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. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

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2. V(zz*) = v(a² + b²)
Rational Number
Polar Coordinates - sin?
Imaginary Numbers
|z| = mod(z)


3. The complex number z representing a+bi.
-1
Affix
z + z*
Complex Numbers: Multiply


4. Like pi
|z| = mod(z)
transcendental
i^2 = -1
z1 ^ (z2)


5. 3rd. Rule of Complex Arithmetic
a real number: (a + bi)(a - bi) = a² + b²
z1 / z2
How to add and subtract complex numbers (2-3i)-(4+6i)
For real a and b - a + bi = 0 if and only if a = b = 0


6. Have radical
Polar Coordinates - Division
radicals
conjugate pairs
Polar Coordinates - cos?


7. 3
How to multiply complex nubers(2+i)(2i-3)
(a + c) + ( b + d)i
i^3
Complex Numbers: Add & subtract


8. All numbers
ln z
v(-1)
complex
Absolute Value of a Complex Number


9. I
Liouville's Theorem -
v(-1)
ln z
interchangeable


10. Root negative - has letter i
|z| = mod(z)
imaginary
Real and Imaginary Parts
(a + c) + ( b + d)i


11. A complex number may be taken to the power of another complex number.
v(-1)
the complex numbers
Complex Exponentiation
Liouville's Theorem -


12. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - cos?
Imaginary Unit


13. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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14. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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15. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex Division
multiplying complex numbers
ln z
v(-1)


16. 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
|z| = mod(z)
How to solve (2i+3)/(9-i)
We say that c+di and c-di are complex conjugates.
Complex Subtraction


17. 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
non-integers
Complex Conjugate
subtracting complex numbers
Polar Coordinates - z?¹


18. For real a and b - a + bi =
0 if and only if a = b = 0
Field
Polar Coordinates - Multiplication by i
can't get out of the complex numbers by adding (or subtracting) or multiplying two


19. Has exactly n roots by the fundamental theorem of algebra
a + bi for some real a and b.
Any polynomial O(xn) - (n > 0)
Imaginary Unit
can't get out of the complex numbers by adding (or subtracting) or multiplying two


20. 2ib
Complex numbers are points in the plane
'i'
Complex Number
z - z*


21. 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
adding complex numbers
|z| = mod(z)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary number


22. When two complex numbers are divided.
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Division
Irrational Number


23. 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: Multiply
De Moivre's Theorem
Complex Numbers: Add & subtract
a real number: (a + bi)(a - bi) = a² + b²


24. Imaginary number

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25. The field of all rational and irrational numbers.
Real Numbers
Polar Coordinates - z
We say that c+di and c-di are complex conjugates.
i^3


26. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
adding complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Every complex number has the 'Standard Form': a + bi for some real a and b.


27. We can also think of the point z= a+ ib as
Every complex number has the 'Standard Form': a + bi for some real a and b.
real
Affix
the vector (a -b)


28. 2a
z + z*
sin iy
Square Root
Integers


29. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
adding complex numbers
z + z*
Subfield
The Complex Numbers


30. 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.
How to solve (2i+3)/(9-i)
Polar Coordinates - Division
Complex Numbers: Multiply
|z-w|


31. A + bi
i^4
subtracting complex numbers
i^2
standard form of complex numbers


32. 2nd. Rule of Complex Arithmetic

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33. Written as fractions - terminating + repeating decimals
How to find any Power
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
rational


34. z1z2* / |z2|²
i²
(a + c) + ( b + d)i
z1 / z2
non-integers


35. A number that can be expressed as a fraction p/q where q is not equal to 0.
imaginary
i^4
Rational Number
conjugate


36. 4th. Rule of Complex Arithmetic
sin z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Arg(z*)
Absolute Value of a Complex Number


37. No i
Polar Coordinates - r
rational
transcendental
real


38. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
ln z
real
Complex numbers are points in the plane
Roots of Unity


39. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Imaginary number
multiplying complex numbers
Complex Subtraction
has a solution.


40. Multiply moduli and add arguments
the vector (a -b)
How to solve (2i+3)/(9-i)
Complex Number
Polar Coordinates - Multiplication


41. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Complex Addition
point of inflection
conjugate pairs


42. A plot of complex numbers as points.
Polar Coordinates - Division
point of inflection
conjugate pairs
Argand diagram


43. 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
cosh²y - sinh²y
z + z*
Every complex number has the 'Standard Form': a + bi for some real a and b.
Rules of Complex Arithmetic


44. To simplify the square root of a negative number
i^0
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
(cos? +isin?)n


45. Equivalent to an Imaginary Unit.
Imaginary number
zz*
subtracting complex numbers
Polar Coordinates - cos?


46. A complex number and its conjugate
Argand diagram
Liouville's Theorem -
conjugate pairs
i²


47. I^2 =
-1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication
adding complex numbers


48. (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
How to multiply complex nubers(2+i)(2i-3)
interchangeable
z - z*
Complex Conjugate


49. Divide moduli and subtract arguments
point of inflection
cos z
Complex Addition
Polar Coordinates - Division


50. (e^(iz) - e^(-iz)) / 2i
sin z
Complex Numbers: Add & subtract
v(-1)
Polar Coordinates - r