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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. 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
Complex Numbers: Add & subtract
How to multiply complex nubers(2+i)(2i-3)
Complex Number Formula
radicals
2. Cos n? + i sin n? (for all n integers)
cosh²y - sinh²y
(cos? +isin?)n
Polar Coordinates - sin?
zz*
3. 2ib
z1 ^ (z2)
cos iy
z - z*
z1 / z2
4. A+bi
Complex Number Formula
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Division
Polar Coordinates - cos?
5. To simplify a complex fraction
adding complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
the vector (a -b)
i^1
6. 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
Polar Coordinates - sin?
sin z
How to solve (2i+3)/(9-i)
the complex numbers
7. All the powers of i can be written as
z1 ^ (z2)
non-integers
Rational Number
four different numbers: i - -i - 1 - and -1.
8. A number that cannot be expressed as a fraction for any integer.
ln z
Irrational Number
Polar Coordinates - cos?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
9. When two complex numbers are subtracted from one another.
Imaginary Unit
rational
conjugate pairs
Complex Subtraction
10. Have radical
Complex Numbers: Multiply
e^(ln z)
radicals
i^3
11. A subset within a field.
rational
Subfield
Complex Exponentiation
z + z*
12. (a + bi)(c + bi) =
Polar Coordinates - z?¹
radicals
non-integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
13. 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
ln z
Polar Coordinates - r
Rational Number
adding complex numbers
14. z1z2* / |z2|²
non-integers
z1 / z2
i^3
Complex Number
15. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
0 if and only if a = b = 0
Field
z + z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
16. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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17. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Addition
-1
18. 1
i^2
Affix
Polar Coordinates - z
Imaginary number
19. 1
Rational Number
Complex Subtraction
i²
Affix
20. When two complex numbers are divided.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
21. (e^(-y) - e^(y)) / 2i = i sinh y
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r
sin iy
Affix
22. A² + b² - real and non negative
z1 ^ (z2)
adding complex numbers
Polar Coordinates - r
zz*
23. Not on the numberline
Imaginary Unit
sin iy
non-integers
Complex Subtraction
24. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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25. 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
Complex Number
i^2
i^3
26. (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)
Euler Formula
Complex Division
Polar Coordinates - sin?
27. 1
non-integers
i^0
Rules of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
28. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
cos iy
sin z
0 if and only if a = b = 0
ln z
29. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Polar Coordinates - sin?
Complex Division
'i'
30. Rotates anticlockwise by p/2
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication by i
'i'
Complex numbers are points in the plane
31. The complex number z representing a+bi.
(cos? +isin?)n
Affix
Rules of Complex Arithmetic
rational
32. Derives z = a+bi
conjugate
Complex Subtraction
Real Numbers
Euler Formula
33. ½(e^(iz) + e^(-iz))
cos z
'i'
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex
34. Imaginary number
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35. When two complex numbers are added together.
Complex Conjugate
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Addition
Absolute Value of a Complex Number
36. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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37. 1st. Rule of Complex Arithmetic
i^2
sin z
z - z*
i^2 = -1
38. 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
We say that c+di and c-di are complex conjugates.
v(-1)
multiplying complex numbers
Euler's Formula
39. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Liouville's Theorem -
has a solution.
Polar Coordinates - z?¹
40. 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 - Multiplication by i
The Complex Numbers
Roots of Unity
point of inflection
41. 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
Square Root
Polar Coordinates - Arg(z*)
Rules of Complex Arithmetic
Complex Addition
42. The product of an imaginary number and its conjugate is
transcendental
Complex Numbers: Multiply
a real number: (a + bi)(a - bi) = a² + b²
i²
43. All numbers
Complex Conjugate
complex
Complex Addition
imaginary
44. The reals are just the
|z-w|
(a + c) + ( b + d)i
x-axis in the complex plane
Imaginary Unit
45. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Division
conjugate
How to add and subtract complex numbers (2-3i)-(4+6i)
46. 3
i^3
Any polynomial O(xn) - (n > 0)
We say that c+di and c-di are complex conjugates.
v(-1)
47. 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.
the distance from z to the origin in the complex plane
Complex numbers are points in the plane
x-axis in the complex plane
Affix
48. V(zz*) = v(a² + b²)
|z| = mod(z)
Complex Numbers: Add & subtract
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin iy
49. x / r
z + z*
ln z
Polar Coordinates - cos?
point of inflection
50. R^2 = x
i^3
Square Root
radicals
Absolute Value of a Complex Number