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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. (a + bi)(c + bi) =
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
2. A+bi
cos iy
Polar Coordinates - sin?
Complex Numbers: Multiply
Complex Number Formula
3. Equivalent to an Imaginary Unit.
standard form of complex numbers
Euler Formula
Imaginary number
Complex Multiplication
4. A subset within a field.
i^2
i^2 = -1
Subfield
complex numbers
5. In this amazing number field every algebraic equation in z with complex coefficients
conjugate pairs
We say that c+di and c-di are complex conjugates.
has a solution.
0 if and only if a = b = 0
6. A complex number and its conjugate
imaginary
conjugate pairs
Integers
multiplying complex numbers
7. 1
conjugate
standard form of complex numbers
i^4
Complex Addition
8. To simplify a complex fraction
Imaginary Numbers
imaginary
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Division
9. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
The Complex Numbers
Complex numbers are points in the plane
subtracting complex numbers
10. (e^(iz) - e^(-iz)) / 2i
sin z
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Number Formula
i^4
11. R^2 = x
Rational Number
i^3
x-axis in the complex plane
Square Root
12. 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
Rules of Complex Arithmetic
Absolute Value of a Complex Number
Rational Number
subtracting complex numbers
13. 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.
Subfield
Euler's Formula
Complex Numbers: Multiply
Polar Coordinates - Multiplication
14. Divide moduli and subtract arguments
Polar Coordinates - Division
Polar Coordinates - Multiplication
How to multiply complex nubers(2+i)(2i-3)
Absolute Value of a Complex Number
15. Numbers on a numberline
Argand diagram
De Moivre's Theorem
integers
Real Numbers
16. Any number not rational
i^2
natural
irrational
Polar Coordinates - Division
17. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - Division
Subfield
De Moivre's Theorem
Field
18. (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
Complex Division
How to solve (2i+3)/(9-i)
0 if and only if a = b = 0
sin z
19. Written as fractions - terminating + repeating decimals
rational
complex
The Complex Numbers
standard form of complex numbers
20. We see in this way that the distance between two points z and w in the complex plane is
i^2 = -1
|z-w|
e^(ln z)
Integers
21. ½(e^(-y) +e^(y)) = cosh y
i^3
cos iy
We say that c+di and c-di are complex conjugates.
a + bi for some real a and b.
22. The square root of -1.
Imaginary Unit
The Complex Numbers
zz*
How to solve (2i+3)/(9-i)
23. Not on the numberline
Imaginary Numbers
non-integers
i^0
Every complex number has the 'Standard Form': a + bi for some real a and b.
24. Starts at 1 - does not include 0
Integers
Polar Coordinates - r
z - z*
natural
25. 1
i^2
sin iy
conjugate pairs
rational
26. 1
real
How to multiply complex nubers(2+i)(2i-3)
i²
complex numbers
27. No i
0 if and only if a = b = 0
real
zz*
ln z
28. 1
standard form of complex numbers
How to solve (2i+3)/(9-i)
sin z
i^0
29. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Imaginary Numbers
Absolute Value of a Complex Number
Complex numbers are points in the plane
Euler's Formula
30. (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
Real and Imaginary Parts
Complex Numbers: Multiply
Imaginary Numbers
How to multiply complex nubers(2+i)(2i-3)
31. A + bi
standard form of complex numbers
i²
interchangeable
Imaginary Unit
32. 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
Complex numbers are points in the plane
Complex Numbers: Add & subtract
z1 ^ (z2)
the complex numbers
33. 1
standard form of complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(cos? +isin?)n
cosh²y - sinh²y
34. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
How to add and subtract complex numbers (2-3i)-(4+6i)
natural
Complex Number
35. 3rd. Rule of Complex Arithmetic
cos iy
How to multiply complex nubers(2+i)(2i-3)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0
36. Has exactly n roots by the fundamental theorem of algebra
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
Any polynomial O(xn) - (n > 0)
37. 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.
ln z
Imaginary number
the vector (a -b)
Complex numbers are points in the plane
38. ½(e^(iz) + e^(-iz))
Imaginary number
cos z
(cos? +isin?)n
Polar Coordinates - r
39. x + iy = r(cos? + isin?) = re^(i?)
imaginary
conjugate
-1
Polar Coordinates - z
40. ? = -tan?
-1
cosh²y - sinh²y
complex
Polar Coordinates - Arg(z*)
41. V(x² + y²) = |z|
Euler's Formula
non-integers
Irrational Number
Polar Coordinates - r
42. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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43. 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
Liouville's Theorem -
Polar Coordinates - r
Complex Numbers: Add & subtract
natural
44. A plot of complex numbers as points.
Complex Multiplication
Complex numbers are points in the plane
Imaginary number
Argand diagram
45. V(zz*) = v(a² + b²)
a + bi for some real a and b.
|z| = mod(z)
We say that c+di and c-di are complex conjugates.
a real number: (a + bi)(a - bi) = a² + b²
46. To simplify the square root of a negative number
z1 ^ (z2)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cosh²y - sinh²y
Polar Coordinates - sin?
47. 2nd. Rule of Complex Arithmetic
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48. E ^ (z2 ln z1)
Argand diagram
z1 ^ (z2)
Euler Formula
i^1
49. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Imaginary Unit
a + bi for some real a and b.
Complex Exponentiation
The Complex Numbers
50. A complex number may be taken to the power of another complex number.
Complex Conjugate
i^4
Complex Exponentiation
sin z