CLEP General Mathematics: Complex Numbers

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. Given (4-2i) the complex conjugate would be (4+2i)
-1
cosh²y - sinh²y
Complex Conjugate
i^3


2. Rotates anticlockwise by p/2
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication by i
Polar Coordinates - z
Field


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

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4. 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
x-axis in the complex plane
cos iy
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i


5. All the powers of i can be written as
transcendental
i^2 = -1
the distance from z to the origin in the complex plane
four different numbers: i - -i - 1 - and -1.


6. 5th. Rule of Complex Arithmetic
i²
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Square Root
cosh²y - sinh²y


7. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Integers
ln z
Polar Coordinates - Multiplication by i


8. 1
'i'
Rules of Complex Arithmetic
i^2
Affix


9. Divide moduli and subtract arguments
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division
Complex Numbers: Add & subtract
standard form of complex numbers


10. 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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11. Imaginary number

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12. 2nd. Rule of Complex Arithmetic

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13. The square root of -1.
Absolute Value of a Complex Number
We say that c+di and c-di are complex conjugates.
Complex Exponentiation
Imaginary Unit


14. x / r
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Numbers: Multiply
Polar Coordinates - cos?
point of inflection


15. (e^(-y) - e^(y)) / 2i = i sinh y
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - cos?
Complex Numbers: Add & subtract
sin iy


16. ? = -tan?
Polar Coordinates - Arg(z*)
i^3
rational
'i'


17. 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.
the complex numbers
Complex Subtraction
Euler Formula
Complex numbers are points in the plane


18. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
i^2 = -1
conjugate pairs
sin iy


19. A plot of complex numbers as points.
Argand diagram
z1 / z2
sin iy
interchangeable


20. Starts at 1 - does not include 0
subtracting complex numbers
Polar Coordinates - Multiplication by i
(a + c) + ( b + d)i
natural


21. A number that can be expressed as a fraction p/q where q is not equal to 0.
Complex Addition
a real number: (a + bi)(a - bi) = a² + b²
Rational Number
z1 / z2


22. For real a and b - a + bi =
Polar Coordinates - cos?
0 if and only if a = b = 0
Imaginary Unit
Complex Numbers: Multiply


23. (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
How to solve (2i+3)/(9-i)
cos iy
conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i


24. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Polar Coordinates - Multiplication by i
real
a + bi for some real a and b.


25. 1
natural
i²
Polar Coordinates - z
De Moivre's Theorem


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

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27. 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
How to add and subtract complex numbers (2-3i)-(4+6i)
the distance from z to the origin in the complex plane
Complex Multiplication
subtracting complex numbers


28. When two complex numbers are added together.
i^3
Complex Addition
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication


29. 1
We say that c+di and c-di are complex conjugates.
rational
cosh²y - sinh²y
(a + c) + ( b + d)i


30. A+bi
Complex Number Formula
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
conjugate pairs
(cos? +isin?)n


31. 3
i^3
cos z
Complex Numbers: Add & subtract
(cos? +isin?)n


32. Root negative - has letter i
Polar Coordinates - z?¹
Polar Coordinates - z
multiplying complex numbers
imaginary


33. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
conjugate
integers
Roots of Unity
Complex Number Formula


34. z1z2* / |z2|²
z1 / z2
ln z
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0


35. To simplify the square root of a negative number
Imaginary number
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Subtraction


36. R^2 = x
Square Root
Complex Exponentiation
Complex Multiplication
i^4


37. y / r
Polar Coordinates - cos?
Complex Exponentiation
Polar Coordinates - sin?
Complex Multiplication


38. I
How to solve (2i+3)/(9-i)
Complex Division
complex numbers
i^1


39. 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
Polar Coordinates - z
Argand diagram
We say that c+di and c-di are complex conjugates.
Polar Coordinates - r


40. 3rd. Rule of Complex Arithmetic
standard form of complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
sin iy
How to find any Power


41. 1
conjugate pairs
irrational
point of inflection
i^4


42. When two complex numbers are multipiled together.
interchangeable
Polar Coordinates - z?¹
Complex Multiplication
i^3


43. A + bi
cos z
Field
standard form of complex numbers
Affix


44. R?¹(cos? - isin?)
Argand diagram
Polar Coordinates - z?¹
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers


45. The field of all rational and irrational numbers.
four different numbers: i - -i - 1 - and -1.
interchangeable
Real Numbers
Imaginary Numbers


46. I^2 =
Every complex number has the 'Standard Form': a + bi for some real a and b.
Subfield
De Moivre's Theorem
-1


47. When two complex numbers are divided.
Absolute Value of a Complex Number
i^0
Euler's Formula
Complex Division


48. V(x² + y²) = |z|
Polar Coordinates - Arg(z*)
Polar Coordinates - cos?
Complex Numbers: Multiply
Polar Coordinates - r


49. x + iy = r(cos? + isin?) = re^(i?)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the vector (a -b)


50. The complex number z representing a+bi.
Affix
-1
Liouville's Theorem -
i^3