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. When two complex numbers are multipiled together.
cos iy
Polar Coordinates - Multiplication by i
Complex Multiplication
Complex numbers are points in the plane


2. y / r
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - sin?
complex
the vector (a -b)


3. 1
Liouville's Theorem -
i²
transcendental
radicals


4. To simplify a complex fraction
has a solution.
cos iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
(cos? +isin?)n


5. Imaginary number

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6. 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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7. Any number not rational
|z| = mod(z)
Polar Coordinates - r
imaginary
irrational


8. (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
We say that c+di and c-di are complex conjugates.
adding complex numbers
zz*
How to multiply complex nubers(2+i)(2i-3)


9. Like pi
ln z
transcendental
complex
cosh²y - sinh²y


10. 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.
Complex numbers are points in the plane
x-axis in the complex plane
Subfield
four different numbers: i - -i - 1 - and -1.


11. 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
four different numbers: i - -i - 1 - and -1.
integers
i^2 = -1


12. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Liouville's Theorem -
Integers
i^0


13. R^2 = x
Square Root
Polar Coordinates - cos?
subtracting complex numbers
Polar Coordinates - Arg(z*)


14. A² + b² - real and non negative
multiplying complex numbers
zz*
Subfield
conjugate pairs


15. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex Subtraction
ln z
0 if and only if a = b = 0
Polar Coordinates - Division


16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
the complex numbers
Argand diagram
Polar Coordinates - z
Roots of Unity


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 Numbers: Add & subtract
Real and Imaginary Parts
Imaginary Numbers
Polar Coordinates - Division


18. (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)
conjugate pairs
The Complex Numbers
z1 / z2


19. 1
i^0
cosh²y - sinh²y
How to add and subtract complex numbers (2-3i)-(4+6i)
z1 / z2


20. 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 Exponentiation
Complex Number
z1 ^ (z2)
natural


21. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Complex Division
-1
zz*


22. Derives z = a+bi
complex
Euler Formula
Complex Number Formula
(a + bi) = (c + bi) = (a + c) + ( b + d)i


23. We see in this way that the distance between two points z and w in the complex plane is
Complex Number
|z-w|
Affix
conjugate pairs


24. I
i^1
Polar Coordinates - Multiplication
Imaginary number
conjugate pairs


25. A+bi
Liouville's Theorem -
Complex Numbers: Multiply
Complex Number Formula
Polar Coordinates - Multiplication


26. To simplify the square root of a negative number
sin iy
integers
Subfield
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)


27. 2nd. Rule of Complex Arithmetic

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28. Have radical
Complex Numbers: Add & subtract
radicals
How to find any Power
Integers


29. The product of an imaginary number and its conjugate is
How to multiply complex nubers(2+i)(2i-3)
Imaginary number
a real number: (a + bi)(a - bi) = a² + b²
sin iy


30. E ^ (z2 ln z1)
Irrational Number
i^4
Polar Coordinates - z
z1 ^ (z2)


31. Every complex number has the 'Standard Form':
a + bi for some real a and b.
interchangeable
(a + c) + ( b + d)i
Imaginary Numbers


32. A complex number and its conjugate
has a solution.
conjugate pairs
Complex Numbers: Add & subtract
How to multiply complex nubers(2+i)(2i-3)


33. No i
rational
Roots of Unity
the vector (a -b)
real


34. (a + bi)(c + bi) =
z + z*
real
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
e^(ln z)


35. ? = -tan?
rational
Complex Numbers: Multiply
point of inflection
Polar Coordinates - Arg(z*)


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

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37. Not on the numberline
non-integers
-1
Polar Coordinates - sin?
|z-w|


38. 1
sin iy
i^4
Argand diagram
has a solution.


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
Euler Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
We say that c+di and c-di are complex conjugates.
Subfield


40. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary number
cos iy
conjugate


41. 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.
Polar Coordinates - z?¹
Euler's Formula
conjugate
Complex Numbers: Multiply


42. Has exactly n roots by the fundamental theorem of algebra
Square Root
Polar Coordinates - Multiplication by i
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)


43. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
cos z
Rules of Complex Arithmetic
multiplying complex numbers
a real number: (a + bi)(a - bi) = a² + b²


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

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45. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to add and subtract complex numbers (2-3i)-(4+6i)


46. All the powers of i can be written as
z1 ^ (z2)
Polar Coordinates - z?¹
sin iy
four different numbers: i - -i - 1 - and -1.


47. x / r
cos z
sin z
De Moivre's Theorem
Polar Coordinates - cos?


48. 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
-1
Rules of Complex Arithmetic
Real Numbers
i^0


49. 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
x-axis in the complex plane
real
subtracting complex numbers
Rational Number


50. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
The Complex Numbers
Complex Division
For real a and b - a + bi = 0 if and only if a = b = 0
Absolute Value of a Complex Number