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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. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
sin z
Complex numbers are points in the plane
radicals
2. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - cos?
subtracting complex numbers
integers
3. The reals are just the
x-axis in the complex plane
z - z*
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to find any Power
4. R?¹(cos? - isin?)
Polar Coordinates - z?¹
multiplying complex numbers
z1 ^ (z2)
i^0
5. 1
Integers
(a + c) + ( b + d)i
sin z
i²
6. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
has a solution.
standard form of complex numbers
Euler's Formula
Absolute Value of a Complex Number
7. 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
complex numbers
subtracting complex numbers
Complex Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
8. Equivalent to an Imaginary Unit.
Imaginary number
Complex Conjugate
a real number: (a + bi)(a - bi) = a² + b²
transcendental
9. Derives z = a+bi
Imaginary Numbers
integers
Euler Formula
Complex Conjugate
10. R^2 = x
point of inflection
the vector (a -b)
Square Root
integers
11. 1
i^2
interchangeable
Argand diagram
standard form of complex numbers
12. 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.
cosh²y - sinh²y
v(-1)
i^4
Complex numbers are points in the plane
13. Have radical
radicals
-1
complex numbers
Complex Subtraction
14. z1z2* / |z2|²
Argand diagram
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 / z2
e^(ln z)
15. A complex number and its conjugate
standard form of complex numbers
Affix
Integers
conjugate pairs
16. (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)
integers
Polar Coordinates - z?¹
Euler Formula
17. x / r
Polar Coordinates - cos?
Polar Coordinates - Division
z1 ^ (z2)
Imaginary number
18. Every complex number has the 'Standard Form':
a real number: (a + bi)(a - bi) = a² + b²
a + bi for some real a and b.
point of inflection
Liouville's Theorem -
19. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
-1
Roots of Unity
Polar Coordinates - z
20. x + iy = r(cos? + isin?) = re^(i?)
standard form of complex numbers
Polar Coordinates - z
irrational
For real a and b - a + bi = 0 if and only if a = b = 0
21. For real a and b - a + bi =
Polar Coordinates - z?¹
z - z*
0 if and only if a = b = 0
the distance from z to the origin in the complex plane
22. When two complex numbers are added together.
Complex Addition
Affix
natural
Irrational Number
23. 3
How to multiply complex nubers(2+i)(2i-3)
i^3
sin z
four different numbers: i - -i - 1 - and -1.
24. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
v(-1)
Polar Coordinates - Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.
25. (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 Multiplication
How to solve (2i+3)/(9-i)
cos iy
Complex Numbers: Multiply
26. Not on the numberline
zz*
Liouville's Theorem -
non-integers
0 if and only if a = b = 0
27. 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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28. Real and imaginary numbers
point of inflection
Imaginary Numbers
complex numbers
i^3
29. To simplify a complex fraction
i^0
Complex Numbers: Add & subtract
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiply the numerator and the denominator by the complex conjugate of the denominator.
30. A+bi
Complex Number Formula
Complex Numbers: Multiply
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - sin?
31. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
standard form of complex numbers
conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
32. Written as fractions - terminating + repeating decimals
Square Root
rational
complex
|z| = mod(z)
33. 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
conjugate pairs
Rules of Complex Arithmetic
complex numbers
34. A² + b² - real and non negative
zz*
'i'
Rational Number
Real and Imaginary Parts
35. No i
real
radicals
Complex numbers are points in the plane
i^2 = -1
36. 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
0 if and only if a = b = 0
Rules of Complex Arithmetic
conjugate
Polar Coordinates - Multiplication
37. 1
imaginary
Integers
cosh²y - sinh²y
i^1
38. The field of all rational and irrational numbers.
Real and Imaginary Parts
Affix
Real Numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
39. y / r
Polar Coordinates - sin?
z - z*
Square Root
v(-1)
40. The modulus of the complex number z= a + ib now can be interpreted as
integers
the distance from z to the origin in the complex plane
|z-w|
irrational
41. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
cosh²y - sinh²y
Every complex number has the 'Standard Form': a + bi for some real a and b.
Any polynomial O(xn) - (n > 0)
42. A plot of complex numbers as points.
conjugate
Polar Coordinates - cos?
Argand diagram
Absolute Value of a Complex Number
43. When two complex numbers are subtracted from one another.
irrational
radicals
conjugate pairs
Complex Subtraction
44. When two complex numbers are multipiled together.
ln z
rational
Complex Multiplication
z + z*
45. (a + bi) = (c + bi) =
z + z*
(a + c) + ( b + d)i
Polar Coordinates - Multiplication
sin iy
46. I
a + bi for some real a and b.
imaginary
the distance from z to the origin in the complex plane
i^1
47. Multiply moduli and add arguments
z1 / z2
Complex Multiplication
Polar Coordinates - Multiplication
De Moivre's Theorem
48. 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 Number
cosh²y - sinh²y
complex numbers
radicals
49. I
How to multiply complex nubers(2+i)(2i-3)
a real number: (a + bi)(a - bi) = a² + b²
v(-1)
(cos? +isin?)n
50. Starts at 1 - does not include 0
non-integers
natural
We say that c+di and c-di are complex conjugates.
a real number: (a + bi)(a - bi) = a² + b²