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• Answer 50 questions in 15 minutes.
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• Match each statement with the correct term.
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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. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Rules of Complex Arithmetic
Real Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i


2. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
i^1
ln z
i^0
Euler Formula


3. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Irrational Number
Integers
z1 ^ (z2)
multiply the numerator and the denominator by the complex conjugate of the denominator.


4. No i
v(-1)
Complex Addition
real
-1


5. A + bi
Field
cosh²y - sinh²y
standard form of complex numbers
adding complex numbers


6. The complex number z representing a+bi.
Affix
standard form of complex numbers
radicals
e^(ln z)


7. All the powers of i can be written as
cosh²y - sinh²y
i²
four different numbers: i - -i - 1 - and -1.
the complex numbers


8. 2ib
radicals
Complex Number Formula
Polar Coordinates - Multiplication by i
z - z*


9. 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
Polar Coordinates - cos?
Real and Imaginary Parts
Real Numbers
a real number: (a + bi)(a - bi) = a² + b²


10. V(zz*) = v(a² + b²)
x-axis in the complex plane
e^(ln z)
|z| = mod(z)
How to multiply complex nubers(2+i)(2i-3)


11. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Euler's Formula
(a + c) + ( b + d)i
Polar Coordinates - Multiplication


12. 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.'
Complex Number
a real number: (a + bi)(a - bi) = a² + b²
Complex Exponentiation
Absolute Value of a Complex Number


13. 4th. Rule of Complex Arithmetic
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
radicals
z1 ^ (z2)


14. x + iy = r(cos? + isin?) = re^(i?)
|z| = mod(z)
conjugate pairs
z1 / z2
Polar Coordinates - z


15. 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
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?
Euler's Formula
the complex numbers


16. We can also think of the point z= a+ ib as
Polar Coordinates - z?¹
Complex Subtraction
Polar Coordinates - Arg(z*)
the vector (a -b)


17. 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 Addition
the complex numbers
conjugate
Complex Numbers: Add & subtract


18. 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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19. A complex number may be taken to the power of another complex number.
How to find any Power
Liouville's Theorem -
Complex Exponentiation
Euler's Formula


20. 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
point of inflection
Rules of Complex Arithmetic
standard form of complex numbers
the distance from z to the origin in the complex plane


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

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22. 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.
Real Numbers
z - z*
Complex Numbers: Multiply
Every complex number has the 'Standard Form': a + bi for some real a and b.


23. Imaginary number

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24. 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.
|z| = mod(z)
Real and Imaginary Parts
Rules of Complex Arithmetic
Complex numbers are points in the plane


25. Given (4-2i) the complex conjugate would be (4+2i)
Polar Coordinates - z?¹
Complex Conjugate
complex numbers
irrational


26. Starts at 1 - does not include 0
Real Numbers
Complex Numbers: Multiply
natural
For real a and b - a + bi = 0 if and only if a = b = 0


27. (e^(iz) - e^(-iz)) / 2i
point of inflection
sin z
(cos? +isin?)n
Polar Coordinates - z


28. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z
Rules of Complex Arithmetic
adding complex numbers


29. 2nd. Rule of Complex Arithmetic

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30. 1
rational
i²
Complex numbers are points in the plane
i^1


31. x / r
Polar Coordinates - z?¹
the complex numbers
Polar Coordinates - cos?
-1


32. Multiply moduli and add arguments
rational
Polar Coordinates - Multiplication
real
Any polynomial O(xn) - (n > 0)


33. 1st. Rule of Complex Arithmetic
Complex Division
Complex numbers are points in the plane
Integers
i^2 = -1


34. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
Complex Multiplication
Imaginary number
|z-w|


35. A plot of complex numbers as points.
Subfield
(a + c) + ( b + d)i
ln z
Argand diagram


36. z1z2* / |z2|²
Every complex number has the 'Standard Form': a + bi for some real a and b.
z1 / z2
x-axis in the complex plane
(cos? +isin?)n


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

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38. A+bi
Field
z1 ^ (z2)
Complex Number Formula
ln z


39. Equivalent to an Imaginary Unit.
transcendental
Imaginary number
e^(ln z)
Complex Conjugate


40. 3rd. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
Complex numbers are points in the plane
Subfield
For real a and b - a + bi = 0 if and only if a = b = 0


41. Like pi
transcendental
Polar Coordinates - sin?
Imaginary Numbers
(cos? +isin?)n


42. 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
Real Numbers
standard form of complex numbers
rational
subtracting complex numbers


43. 1
rational
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4
Imaginary Numbers


44. 5th. Rule of Complex Arithmetic
Polar Coordinates - Multiplication
four different numbers: i - -i - 1 - and -1.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Unit


45. Real and imaginary numbers
complex numbers
Complex Number
point of inflection
How to multiply complex nubers(2+i)(2i-3)


46. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Imaginary number
sin iy
How to add and subtract complex numbers (2-3i)-(4+6i)
i^2 = -1


47. 1
i^0
Polar Coordinates - z?¹
cos iy
i^4


48. (a + bi) = (c + bi) =
cos iy
z + z*
radicals
(a + c) + ( b + d)i


49. Where the curvature of the graph changes
point of inflection
Absolute Value of a Complex Number
zz*
non-integers


50. I
i^1
-1
Polar Coordinates - Arg(z*)
v(-1)