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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. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Euler's Formula
'i'
cos z
2. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
i^2 = -1
Polar Coordinates - cos?
Field
Any polynomial O(xn) - (n > 0)
3. Root negative - has letter i
a real number: (a + bi)(a - bi) = a² + b²
z1 / z2
imaginary
conjugate
4. (a + bi) = (c + bi) =
rational
Complex Exponentiation
(a + c) + ( b + d)i
real
5. V(x² + y²) = |z|
Polar Coordinates - r
z + z*
Irrational Number
i^2 = -1
6. Derives z = a+bi
zz*
Euler Formula
Real and Imaginary Parts
'i'
7. 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.
Subfield
irrational
How to find any Power
subtracting complex numbers
8. (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)
transcendental
z1 ^ (z2)
0 if and only if a = b = 0
9. 3rd. Rule of Complex Arithmetic
Rational Number
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Exponentiation
integers
10. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
real
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
subtracting complex numbers
11. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i^3
can't get out of the complex numbers by adding (or subtracting) or multiplying two
sin z
Integers
12. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Multiplication
z1 / z2
Every complex number has the 'Standard Form': a + bi for some real a and b.
The Complex Numbers
13. A complex number and its conjugate
How to find any Power
conjugate pairs
Real and Imaginary Parts
rational
14. To simplify the square root of a negative number
Rules of Complex Arithmetic
complex
Liouville's Theorem -
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
15. For real a and b - a + bi =
Complex Subtraction
Affix
0 if and only if a = b = 0
Argand diagram
16. The modulus of the complex number z= a + ib now can be interpreted as
Complex Conjugate
Complex Division
the distance from z to the origin in the complex plane
Polar Coordinates - z?¹
17. 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
x-axis in the complex plane
z1 / z2
Real Numbers
18. Every complex number has the 'Standard Form':
multiplying complex numbers
a + bi for some real a and b.
i^1
point of inflection
19. A subset within a field.
i^3
a + bi for some real a and b.
Subfield
standard form of complex numbers
20. Any number not rational
Polar Coordinates - z?¹
irrational
Square Root
The Complex Numbers
21. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiplying complex numbers
i²
point of inflection
22. Where the curvature of the graph changes
How to multiply complex nubers(2+i)(2i-3)
point of inflection
Irrational Number
ln z
23. Numbers on a numberline
imaginary
Rational Number
integers
-1
24. (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
0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)
non-integers
Complex Conjugate
25. We see in this way that the distance between two points z and w in the complex plane is
Any polynomial O(xn) - (n > 0)
|z-w|
subtracting complex numbers
cosh²y - sinh²y
26. Written as fractions - terminating + repeating decimals
rational
Complex Division
complex numbers
i^1
27. ½(e^(-y) +e^(y)) = cosh y
cos iy
Polar Coordinates - Arg(z*)
sin z
ln z
28. I^2 =
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
-1
Polar Coordinates - r
Euler's Formula
29. 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.
sin iy
transcendental
point of inflection
30. 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.
Euler Formula
Complex Addition
Polar Coordinates - z?¹
Complex Numbers: Multiply
31. xpressions such as ``the complex number z'' - and ``the point z'' are now
i^1
Subfield
interchangeable
Complex Subtraction
32. Imaginary number
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33. x / r
cos z
Polar Coordinates - cos?
Polar Coordinates - sin?
Complex Numbers: Add & subtract
34. A number that can be expressed as a fraction p/q where q is not equal to 0.
Roots of Unity
Rational Number
rational
z1 ^ (z2)
35. Divide moduli and subtract arguments
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division
Polar Coordinates - z?¹
Every complex number has the 'Standard Form': a + bi for some real a and b.
36. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
real
conjugate
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to solve (2i+3)/(9-i)
37. 1
(a + c) + ( b + d)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
the complex numbers
i^0
38. The square root of -1.
How to multiply complex nubers(2+i)(2i-3)
Imaginary Unit
a + bi for some real a and b.
Subfield
39. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
cosh²y - sinh²y
Complex Number Formula
adding complex numbers
40. I
We say that c+di and c-di are complex conjugates.
i^2
v(-1)
'i'
41. Equivalent to an Imaginary Unit.
Polar Coordinates - Arg(z*)
0 if and only if a = b = 0
Absolute Value of a Complex Number
Imaginary number
42. Has exactly n roots by the fundamental theorem of algebra
the distance from z to the origin in the complex plane
Any polynomial O(xn) - (n > 0)
integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
43. 5th. Rule of Complex Arithmetic
i^2
Real Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|
44. y / r
(cos? +isin?)n
z - z*
Liouville's Theorem -
Polar Coordinates - sin?
45. When two complex numbers are divided.
Complex Division
z1 ^ (z2)
Complex Multiplication
subtracting complex numbers
46. A+bi
Complex Number Formula
adding complex numbers
Rules of Complex Arithmetic
Complex Exponentiation
47. A² + b² - real and non negative
-1
e^(ln z)
zz*
sin iy
48. When two complex numbers are added together.
the distance from z to the origin in the complex plane
ln z
Rules of Complex Arithmetic
Complex Addition
49. Not on the numberline
transcendental
non-integers
Rules of Complex Arithmetic
cos iy
50. (e^(iz) - e^(-iz)) / 2i
interchangeable
Euler's Formula
sin z
The Complex Numbers