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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. 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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2. Cos n? + i sin n? (for all n integers)
complex
(cos? +isin?)n
multiply the numerator and the denominator by the complex conjugate of the denominator.
Field
3. A number that can be expressed as a fraction p/q where q is not equal to 0.
i^0
Rational Number
complex
i^2
4. 4th. Rule of Complex Arithmetic
Polar Coordinates - r
cos iy
point of inflection
(a + bi) = (c + bi) = (a + c) + ( b + d)i
5. Derives z = a+bi
adding complex numbers
natural
Euler Formula
Complex Exponentiation
6. The square root of -1.
the vector (a -b)
Polar Coordinates - z?¹
Imaginary Unit
point of inflection
7. We can also think of the point z= a+ ib as
-1
Complex Addition
the vector (a -b)
For real a and b - a + bi = 0 if and only if a = b = 0
8. 1
Polar Coordinates - Division
i²
complex
Complex Numbers: Multiply
9. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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10. No i
sin z
real
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
11. ½(e^(-y) +e^(y)) = cosh y
cos iy
radicals
Any polynomial O(xn) - (n > 0)
z + z*
12. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^0
Complex numbers are points in the plane
Real and Imaginary Parts
13. The modulus of the complex number z= a + ib now can be interpreted as
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos z
Argand diagram
the distance from z to the origin in the complex plane
14. Written as fractions - terminating + repeating decimals
rational
Imaginary Unit
De Moivre's Theorem
the complex numbers
15. Root negative - has letter i
(a + c) + ( b + d)i
imaginary
sin iy
natural
16. Like pi
transcendental
(cos? +isin?)n
sin z
Rational Number
17. When two complex numbers are divided.
Complex Division
radicals
Square Root
e^(ln z)
18. Has exactly n roots by the fundamental theorem of algebra
z1 ^ (z2)
Euler's Formula
natural
Any polynomial O(xn) - (n > 0)
19. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
conjugate pairs
-1
How to multiply complex nubers(2+i)(2i-3)
How to add and subtract complex numbers (2-3i)-(4+6i)
20. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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21. Given (4-2i) the complex conjugate would be (4+2i)
0 if and only if a = b = 0
Complex Conjugate
subtracting complex numbers
i^0
22. ½(e^(iz) + e^(-iz))
Complex Numbers: Add & subtract
Euler Formula
cos z
Polar Coordinates - r
23. For real a and b - a + bi =
rational
sin z
0 if and only if a = b = 0
Euler's Formula
24. The field of all rational and irrational numbers.
Real Numbers
Euler Formula
imaginary
Polar Coordinates - r
25. Numbers on a numberline
Complex Addition
irrational
Argand diagram
integers
26. In this amazing number field every algebraic equation in z with complex coefficients
point of inflection
Rules of Complex Arithmetic
has a solution.
a real number: (a + bi)(a - bi) = a² + b²
27. I = imaginary unit - i² = -1 or i = v-1
Euler Formula
Imaginary Numbers
Polar Coordinates - Arg(z*)
Roots of Unity
28. 5th. Rule of Complex Arithmetic
|z| = mod(z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
sin iy
complex
29. A number that cannot be expressed as a fraction for any integer.
Complex Numbers: Multiply
Irrational Number
Subfield
subtracting complex numbers
30. ? = -tan?
De Moivre's Theorem
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Arg(z*)
31. Real and imaginary numbers
complex numbers
cos iy
|z| = mod(z)
Affix
32. A complex number and its conjugate
0 if and only if a = b = 0
complex
conjugate pairs
How to add and subtract complex numbers (2-3i)-(4+6i)
33. 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.
Complex Exponentiation
complex numbers
Polar Coordinates - z
Complex numbers are points in the plane
34. 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
How to solve (2i+3)/(9-i)
a real number: (a + bi)(a - bi) = a² + b²
Rules of Complex Arithmetic
Polar Coordinates - z?¹
35. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
'i'
Complex Addition
cosh²y - sinh²y
multiplying complex numbers
36. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
i^1
rational
The Complex Numbers
How to multiply complex nubers(2+i)(2i-3)
37. A+bi
standard form of complex numbers
Field
sin iy
Complex Number Formula
38. When two complex numbers are added together.
a + bi for some real a and b.
natural
Complex Addition
rational
39. A plot of complex numbers as points.
Argand diagram
rational
|z-w|
Rules of Complex Arithmetic
40. Any number not rational
Polar Coordinates - z
Imaginary Numbers
irrational
Rules of Complex Arithmetic
41. Rotates anticlockwise by p/2
Field
i^4
(cos? +isin?)n
Polar Coordinates - Multiplication by i
42. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
cos iy
Complex Division
Absolute Value of a Complex Number
Polar Coordinates - r
43. A subset within a field.
Imaginary Unit
Every complex number has the 'Standard Form': a + bi for some real a and b.
Subfield
the vector (a -b)
44. Multiply moduli and add arguments
De Moivre's Theorem
Polar Coordinates - Multiplication
v(-1)
Complex Division
45. z1z2* / |z2|²
complex
z1 / z2
Argand diagram
real
46. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*
Imaginary Numbers
the vector (a -b)
47. E ^ (z2 ln z1)
Complex numbers are points in the plane
(cos? +isin?)n
interchangeable
z1 ^ (z2)
48. E^(ln r) e^(i?) e^(2pin)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)
standard form of complex numbers
0 if and only if a = b = 0
49. Have radical
point of inflection
conjugate pairs
radicals
e^(ln z)
50. V(zz*) = v(a² + b²)
Field
Rules of Complex Arithmetic
|z| = mod(z)
|z-w|