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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.
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. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
|z-w|
i^2
irrational
The Complex Numbers
2. 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.
ln z
Polar Coordinates - Division
0 if and only if a = b = 0
How to find any Power
3. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
cos z
conjugate
Euler's Formula
Complex Multiplication
4. I
Integers
v(-1)
Imaginary Numbers
-1
5. x / r
Polar Coordinates - cos?
Polar Coordinates - z?¹
v(-1)
Rational Number
6. When two complex numbers are subtracted from one another.
e^(ln z)
Complex Subtraction
z - z*
Liouville's Theorem -
7. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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8. 2nd. Rule of Complex Arithmetic
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9. In this amazing number field every algebraic equation in z with complex coefficients
z1 / z2
Argand diagram
has a solution.
Complex Numbers: Add & subtract
10. A complex number may be taken to the power of another complex number.
rational
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Exponentiation
e^(ln z)
11. Where the curvature of the graph changes
point of inflection
Real Numbers
Polar Coordinates - sin?
How to multiply complex nubers(2+i)(2i-3)
12. A complex number and its conjugate
imaginary
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
conjugate pairs
13. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
integers
How to find any Power
ln z
interchangeable
14. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
irrational
i²
subtracting complex numbers
15. 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
Imaginary Unit
Polar Coordinates - z
Real and Imaginary Parts
Polar Coordinates - Multiplication by i
16. To simplify the square root of a negative number
i^4
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
zz*
a real number: (a + bi)(a - bi) = a² + b²
17. For real a and b - a + bi =
i²
0 if and only if a = b = 0
the distance from z to the origin in the complex plane
a + bi for some real a and b.
18. E ^ (z2 ln z1)
cos iy
standard form of complex numbers
z1 ^ (z2)
rational
19. 1
Field
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - sin?
i^0
20. Derives z = a+bi
Euler Formula
Affix
subtracting complex numbers
Rational Number
21. R?¹(cos? - isin?)
rational
Polar Coordinates - z?¹
the distance from z to the origin in the complex plane
Polar Coordinates - Division
22. 1st. Rule of Complex Arithmetic
Polar Coordinates - sin?
sin z
i^2 = -1
Complex Addition
23. 2a
Real Numbers
complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
z + z*
24. Like pi
cosh²y - sinh²y
transcendental
How to multiply complex nubers(2+i)(2i-3)
Subfield
25. (a + bi)(c + bi) =
Subfield
Euler Formula
i^2 = -1
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
26. A subset within a field.
Real and Imaginary Parts
Subfield
Polar Coordinates - Multiplication
multiplying complex numbers
27. I = imaginary unit - i² = -1 or i = v-1
i^1
Imaginary Numbers
z1 / z2
Complex Multiplication
28. The product of an imaginary number and its conjugate is
point of inflection
imaginary
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - sin?
29. 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 complex numbers
Irrational Number
Subfield
Complex numbers are points in the plane
30. The complex number z representing a+bi.
Polar Coordinates - z
(a + c) + ( b + d)i
Imaginary Numbers
Affix
31. 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
Argand diagram
How to find any Power
Affix
the complex numbers
32. R^2 = x
Polar Coordinates - Multiplication by i
0 if and only if a = b = 0
Square Root
rational
33. 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
adding complex numbers
0 if and only if a = b = 0
non-integers
has a solution.
34. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
the distance from z to the origin in the complex plane
the complex numbers
x-axis in the complex plane
multiplying complex numbers
35. y / r
Polar Coordinates - sin?
point of inflection
the complex numbers
Rational Number
36. To simplify a complex fraction
cosh²y - sinh²y
Argand diagram
Absolute Value of a Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
37. A² + b² - real and non negative
imaginary
the vector (a -b)
complex numbers
zz*
38. A+bi
conjugate
subtracting complex numbers
Complex Number Formula
How to solve (2i+3)/(9-i)
39. x + iy = r(cos? + isin?) = re^(i?)
Complex Subtraction
Polar Coordinates - z
complex
i^2
40. The square root of -1.
Imaginary Unit
ln z
Complex Exponentiation
natural
41. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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42. A plot of complex numbers as points.
Rational Number
Argand diagram
z + z*
multiply the numerator and the denominator by the complex conjugate of the denominator.
43. We see in this way that the distance between two points z and w in the complex plane is
Field
|z-w|
0 if and only if a = b = 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
44. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
non-integers
How to solve (2i+3)/(9-i)
cos iy
45. Starts at 1 - does not include 0
z - z*
conjugate
natural
conjugate pairs
46. z1z2* / |z2|²
|z| = mod(z)
interchangeable
integers
z1 / z2
47. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Affix
Imaginary Unit
Square Root
48. (e^(iz) - e^(-iz)) / 2i
sin z
Polar Coordinates - Multiplication
Complex numbers are points in the plane
zz*
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 Conjugate
Complex Number
i^2
Polar Coordinates - cos?
50. 5th. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division
multiply the numerator and the denominator by the complex conjugate of the denominator.