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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. 3rd. Rule of Complex Arithmetic
Complex Division
Euler's Formula
How to solve (2i+3)/(9-i)
For real a and b - a + bi = 0 if and only if a = b = 0
2. I
Imaginary Unit
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Subfield
i^1
3. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
multiplying complex numbers
Field
For real a and b - a + bi = 0 if and only if a = b = 0
How to find any Power
4. 1
i^4
Complex numbers are points in the plane
e^(ln z)
Polar Coordinates - Multiplication
5. Not on the numberline
Rules of Complex Arithmetic
non-integers
i^2 = -1
Polar Coordinates - r
6. Real and imaginary numbers
conjugate pairs
complex numbers
Liouville's Theorem -
subtracting complex numbers
7. 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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Arg(z*)
Real and Imaginary Parts
Any polynomial O(xn) - (n > 0)
8. (a + bi) = (c + bi) =
conjugate pairs
0 if and only if a = b = 0
Argand diagram
(a + c) + ( b + d)i
9. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0
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10. Written as fractions - terminating + repeating decimals
Euler Formula
point of inflection
rational
Any polynomial O(xn) - (n > 0)
11. Derives z = a+bi
Euler Formula
cos iy
How to multiply complex nubers(2+i)(2i-3)
x-axis in the complex plane
12. I = imaginary unit - i² = -1 or i = v-1
Rules of Complex Arithmetic
Polar Coordinates - z
integers
Imaginary Numbers
13. Multiply moduli and add arguments
a + bi for some real a and b.
standard form of complex numbers
Euler Formula
Polar Coordinates - Multiplication
14. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
point of inflection
cosh²y - sinh²y
De Moivre's Theorem
Integers
15. A plot of complex numbers as points.
Argand diagram
Field
sin iy
Any polynomial O(xn) - (n > 0)
16. 1
Complex Numbers: Multiply
Euler Formula
v(-1)
cosh²y - sinh²y
17. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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18. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
i^1
0 if and only if a = b = 0
Complex Numbers: Multiply
19. ½(e^(iz) + e^(-iz))
Complex numbers are points in the plane
cos z
standard form of complex numbers
point of inflection
20. Cos n? + i sin n? (for all n integers)
radicals
(cos? +isin?)n
Square Root
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
21. A + bi
standard form of complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Subfield
Complex Multiplication
22. 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.
sin z
a + bi for some real a and b.
Complex numbers are points in the plane
|z| = mod(z)
23. ? = -tan?
i^4
Polar Coordinates - Arg(z*)
i^3
Complex Number Formula
24. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Euler Formula
ln z
subtracting complex numbers
i²
25. Have radical
conjugate pairs
radicals
i^2
rational
26. V(x² + y²) = |z|
subtracting complex numbers
a + bi for some real a and b.
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - r
27. 1
|z| = mod(z)
Polar Coordinates - sin?
interchangeable
i²
28. A complex number may be taken to the power of another complex number.
0 if and only if a = b = 0
imaginary
Complex Exponentiation
Any polynomial O(xn) - (n > 0)
29. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Affix
multiplying complex numbers
the vector (a -b)
i^2
30. 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
cos iy
a + bi for some real a and b.
Polar Coordinates - r
the complex numbers
31. E ^ (z2 ln z1)
e^(ln z)
z1 ^ (z2)
Polar Coordinates - z?¹
i^2
32. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
ln z
|z| = mod(z)
Polar Coordinates - r
33. 4th. Rule of Complex Arithmetic
standard form of complex numbers
Polar Coordinates - z?¹
How to solve (2i+3)/(9-i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
34. For real a and b - a + bi =
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
(cos? +isin?)n
complex
35. A+bi
cosh²y - sinh²y
Complex Number Formula
Polar Coordinates - Multiplication by i
natural
36. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
De Moivre's Theorem
integers
Every complex number has the 'Standard Form': a + bi for some real a and b.
37. The field of all rational and irrational numbers.
Rules of Complex Arithmetic
Subfield
How to add and subtract complex numbers (2-3i)-(4+6i)
Real Numbers
38. 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.'
conjugate pairs
Imaginary Numbers
point of inflection
Complex Number
39. R?¹(cos? - isin?)
Irrational Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers
Polar Coordinates - z?¹
40. 1st. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2 = -1
Polar Coordinates - z?¹
Polar Coordinates - Arg(z*)
41. 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
zz*
Rules of Complex Arithmetic
Liouville's Theorem -
transcendental
42. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
the complex numbers
Subfield
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
43. The product of an imaginary number and its conjugate is
the complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
a real number: (a + bi)(a - bi) = a² + b²
44. A² + b² - real and non negative
complex numbers
Polar Coordinates - z
zz*
z - z*
45. (a + bi)(c + bi) =
i^2
a + bi for some real a and b.
Polar Coordinates - Division
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
46. (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 Number
standard form of complex numbers
47. 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
a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.
Subfield
48. All numbers
the distance from z to the origin in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex
Every complex number has the 'Standard Form': a + bi for some real a and b.
49. Rotates anticlockwise by p/2
Absolute Value of a Complex Number
Complex Division
Polar Coordinates - Multiplication by i
0 if and only if a = b = 0
50. To simplify a complex fraction
imaginary
multiply the numerator and the denominator by the complex conjugate of the denominator.
has a solution.
cosh²y - sinh²y
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