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. Written as fractions - terminating + repeating decimals
rational
|z-w|
The Complex Numbers
zz*


2. 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
Imaginary Unit
the complex numbers
Liouville's Theorem -
Complex Conjugate


3. Divide moduli and subtract arguments
Polar Coordinates - Division
Integers
(cos? +isin?)n
Euler's Formula


4. 1st. Rule of Complex Arithmetic
integers
transcendental
i^2 = -1
Square Root


5. The complex number z representing a+bi.
(a + c) + ( b + d)i
i^0
Affix
Real and Imaginary Parts


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

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7. (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
For real a and b - a + bi = 0 if and only if a = b = 0
Real and Imaginary Parts
Complex Number Formula
How to solve (2i+3)/(9-i)


8. E ^ (z2 ln z1)
i^4
z1 ^ (z2)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Field


9. 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
complex numbers
Roots of Unity
adding complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.


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

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11. 2ib
Polar Coordinates - Multiplication by i
|z| = mod(z)
z - z*
ln z


12. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to add and subtract complex numbers (2-3i)-(4+6i)
a + bi for some real a and b.
Argand diagram


13. E^(ln r) e^(i?) e^(2pin)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
e^(ln z)
i^2


14. Every complex number has the 'Standard Form':
conjugate
Any polynomial O(xn) - (n > 0)
Imaginary Numbers
a + bi for some real a and b.


15. In this amazing number field every algebraic equation in z with complex coefficients
z1 ^ (z2)
i^3
Square Root
has a solution.


16. A number that cannot be expressed as a fraction for any integer.
We say that c+di and c-di are complex conjugates.
Complex Numbers: Multiply
Irrational Number
the distance from z to the origin in the complex plane


17. I
Imaginary Unit
complex numbers
Real and Imaginary Parts
i^1


18. I^2 =
The Complex Numbers
0 if and only if a = b = 0
-1
z1 ^ (z2)


19. y / r
Polar Coordinates - sin?
transcendental
How to multiply complex nubers(2+i)(2i-3)
Complex numbers are points in the plane


20. Numbers on a numberline
Complex Division
Imaginary number
integers
Square Root


21. xpressions such as ``the complex number z'' - and ``the point z'' are now
irrational
interchangeable
Complex Numbers: Multiply
The Complex Numbers


22. For real a and b - a + bi =
standard form of complex numbers
'i'
Absolute Value of a Complex Number
0 if and only if a = b = 0


23. 2nd. Rule of Complex Arithmetic

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24. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
cos z
imaginary
Complex Conjugate


25. 3rd. Rule of Complex Arithmetic
The Complex Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
conjugate pairs
i^4


26. V(x² + y²) = |z|
Polar Coordinates - r
cos iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z1 ^ (z2)


27. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
point of inflection
x-axis in the complex plane
Field


28. Derives z = a+bi
Polar Coordinates - sin?
Euler Formula
i^4
rational


29. A + bi
radicals
i^0
standard form of complex numbers
the distance from z to the origin in the complex plane


30. To simplify the square root of a negative number
Complex Number
Imaginary Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
'i'


31. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Number Formula
Absolute Value of a Complex Number
Complex numbers are points in the plane
Every complex number has the 'Standard Form': a + bi for some real a and b.


32. z1z2* / |z2|²
Complex Number
radicals
Polar Coordinates - cos?
z1 / z2


33. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
imaginary
sin z
Complex Division


34. The reals are just the
x-axis in the complex plane
a + bi for some real a and b.
e^(ln z)
Real and Imaginary Parts


35. Starts at 1 - does not include 0
Absolute Value of a Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
natural
ln z


36. 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.'
'i'
Complex Number
Complex Numbers: Add & subtract
interchangeable


37. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
z + z*
i^0
Complex Addition


38. Where the curvature of the graph changes
i^3
For real a and b - a + bi = 0 if and only if a = b = 0
0 if and only if a = b = 0
point of inflection


39. 1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Imaginary Unit
i^0
rational


40. x / r
Euler's Formula
We say that c+di and c-di are complex conjugates.
Polar Coordinates - cos?
Complex numbers are points in the plane


41. 1
Complex Number
Polar Coordinates - z
rational
i^4


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

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43. A number that can be expressed as a fraction p/q where q is not equal to 0.
multiply the numerator and the denominator by the complex conjugate of the denominator.
Rational Number
conjugate pairs
z + z*


44. (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
adding complex numbers
i^3
Roots of Unity
How to multiply complex nubers(2+i)(2i-3)


45. Cos n? + i sin n? (for all n integers)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
Polar Coordinates - z?¹
(cos? +isin?)n


46. 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
adding complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z


47. A subset within a field.
Subfield
multiplying complex numbers
Real and Imaginary Parts
adding complex numbers


48. Like pi
z + z*
transcendental
Complex Numbers: Add & subtract
i²


49. 1
conjugate pairs
Complex Multiplication
cosh²y - sinh²y
For real a and b - a + bi = 0 if and only if a = b = 0


50. Imaginary number

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