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. Imaginary number

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2. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
De Moivre's Theorem
Imaginary Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
Real and Imaginary Parts


3. z1z2* / |z2|²
Complex numbers are points in the plane
Roots of Unity
z1 / z2
four different numbers: i - -i - 1 - and -1.


4. 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
has a solution.
Rules of Complex Arithmetic
Complex Multiplication
the distance from z to the origin in the complex plane


5. x + iy = r(cos? + isin?) = re^(i?)
complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
How to solve (2i+3)/(9-i)
Polar Coordinates - z


6. A² + b² - real and non negative
Imaginary Numbers
Absolute Value of a Complex Number
zz*
-1


7. Divide moduli and subtract arguments
Imaginary Unit
Polar Coordinates - Division
Polar Coordinates - z?¹
Real and Imaginary Parts


8. 2nd. Rule of Complex Arithmetic

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9. Derives z = a+bi
Euler Formula
multiplying complex numbers
cos iy
i^0


10. R^2 = x
conjugate
cos iy
Complex Subtraction
Square Root


11. When two complex numbers are multipiled together.
Complex Addition
Complex Conjugate
Complex Multiplication
Absolute Value of a Complex Number


12. V(zz*) = v(a² + b²)
conjugate pairs
Polar Coordinates - Arg(z*)
Complex Division
|z| = mod(z)


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

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14. 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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15. A complex number may be taken to the power of another complex number.
Irrational Number
Complex Exponentiation
0 if and only if a = b = 0
Complex Number Formula


16. Real and imaginary numbers
x-axis in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.
complex numbers
z1 / z2


17. Any number not rational
i²
Euler Formula
irrational
Complex numbers are points in the plane


18. x / r
Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.
sin z
Polar Coordinates - cos?


19. (e^(-y) - e^(y)) / 2i = i sinh y
Complex Number Formula
sin iy
standard form of complex numbers
Euler's Formula


20. I
Complex Division
Euler Formula
sin z
v(-1)


21. All numbers
|z| = mod(z)
Complex Multiplication
multiplying complex numbers
complex


22. A + bi
imaginary
standard form of complex numbers
Complex Number
i^3


23. E^(ln r) e^(i?) e^(2pin)
radicals
Imaginary Numbers
e^(ln z)
Rational Number


24. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers
0 if and only if a = b = 0
i^4


25. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
'i'
Liouville's Theorem -
i^4


26. A+bi
Complex Number Formula
a real number: (a + bi)(a - bi) = a² + b²
z1 ^ (z2)
Liouville's Theorem -


27. Every complex number has the 'Standard Form':
Real Numbers
Rules of Complex Arithmetic
How to find any Power
a + bi for some real a and b.


28. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Number Formula
interchangeable
cos z
Complex Multiplication


29. The field of all rational and irrational numbers.
i²
Euler Formula
Real Numbers
-1


30. E ^ (z2 ln z1)
v(-1)
Affix
z1 ^ (z2)
Complex Conjugate


31. Have radical
real
radicals
(a + c) + ( b + d)i
interchangeable


32. 4th. Rule of Complex Arithmetic
Complex Division
Polar Coordinates - Arg(z*)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex numbers are points in the plane


33. y / r
subtracting complex numbers
i^2
Irrational Number
Polar Coordinates - sin?


34. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

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

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36. The reals are just the
can't get out of the complex numbers by adding (or subtracting) or multiplying two
a real number: (a + bi)(a - bi) = a² + b²
x-axis in the complex plane
sin z


37. The modulus of the complex number z= a + ib now can be interpreted as
rational
Complex Numbers: Multiply
a + bi for some real a and b.
the distance from z to the origin in the complex plane


38. ? = -tan?
Polar Coordinates - Arg(z*)
Euler Formula
e^(ln z)
Imaginary number


39. 2a
Roots of Unity
v(-1)
How to solve (2i+3)/(9-i)
z + z*


40. ½(e^(iz) + e^(-iz))
cos z
How to find any Power
(a + c) + ( b + d)i
i^2 = -1


41. Rotates anticlockwise by p/2
the complex numbers
Complex Division
Polar Coordinates - Multiplication by i
Polar Coordinates - sin?


42. ½(e^(-y) +e^(y)) = cosh y
cos iy
complex
the distance from z to the origin in the complex plane
has a solution.


43. Numbers on a numberline
cosh²y - sinh²y
Irrational Number
integers
cos z


44. 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
the complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication
non-integers


45. 2ib
sin iy
Argand diagram
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*


46. Has exactly n roots by the fundamental theorem of algebra
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Any polynomial O(xn) - (n > 0)
Imaginary Unit
Real and Imaginary Parts


47. 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.
Complex Numbers: Multiply
subtracting complex numbers
i^0
z1 ^ (z2)


48. Root negative - has letter i
z1 / z2
imaginary
Absolute Value of a Complex Number
Real Numbers


49. The product of an imaginary number and its conjugate is
cos iy
a real number: (a + bi)(a - bi) = a² + b²
point of inflection
The Complex Numbers


50. Not on the numberline
sin iy
x-axis in the complex plane
non-integers
multiply the numerator and the denominator by the complex conjugate of the denominator.