and c-di are complex conjugates.
32. A plot of complex numbers as points.
Argand diagram
i^4
rational
Imaginary Unit
33. Numbers on a numberline
irrational
Polar Coordinates - Arg(z*)
integers
Roots of Unity
34. The square root of -1.
Polar Coordinates - cos?
Real Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Unit
35. 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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36. 5th. Rule of Complex Arithmetic
Complex Number Formula
The Complex Numbers
Imaginary Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
37. I^2 =
Polar Coordinates - cos?
-1
Polar Coordinates - Arg(z*)
Complex Numbers: Add & subtract
38. R?¹(cos? - isin?)
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z?¹
i^0
0 if and only if a = b = 0
39. We see in this way that the distance between two points z and w in the complex plane is
Complex Numbers: Add & subtract
|z-w|
Subfield
natural
40. 2a
z + z*
Euler Formula
i^0
Square Root
41. Starts at 1 - does not include 0
conjugate pairs
the complex numbers
adding complex numbers
natural
42. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Complex Multiplication
the vector (a -b)
Complex Numbers: Multiply
43. V(x² + y²) = |z|
Roots of Unity
Polar Coordinates - r
Complex Subtraction
Irrational Number
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
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
45. 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
cosh²y - sinh²y
Rules of Complex Arithmetic
x-axis in the complex plane
a + bi for some real a and b.
46. To simplify a complex fraction
imaginary
Complex Exponentiation
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers
47. We can also think of the point z= a+ ib as
Every complex number has the 'Standard Form': a + bi for some real a and b.
Field
The Complex Numbers
the vector (a -b)
48. The field of all rational and irrational numbers.
conjugate pairs
|z-w|
transcendental
Real Numbers
49. The complex number z representing a+bi.
i^3
a + bi for some real a and b.
Affix
complex numbers
50. Not on the numberline
cos z
Real Numbers
non-integers
v(-1)
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