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Test your basic knowledge |
CLEP General Mathematics: Complex Numbers
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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. The complex number z representing a+bi.
Complex Multiplication
Affix
transcendental
complex numbers
2. z1z2* / |z2|²
Every complex number has the 'Standard Form': a + bi for some real a and b.
Liouville's Theorem -
the complex numbers
z1 / z2
3. 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.
Complex numbers are points in the plane
How to multiply complex nubers(2+i)(2i-3)
standard form of complex numbers
Polar Coordinates - cos?
4. 1
z1 ^ (z2)
Complex Numbers: Multiply
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i²
5. V(zz*) = v(a² + b²)
Complex Numbers: Add & subtract
Imaginary Unit
cos iy
|z| = mod(z)
6. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex Division
Complex Numbers: Multiply
ln z
Rules of Complex Arithmetic
7. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Rational Number
Imaginary Unit
subtracting complex numbers
The Complex Numbers
8. 1
Polar Coordinates - cos?
Complex numbers are points in the plane
irrational
i^2
9. Every complex number has the 'Standard Form':
a + bi for some real a and b.
standard form of complex numbers
Polar Coordinates - Division
Every complex number has the 'Standard Form': a + bi for some real a and b.
10. All the powers of i can be written as
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Square Root
four different numbers: i - -i - 1 - and -1.
cos z
11. R?¹(cos? - isin?)
Complex Numbers: Add & subtract
Any polynomial O(xn) - (n > 0)
Euler Formula
Polar Coordinates - z?¹
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
Subfield
i^0
Rules of Complex Arithmetic
cos z
13. E ^ (z2 ln z1)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
transcendental
z1 ^ (z2)
Every complex number has the 'Standard Form': a + bi for some real a and b.
14. Not on the numberline
non-integers
conjugate pairs
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Multiplication
15. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
Imaginary Numbers
Complex Multiplication
adding complex numbers
i^4
16. A+bi
Field
Any polynomial O(xn) - (n > 0)
Affix
Complex Number Formula
17. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^1
18. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - Arg(z*)
|z-w|
The Complex Numbers
Rational Number
19. 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
the distance from z to the origin in the complex plane
Argand diagram
v(-1)
Complex Numbers: Add & subtract
20. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Polar Coordinates - Multiplication by i
adding complex numbers
Complex Exponentiation
21. Multiply moduli and add arguments
Polar Coordinates - Multiplication
How to solve (2i+3)/(9-i)
Complex Conjugate
ln z
22. When two complex numbers are added together.
Complex Addition
four different numbers: i - -i - 1 - and -1.
i²
a + bi for some real a and b.
23. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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24. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane
i^0
(cos? +isin?)n
25. All numbers
Affix
Euler's Formula
We say that c+di and c-di are complex conjugates.
complex
26. Any number not rational
Complex Addition
imaginary
sin iy
irrational
27. 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
z1 ^ (z2)
Complex Exponentiation
the complex numbers
Euler Formula
28. A² + b² - real and non negative
Liouville's Theorem -
zz*
z1 ^ (z2)
point of inflection
29. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Subfield
Polar Coordinates - z?¹
multiplying complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
30. xpressions such as ``the complex number z'' - and ``the point z'' are now
ln z
i^0
interchangeable
e^(ln z)
31. In this amazing number field every algebraic equation in z with complex coefficients
i^3
How to find any Power
x-axis in the complex plane
has a solution.
32. R^2 = x
De Moivre's Theorem
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)
Square Root
33. Like pi
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
z1 / z2
transcendental
34. A subset within a field.
Imaginary Numbers
Subfield
complex
radicals
35. Derives z = a+bi
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler Formula
Real and Imaginary Parts
0 if and only if a = b = 0
36. 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
Real and Imaginary Parts
How to solve (2i+3)/(9-i)
sin z
zz*
37. Real and imaginary numbers
-1
Rules of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers
38. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Integers
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Arg(z*)
Field
39. I = imaginary unit - i² = -1 or i = v-1
Argand diagram
Imaginary Numbers
How to multiply complex nubers(2+i)(2i-3)
i^1
40. Starts at 1 - does not include 0
point of inflection
natural
integers
Affix
41. For real a and b - a + bi =
Liouville's Theorem -
point of inflection
Complex Number
0 if and only if a = b = 0
42. ½(e^(-y) +e^(y)) = cosh y
Complex Exponentiation
cos iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - cos?
43. Given (4-2i) the complex conjugate would be (4+2i)
We say that c+di and c-di are complex conjugates.
Complex Division
natural
Complex Conjugate
44. When two complex numbers are subtracted from one another.
Complex Subtraction
Imaginary Unit
the distance from z to the origin in the complex plane
Complex Number
45. When two complex numbers are multipiled together.
Complex Multiplication
Polar Coordinates - Arg(z*)
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
46. I
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z?¹
Imaginary number
v(-1)
47. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
has a solution.
transcendental
Absolute Value of a Complex Number
adding complex numbers
48. 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
The Complex Numbers
Complex Conjugate
Complex Number Formula
subtracting complex numbers
49. Has exactly n roots by the fundamental theorem of algebra
Rules of Complex Arithmetic
Real and Imaginary Parts
Any polynomial O(xn) - (n > 0)
integers
50. A + bi
Complex Addition
Complex Multiplication
standard form of complex numbers
radicals