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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
Start Test
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. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
subtracting complex numbers
e^(ln z)
How to multiply complex nubers(2+i)(2i-3)
2. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
i^2
interchangeable
|z-w|
3. Any number not rational
How to solve (2i+3)/(9-i)
z1 / z2
adding complex numbers
irrational
4. 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.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
0 if and only if a = b = 0
Complex numbers are points in the plane
zz*
5. A + bi
standard form of complex numbers
has a solution.
Polar Coordinates - Division
z + z*
6. When two complex numbers are divided.
Imaginary Numbers
sin z
interchangeable
Complex Division
7. The modulus of the complex number z= a + ib now can be interpreted as
adding complex numbers
the distance from z to the origin in the complex plane
a real number: (a + bi)(a - bi) = a² + b²
standard form of complex numbers
8. In this amazing number field every algebraic equation in z with complex coefficients
rational
has a solution.
v(-1)
De Moivre's Theorem
9. V(zz*) = v(a² + b²)
|z| = mod(z)
(cos? +isin?)n
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Field
10. A subset within a field.
'i'
Subfield
zz*
e^(ln z)
11. (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
Polar Coordinates - sin?
multiplying complex numbers
Rules of Complex Arithmetic
How to multiply complex nubers(2+i)(2i-3)
12. x / r
Complex Subtraction
Polar Coordinates - cos?
Polar Coordinates - Multiplication by i
integers
13. No i
interchangeable
Imaginary Unit
real
How to find any Power
14. 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.
Square Root
Complex Subtraction
rational
15. xpressions such as ``the complex number z'' - and ``the point z'' are now
ln z
cos iy
0 if and only if a = b = 0
interchangeable
16. Starts at 1 - does not include 0
How to solve (2i+3)/(9-i)
Complex Division
natural
-1
17. 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
Complex Numbers: Multiply
adding complex numbers
Rules of Complex Arithmetic
Integers
18. Real and imaginary numbers
complex numbers
Polar Coordinates - Multiplication
Imaginary number
cosh²y - sinh²y
19. A² + b² - real and non negative
z + z*
cos iy
zz*
How to multiply complex nubers(2+i)(2i-3)
20. Have radical
Polar Coordinates - sin?
has a solution.
Square Root
radicals
21. (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
non-integers
How to solve (2i+3)/(9-i)
Imaginary Unit
Complex Number Formula
22. ½(e^(iz) + e^(-iz))
i^2 = -1
Field
cos z
sin iy
23. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Multiplication by i
conjugate pairs
natural
24. x + iy = r(cos? + isin?) = re^(i?)
Complex numbers are points in the plane
Complex Number
Polar Coordinates - z
Complex Addition
25. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Multiplication
sin z
Complex Multiplication
Polar Coordinates - Multiplication by i
26. 2nd. Rule of Complex Arithmetic
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27. Every complex number has the 'Standard Form':
Square Root
Imaginary Numbers
x-axis in the complex plane
a + bi for some real a and b.
28. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)
Rules of Complex Arithmetic
ln z
29. The field of all rational and irrational numbers.
conjugate
sin z
Real Numbers
conjugate pairs
30. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
z1 / z2
We say that c+di and c-di are complex conjugates.
Complex Number Formula
31. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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32. A number that cannot be expressed as a fraction for any integer.
conjugate pairs
(cos? +isin?)n
Liouville's Theorem -
Irrational Number
33. The complex number z representing a+bi.
Affix
Complex Division
|z-w|
Polar Coordinates - r
34. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Argand diagram
Absolute Value of a Complex Number
How to solve (2i+3)/(9-i)
35. 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
subtracting complex numbers
Imaginary Unit
has a solution.
How to solve (2i+3)/(9-i)
36. The square root of -1.
multiplying complex numbers
i^3
Imaginary Unit
|z| = mod(z)
37. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - z?¹
|z| = mod(z)
i^0
38. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - z?¹
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Unit
Field
39. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex numbers are points in the plane
We say that c+di and c-di are complex conjugates.
Absolute Value of a Complex Number
rational
40. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
i^0
Irrational Number
subtracting complex numbers
41. When two complex numbers are added together.
Polar Coordinates - Arg(z*)
Euler Formula
sin iy
Complex Addition
42. 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
transcendental
Complex Division
Integers
43. I
cos iy
i^1
Irrational Number
Every complex number has the 'Standard Form': a + bi for some real a and b.
44. Root negative - has letter i
Rules of Complex Arithmetic
imaginary
Argand diagram
How to solve (2i+3)/(9-i)
45. 1
i^2
point of inflection
How to multiply complex nubers(2+i)(2i-3)
Complex Subtraction
46. Equivalent to an Imaginary Unit.
Complex Addition
Field
Imaginary number
Complex Subtraction
47. 1
Rational Number
Complex Subtraction
i²
Polar Coordinates - Multiplication
48. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
We say that c+di and c-di are complex conjugates.
conjugate
Rational Number
'i'
49. I^2 =
Affix
radicals
-1
rational
50. A number that can be expressed as a fraction p/q where q is not equal to 0.
How to find any Power
Rational Number
the distance from z to the origin in the complex plane
Real and Imaginary Parts