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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.
Affix
adding complex numbers
conjugate
the distance from z to the origin in the complex plane
2. The field of all rational and irrational numbers.
|z-w|
Absolute Value of a Complex Number
Real Numbers
z1 ^ (z2)
3. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
-1
conjugate
point of inflection
4. Multiply moduli and add arguments
Polar Coordinates - r
i²
Polar Coordinates - Multiplication
cos z
5. Like pi
(a + c) + ( b + d)i
transcendental
i^3
Polar Coordinates - Multiplication by i
6. (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)
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers
i^1
7. 1
Square Root
i^0
cosh²y - sinh²y
We say that c+di and c-di are complex conjugates.
8. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate
sin z
i^2 = -1
9. xpressions such as ``the complex number z'' - and ``the point z'' are now
For real a and b - a + bi = 0 if and only if a = b = 0
interchangeable
Polar Coordinates - Division
Imaginary Numbers
10. Numbers on a numberline
|z| = mod(z)
integers
ln z
cos z
11. ½(e^(-y) +e^(y)) = cosh y
radicals
Complex Number Formula
point of inflection
cos iy
12. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
i^3
Polar Coordinates - r
multiplying complex numbers
four different numbers: i - -i - 1 - and -1.
13. Rotates anticlockwise by p/2
Imaginary Numbers
Polar Coordinates - Multiplication by i
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
14. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Complex Numbers: Multiply
multiplying complex numbers
Polar Coordinates - cos?
15. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Complex Exponentiation
transcendental
Complex Numbers: Add & subtract
16. The product of an imaginary number and its conjugate is
cos iy
Complex Multiplication
z + z*
a real number: (a + bi)(a - bi) = a² + b²
17. A subset within a field.
Liouville's Theorem -
Subfield
natural
subtracting complex numbers
18. 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
subtracting complex numbers
How to multiply complex nubers(2+i)(2i-3)
Complex numbers are points in the plane
x-axis in the complex plane
19. Starts at 1 - does not include 0
How to find any Power
Liouville's Theorem -
e^(ln z)
natural
20. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Square Root
Complex Number Formula
Polar Coordinates - Arg(z*)
21. A + bi
multiplying complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
conjugate
standard form of complex numbers
22. When two complex numbers are multipiled together.
subtracting complex numbers
Complex Multiplication
v(-1)
|z-w|
23. All the powers of i can be written as
Euler Formula
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
24. 1st. Rule of Complex Arithmetic
cosh²y - sinh²y
conjugate
Polar Coordinates - Arg(z*)
i^2 = -1
25. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Number
transcendental
i²
The Complex Numbers
26. y / r
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - sin?
How to find any Power
four different numbers: i - -i - 1 - and -1.
27. No i
Rules of Complex Arithmetic
i²
real
Polar Coordinates - cos?
28. 3
Complex Numbers: Add & subtract
Polar Coordinates - Multiplication
|z-w|
i^3
29. 1
irrational
v(-1)
i^0
Integers
30. All numbers
Rules of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex
i^1
31. A plot of complex numbers as points.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
irrational
Argand diagram
zz*
32. z1z2* / |z2|²
z1 / z2
rational
cos iy
Complex Addition
33. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - sin?
sin iy
(cos? +isin?)n
natural
34. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
v(-1)
integers
Square Root
35. x / r
Polar Coordinates - cos?
z - z*
imaginary
Polar Coordinates - Multiplication by i
36. 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.
Euler's Formula
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex numbers are points in the plane
37. ½(e^(iz) + e^(-iz))
cos z
conjugate
Roots of Unity
can't get out of the complex numbers by adding (or subtracting) or multiplying two
38. A complex number and its conjugate
conjugate pairs
How to multiply complex nubers(2+i)(2i-3)
z + z*
Euler's Formula
39. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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40. 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)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Numbers: Multiply
Absolute Value of a Complex Number
41. When two complex numbers are subtracted from one another.
Complex Subtraction
Complex Number Formula
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)
42. (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 Number
How to solve (2i+3)/(9-i)
Every complex number has the 'Standard Form': a + bi for some real a and b.
cosh²y - sinh²y
43. We can also think of the point z= a+ ib as
z + z*
complex numbers
the vector (a -b)
How to multiply complex nubers(2+i)(2i-3)
44. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Square Root
i^3
Polar Coordinates - Multiplication
Roots of Unity
45. Have radical
Euler Formula
Field
Euler's Formula
radicals
46. R?¹(cos? - isin?)
Rational Number
transcendental
Polar Coordinates - sin?
Polar Coordinates - z?¹
47. Divide moduli and subtract arguments
Complex Division
Square Root
Polar Coordinates - Division
conjugate pairs
48. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Subtraction
Complex Numbers: Multiply
Integers
Euler Formula
49. (e^(iz) - e^(-iz)) / 2i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^4
sin z
i^2
50. 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
Roots of Unity
Complex Numbers: Add & subtract
(cos? +isin?)n
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i