CLEP General Mathematics: Complex Numbers

Instructions:

• Answer 50 questions in 15 minutes.
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• 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. 2a
zz*
Euler's Formula
Real and Imaginary Parts
z + z*


2. 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.'
-1
sin z
How to find any Power
Complex Number


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

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4. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
Affix
i^2
Field


5. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Imaginary Numbers
Euler Formula
i^4
Roots of Unity


6. Have radical
Complex Subtraction
Irrational Number
Polar Coordinates - sin?
radicals


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

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8. A+bi
Complex Number Formula
Complex Division
ln z
z - z*


9. 1st. Rule of Complex Arithmetic
Field
(a + bi) = (c + bi) = (a + c) + ( b + d)i
transcendental
i^2 = -1


10. Imaginary number

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11. When two complex numbers are subtracted from one another.
Complex Subtraction
Liouville's Theorem -
interchangeable
We say that c+di and c-di are complex conjugates.


12. When two complex numbers are multipiled together.
Complex Multiplication
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Irrational Number
Complex Number Formula


13. 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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14. To simplify a complex fraction
a real number: (a + bi)(a - bi) = a² + b²
Irrational Number
Polar Coordinates - r
multiply the numerator and the denominator by the complex conjugate of the denominator.


15. 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
We say that c+di and c-di are complex conjugates.
the complex numbers
imaginary
has a solution.


16. 2nd. Rule of Complex Arithmetic

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17. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Complex Number
cos iy
-1


18. Like pi
i^2 = -1
Real and Imaginary Parts
transcendental
Polar Coordinates - r


19. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
four different numbers: i - -i - 1 - and -1.
Complex numbers are points in the plane
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Field


20. x + iy = r(cos? + isin?) = re^(i?)
conjugate pairs
Polar Coordinates - z
Imaginary Numbers
Complex Conjugate


21. 1
Any polynomial O(xn) - (n > 0)
adding complex numbers
Complex Conjugate
i^2


22. (e^(-y) - e^(y)) / 2i = i sinh y
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Division
Absolute Value of a Complex Number
sin iy


23. I = imaginary unit - i² = -1 or i = v-1
rational
-1
Imaginary Numbers
complex numbers


24. Every complex number has the 'Standard Form':
adding complex numbers
a + bi for some real a and b.
|z-w|
i^0


25. ½(e^(iz) + e^(-iz))
Euler's Formula
cos z
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic


26. We see in this way that the distance between two points z and w in the complex plane is
Imaginary Unit
z + z*
i^3
|z-w|


27. xpressions such as ``the complex number z'' - and ``the point z'' are now
the distance from z to the origin in the complex plane
interchangeable
Polar Coordinates - Multiplication by i
conjugate pairs


28. ½(e^(-y) +e^(y)) = cosh y
-1
Argand diagram
Complex Division
cos iy


29. 1
i²
rational
the vector (a -b)
four different numbers: i - -i - 1 - and -1.


30. 1
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Multiply


31. ? = -tan?
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Arg(z*)
Roots of Unity
sin iy


32. A complex number may be taken to the power of another complex number.
Euler's Formula
Complex Exponentiation
adding complex numbers
imaginary


33. Real and imaginary numbers
Imaginary number
i^3
radicals
complex numbers


34. When two complex numbers are divided.
non-integers
natural
Complex Division
Square Root


35. V(zz*) = v(a² + b²)
|z| = mod(z)
z + z*
multiplying complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.


36. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Numbers: Multiply
De Moivre's Theorem
multiplying complex numbers
has a solution.


37. 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
How to multiply complex nubers(2+i)(2i-3)
the complex numbers
x-axis in the complex plane
Complex Multiplication


38. Root negative - has letter i
imaginary
Polar Coordinates - r
The Complex Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)


39. All numbers
complex
z - z*
Real Numbers
Imaginary Unit


40. Any number not rational
Field
interchangeable
z1 ^ (z2)
irrational


41. 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
x-axis in the complex plane
Rules of Complex Arithmetic
i^3
a + bi for some real a and b.


42. Has exactly n roots by the fundamental theorem of algebra
Complex numbers are points in the plane
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)
sin iy


43. (e^(iz) - e^(-iz)) / 2i
complex
can't get out of the complex numbers by adding (or subtracting) or multiplying two
sin z
Imaginary Unit


44. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - r
the distance from z to the origin in the complex plane
Polar Coordinates - Arg(z*)
cosh²y - sinh²y


45. 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.
We say that c+di and c-di are complex conjugates.
Complex Numbers: Multiply
integers
Real Numbers


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
Euler's Formula
Real and Imaginary Parts
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Multiplication


47. 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
(cos? +isin?)n
z1 / z2
Complex Numbers: Add & subtract
standard form of complex numbers


48. x / r
has a solution.
i^0
Polar Coordinates - cos?
i^3


49. A number that cannot be expressed as a fraction for any integer.
Roots of Unity
v(-1)
De Moivre's Theorem
Irrational Number


50. V(x² + y²) = |z|
adding complex numbers
z1 ^ (z2)
Polar Coordinates - r
Euler Formula