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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.
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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. Multiply moduli and add arguments
conjugate
Polar Coordinates - Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i
rational
2. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
-1
has a solution.
3. 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
Liouville's Theorem -
the complex numbers
Polar Coordinates - cos?
ln z
4. (a + bi)(c + bi) =
Roots of Unity
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to multiply complex nubers(2+i)(2i-3)
5. I = imaginary unit - i² = -1 or i = v-1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Imaginary Numbers
Square Root
Polar Coordinates - Arg(z*)
6. Any number not rational
cos iy
natural
irrational
Complex Multiplication
7. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n
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8. 2nd. Rule of Complex Arithmetic
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9. Equivalent to an Imaginary Unit.
Polar Coordinates - Multiplication by i
How to add and subtract complex numbers (2-3i)-(4+6i)
standard form of complex numbers
Imaginary number
10. For real a and b - a + bi =
multiplying complex numbers
Euler Formula
e^(ln z)
0 if and only if a = b = 0
11. 1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
cosh²y - sinh²y
multiply the numerator and the denominator by the complex conjugate of the denominator.
Euler's Formula
12. Root negative - has letter i
v(-1)
imaginary
Argand diagram
Square Root
13. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
How to add and subtract complex numbers (2-3i)-(4+6i)
We say that c+di and c-di are complex conjugates.
Euler Formula
14. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
Any polynomial O(xn) - (n > 0)
Rules of Complex Arithmetic
sin iy
15. A+bi
z - z*
subtracting complex numbers
Argand diagram
Complex Number Formula
16. We can also think of the point z= a+ ib as
z - z*
Complex Division
the vector (a -b)
Polar Coordinates - cos?
17. Numbers on a numberline
integers
Real and Imaginary Parts
sin z
irrational
18. z1z2* / |z2|²
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - r
z1 / z2
i^1
19. 2ib
For real a and b - a + bi = 0 if and only if a = b = 0
ln z
z - z*
four different numbers: i - -i - 1 - and -1.
20. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
natural
i^1
Absolute Value of a Complex Number
Real Numbers
21. 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.
Complex Numbers: Add & subtract
Integers
Polar Coordinates - Arg(z*)
How to find any Power
22. 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.'
adding complex numbers
i^3
Complex Number
0 if and only if a = b = 0
23. The reals are just the
x-axis in the complex plane
i^3
(a + c) + ( b + d)i
transcendental
24. 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
Any polynomial O(xn) - (n > 0)
radicals
Real and Imaginary Parts
Euler's Formula
25. 1st. Rule of Complex Arithmetic
v(-1)
x-axis in the complex plane
i^2 = -1
Irrational Number
26. E^(ln r) e^(i?) e^(2pin)
e^(ln z)
Liouville's Theorem -
Complex Subtraction
Complex Addition
27. Real and imaginary numbers
complex numbers
four different numbers: i - -i - 1 - and -1.
Imaginary Numbers
integers
28. 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
i^2
-1
Square Root
subtracting complex numbers
29. When two complex numbers are subtracted from one another.
Complex Subtraction
z - z*
For real a and b - a + bi = 0 if and only if a = b = 0
-1
30. To simplify a complex fraction
point of inflection
Rational Number
Polar Coordinates - Division
multiply the numerator and the denominator by the complex conjugate of the denominator.
31. x + iy = r(cos? + isin?) = re^(i?)
Complex Addition
real
Polar Coordinates - z
adding complex numbers
32. Has exactly n roots by the fundamental theorem of algebra
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)
zz*
Roots of Unity
33. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Division
imaginary
34. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
conjugate pairs
i^2 = -1
Field
i^3
35. ½(e^(-y) +e^(y)) = cosh y
radicals
How to solve (2i+3)/(9-i)
cos iy
Complex Multiplication
36. 4th. Rule of Complex Arithmetic
Polar Coordinates - cos?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a real number: (a + bi)(a - bi) = a² + b²
37. Imaginary number
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38. We see in this way that the distance between two points z and w in the complex plane is
Complex Numbers: Add & subtract
z1 / z2
|z-w|
cos iy
39. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
'i'
Complex Addition
i^4
How to add and subtract complex numbers (2-3i)-(4+6i)
40. Cos n? + i sin n? (for all n integers)
v(-1)
(cos? +isin?)n
complex
-1
41. I
Complex Numbers: Multiply
Imaginary Numbers
i^1
Roots of Unity
42. A plot of complex numbers as points.
Argand diagram
(a + bi) = (c + bi) = (a + c) + ( b + d)i
transcendental
Integers
43. V(x² + y²) = |z|
Polar Coordinates - Division
Polar Coordinates - r
zz*
non-integers
44. (e^(-y) - e^(y)) / 2i = i sinh y
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + c) + ( b + d)i
sin iy
subtracting complex numbers
45. A complex number may be taken to the power of another complex number.
Complex Numbers: Add & subtract
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Exponentiation
multiplying complex numbers
46. When two complex numbers are divided.
Imaginary Unit
Complex Number
zz*
Complex Division
47. 1
ln z
Complex Subtraction
Complex Division
i^0
48. R^2 = x
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i²
Affix
Square Root
49. A² + b² - real and non negative
x-axis in the complex plane
sin z
complex
zz*
50. R?¹(cos? - isin?)
i^2 = -1
radicals
interchangeable
Polar Coordinates - z?¹
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