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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. Any number not rational
Complex Conjugate
irrational
How to find any Power
sin z
2. A complex number and its conjugate
non-integers
Square Root
How to find any Power
conjugate pairs
3. A number that cannot be expressed as a fraction for any integer.
Rational Number
sin z
ln z
Irrational Number
4. Every complex number has the 'Standard Form':
interchangeable
a + bi for some real a and b.
Polar Coordinates - z?¹
zz*
5. Numbers on a numberline
integers
Real and Imaginary Parts
Complex Numbers: Add & subtract
the distance from z to the origin in the complex plane
6. 2ib
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
i^2
z - z*
7. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
i^3
sin iy
How to multiply complex nubers(2+i)(2i-3)
non-integers
8. A + bi
How to solve (2i+3)/(9-i)
Roots of Unity
real
standard form of complex numbers
9. x + iy = r(cos? + isin?) = re^(i?)
Irrational Number
(cos? +isin?)n
Polar Coordinates - z
z1 / z2
10. (a + bi)(c + bi) =
Complex Numbers: Multiply
Complex Exponentiation
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2 = -1
11. 1
Rational Number
How to add and subtract complex numbers (2-3i)-(4+6i)
ln z
i^4
12. V(x² + y²) = |z|
Polar Coordinates - z
Complex Division
Square Root
Polar Coordinates - r
13. 1
z + z*
i^0
a + bi for some real a and b.
How to multiply complex nubers(2+i)(2i-3)
14. y / r
Imaginary number
i²
Polar Coordinates - sin?
i^2 = -1
15. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - z?¹
sin z
The Complex Numbers
irrational
16. ? = -tan?
Polar Coordinates - z?¹
conjugate
irrational
Polar Coordinates - Arg(z*)
17. A complex number may be taken to the power of another complex number.
Real and Imaginary Parts
the distance from z to the origin in the complex plane
Complex Exponentiation
cos z
18. The reals are just the
Complex numbers are points in the plane
x-axis in the complex plane
Polar Coordinates - cos?
imaginary
19. Multiply moduli and add arguments
Polar Coordinates - Division
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Multiplication
Complex numbers are points in the plane
20. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
|z| = mod(z)
Roots of Unity
e^(ln z)
Polar Coordinates - z
21. 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
complex
Euler Formula
the complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
22. Have radical
Polar Coordinates - Multiplication by i
radicals
Euler Formula
four different numbers: i - -i - 1 - and -1.
23. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - z
Polar Coordinates - cos?
How to add and subtract complex numbers (2-3i)-(4+6i)
Rules of Complex Arithmetic
24. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Rules of Complex Arithmetic
Complex Numbers: Multiply
Euler's Formula
Irrational Number
25. The modulus of the complex number z= a + ib now can be interpreted as
real
a + bi for some real a and b.
natural
the distance from z to the origin in the complex plane
26. Starts at 1 - does not include 0
standard form of complex numbers
Polar Coordinates - z?¹
four different numbers: i - -i - 1 - and -1.
natural
27. Not on the numberline
Polar Coordinates - Multiplication by i
non-integers
Complex numbers are points in the plane
the distance from z to the origin in the complex plane
28. E ^ (z2 ln z1)
z1 ^ (z2)
cos iy
Complex Number Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
29. I = imaginary unit - i² = -1 or i = v-1
z1 ^ (z2)
Imaginary Numbers
Field
Complex Subtraction
30. I^2 =
Complex Numbers: Add & subtract
-1
Argand diagram
interchangeable
31. R?¹(cos? - isin?)
integers
How to find any Power
Polar Coordinates - z?¹
Polar Coordinates - Division
32. When two complex numbers are divided.
zz*
Complex Number
a real number: (a + bi)(a - bi) = a² + b²
Complex Division
33. A² + b² - real and non negative
How to multiply complex nubers(2+i)(2i-3)
zz*
sin iy
Polar Coordinates - sin?
34. 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
i^3
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers
35. For real a and b - a + bi =
ln z
multiplying complex numbers
i^2 = -1
0 if and only if a = b = 0
36. To prove that number field every algebraic equation in z with complex coefficients has a solution we need
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37. The field of all rational and irrational numbers.
i^2
Polar Coordinates - Arg(z*)
Real Numbers
conjugate pairs
38. Divide moduli and subtract arguments
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Division
Rules of Complex Arithmetic
How to add and subtract complex numbers (2-3i)-(4+6i)
39. To simplify a complex fraction
Imaginary number
x-axis in the complex plane
Euler's Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
40. A subset within a field.
We say that c+di and c-di are complex conjugates.
Subfield
e^(ln z)
non-integers
41. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
We say that c+di and c-di are complex conjugates.
'i'
Euler's Formula
Polar Coordinates - Multiplication by i
42. Root negative - has letter i
imaginary
Complex Number
Field
Complex Exponentiation
43. (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
We say that c+di and c-di are complex conjugates.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Arg(z*)
How to solve (2i+3)/(9-i)
44. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
The Complex Numbers
Integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rules of Complex Arithmetic
45. A+bi
Complex Number Formula
Polar Coordinates - Multiplication
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
a + bi for some real a and b.
46. 1
integers
Complex Numbers: Multiply
i²
Polar Coordinates - Division
47. E^(ln r) e^(i?) e^(2pin)
the distance from z to the origin in the complex plane
e^(ln z)
z1 ^ (z2)
'i'
48. We can also think of the point z= a+ ib as
the vector (a -b)
z + z*
sin z
transcendental
49. 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
conjugate
Complex Number Formula
Square Root
subtracting complex numbers
50. The product of an imaginary number and its conjugate is
How to add and subtract complex numbers (2-3i)-(4+6i)
a real number: (a + bi)(a - bi) = a² + b²
sin z
-1