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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
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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. Multiply moduli and add arguments
Polar Coordinates - Multiplication
radicals
Complex Division
Polar Coordinates - Multiplication by i
2. We can also think of the point z= a+ ib as
radicals
complex
the vector (a -b)
rational
3. 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.
rational
|z| = mod(z)
Complex Number Formula
How to find any Power
4. To simplify the square root of a negative number
complex numbers
z - z*
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
irrational
5. When two complex numbers are subtracted from one another.
rational
point of inflection
four different numbers: i - -i - 1 - and -1.
Complex Subtraction
6. 2nd. Rule of Complex Arithmetic
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7. x / r
Polar Coordinates - cos?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z1 ^ (z2)
the vector (a -b)
8. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
z + z*
the vector (a -b)
|z| = mod(z)
9. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Real and Imaginary Parts
conjugate
i^3
10. Numbers on a numberline
integers
cos z
Complex Number Formula
Complex Conjugate
11. The complex number z representing a+bi.
Complex Division
Liouville's Theorem -
Affix
How to find any Power
12. The product of an imaginary number and its conjugate is
i^3
natural
a real number: (a + bi)(a - bi) = a² + b²
Complex Division
13. (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
'i'
How to solve (2i+3)/(9-i)
How to multiply complex nubers(2+i)(2i-3)
z1 / z2
14. 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 Number
(a + c) + ( b + d)i
Field
Complex Numbers: Multiply
15. 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
Square Root
adding complex numbers
Complex Subtraction
Complex Numbers: Add & subtract
16. I^2 =
-1
zz*
De Moivre's Theorem
Complex Conjugate
17. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Numbers: Add & subtract
ln z
Euler's Formula
18. 4th. Rule of Complex Arithmetic
i^2
Every complex number has the 'Standard Form': a + bi for some real a and b.
z1 / z2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
19. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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20. V(x² + y²) = |z|
imaginary
'i'
Rules of Complex Arithmetic
Polar Coordinates - r
21. Have radical
radicals
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a + bi for some real a and b.
22. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
ln z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
23. Starts at 1 - does not include 0
Euler's Formula
rational
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
24. Every complex number has the 'Standard Form':
Liouville's Theorem -
non-integers
Polar Coordinates - z?¹
a + bi for some real a and b.
25. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Irrational Number
conjugate
v(-1)
point of inflection
26. 1
natural
i^2
Complex Numbers: Multiply
irrational
27. When two complex numbers are added together.
Affix
z - z*
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Addition
28. Root negative - has letter i
v(-1)
How to multiply complex nubers(2+i)(2i-3)
imaginary
Absolute Value of a Complex Number
29. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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30. Given (4-2i) the complex conjugate would be (4+2i)
z + z*
Complex Conjugate
Complex Numbers: Multiply
Imaginary Numbers
31. 3
i^3
standard form of complex numbers
subtracting complex numbers
e^(ln z)
32. ½(e^(iz) + e^(-iz))
x-axis in the complex plane
Rules of Complex Arithmetic
cos z
Polar Coordinates - Multiplication
33. A + bi
real
Polar Coordinates - sin?
Euler's Formula
standard form of complex numbers
34. 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
Euler's Formula
adding complex numbers
Imaginary Unit
conjugate pairs
35. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - r
(cos? +isin?)n
Absolute Value of a Complex Number
Roots of Unity
36. x + iy = r(cos? + isin?) = re^(i?)
conjugate
Polar Coordinates - z
For real a and b - a + bi = 0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
37. A subset within a field.
We say that c+di and c-di are complex conjugates.
point of inflection
Subfield
Complex Number Formula
38. 1
Euler's Formula
cosh²y - sinh²y
i^2
Complex Subtraction
39. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Polar Coordinates - Division
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
40. The modulus of the complex number z= a + ib now can be interpreted as
Imaginary Numbers
i^4
the distance from z to the origin in the complex plane
|z| = mod(z)
41. A² + b² - real and non negative
four different numbers: i - -i - 1 - and -1.
(a + c) + ( b + d)i
zz*
interchangeable
42. When two complex numbers are multipiled together.
-1
Polar Coordinates - Multiplication
Polar Coordinates - z?¹
Complex Multiplication
43. Cos n? + i sin n? (for all n integers)
Roots of Unity
Any polynomial O(xn) - (n > 0)
(cos? +isin?)n
Polar Coordinates - Division
44. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
0 if and only if a = b = 0
x-axis in the complex plane
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary number
45. A plot of complex numbers as points.
i^0
non-integers
Argand diagram
point of inflection
46. R?¹(cos? - isin?)
Complex Conjugate
cosh²y - sinh²y
conjugate
Polar Coordinates - z?¹
47. Real and imaginary numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Rules of Complex Arithmetic
Euler's Formula
complex numbers
48. The square root of -1.
z + z*
Imaginary Unit
Complex Division
'i'
49. 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.'
Complex Numbers: Multiply
Complex Number
i²
'i'
50. 3rd. Rule of Complex Arithmetic
Polar Coordinates - z
For real a and b - a + bi = 0 if and only if a = b = 0
multiplying complex numbers
Real and Imaginary Parts