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. Like pi
Imaginary Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
transcendental
Any polynomial O(xn) - (n > 0)


2. When two complex numbers are subtracted from one another.
Complex Subtraction
the vector (a -b)
Complex Conjugate
transcendental


3. Multiply moduli and add arguments
(a + c) + ( b + d)i
Subfield
v(-1)
Polar Coordinates - Multiplication


4. R^2 = x
rational
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication
Square Root


5. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Rules of Complex Arithmetic
Complex Number
Subfield


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

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7. A complex number may be taken to the power of another complex number.
Complex Exponentiation
radicals
We say that c+di and c-di are complex conjugates.
z - z*


8. Starts at 1 - does not include 0
De Moivre's Theorem
e^(ln z)
four different numbers: i - -i - 1 - and -1.
natural


9. Has exactly n roots by the fundamental theorem of algebra
i^2 = -1
Any polynomial O(xn) - (n > 0)
z1 / z2
integers


10. 1
i^0
The Complex Numbers
Affix
Complex Addition


11. We can also think of the point z= a+ ib as
sin iy
has a solution.
the vector (a -b)
Roots of Unity


12. Derives z = a+bi
Euler Formula
Real Numbers
irrational
sin z


13. 1
i^4
radicals
Polar Coordinates - Arg(z*)
Polar Coordinates - z?¹


14. ½(e^(iz) + e^(-iz))
Complex Addition
cos z
Polar Coordinates - r
the complex numbers


15. A subset within a field.
Complex Addition
Subfield
Complex Conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i


16. 1
Complex Addition
(a + bi) = (c + bi) = (a + c) + ( b + d)i
transcendental
cosh²y - sinh²y


17. 1
sin z
i²
interchangeable
-1


18. 2nd. Rule of Complex Arithmetic

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19. V(zz*) = v(a² + b²)
De Moivre's Theorem
We say that c+di and c-di are complex conjugates.
z + z*
|z| = mod(z)


20. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two
interchangeable
Complex numbers are points in the plane


21. 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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22. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
How to find any Power
Integers
Euler's Formula
integers


23. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Integers
Complex Subtraction
complex


24. ½(e^(-y) +e^(y)) = cosh y
a + bi for some real a and b.
x-axis in the complex plane
cos iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i


25. To simplify a complex fraction
Integers
Polar Coordinates - z?¹
interchangeable
multiply the numerator and the denominator by the complex conjugate of the denominator.


26. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
radicals
Field
Imaginary Unit
i^0


27. A number that can be expressed as a fraction p/q where q is not equal to 0.
four different numbers: i - -i - 1 - and -1.
|z-w|
i^4
Rational Number


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

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29. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Arg(z*)
multiplying complex numbers
e^(ln z)
(cos? +isin?)n


30. A complex number and its conjugate
conjugate pairs
Polar Coordinates - Multiplication by i
a real number: (a + bi)(a - bi) = a² + b²
Complex Multiplication


31. Every complex number has the 'Standard Form':
natural
a + bi for some real a and b.
How to find any Power
i^3


32. When two complex numbers are multipiled together.
Complex Multiplication
i^1
Imaginary Numbers
cos iy


33. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
Polar Coordinates - z?¹
a real number: (a + bi)(a - bi) = a² + b²
natural
Complex Number


34. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
-1
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary number
cosh²y - sinh²y


35. 3
transcendental
i^3
Polar Coordinates - Multiplication
Real Numbers


36. 2ib
|z| = mod(z)
imaginary
complex numbers
z - z*


37. Numbers on a numberline
0 if and only if a = b = 0
integers
Complex Addition
|z-w|


38. A plot of complex numbers as points.
adding complex numbers
x-axis in the complex plane
Argand diagram
How to solve (2i+3)/(9-i)


39. A + bi
(a + c) + ( b + d)i
standard form of complex numbers
Polar Coordinates - Multiplication
Polar Coordinates - r


40. 4th. Rule of Complex Arithmetic
Complex Numbers: Multiply
standard form of complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)


41. A+bi
Complex Number Formula
Complex Numbers: Multiply
Complex Number
four different numbers: i - -i - 1 - and -1.


42. I
-1
Every complex number has the 'Standard Form': a + bi for some real a and b.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^1


43. 2a
Polar Coordinates - Division
a + bi for some real a and b.
z + z*
Absolute Value of a Complex Number


44. Imaginary number

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45. No i
real
Absolute Value of a Complex Number
z1 ^ (z2)
Any polynomial O(xn) - (n > 0)


46. 5th. Rule of Complex Arithmetic
Complex Multiplication
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)
De Moivre's Theorem


47. 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
standard form of complex numbers
conjugate
Imaginary number


48. When two complex numbers are divided.
Rules of Complex Arithmetic
x-axis in the complex plane
multiplying complex numbers
Complex Division


49. Real and imaginary numbers
Euler's Formula
i^1
i^2 = -1
complex numbers


50. V(x² + y²) = |z|
Polar Coordinates - Arg(z*)
Polar Coordinates - r
Complex Numbers: Add & subtract
Liouville's Theorem -