CLEP General Mathematics: Complex Numbers

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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. 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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2. We see in this way that the distance between two points z and w in the complex plane is
Rational Number
the distance from z to the origin in the complex plane
|z-w|
Polar Coordinates - Arg(z*)


3. The complex number z representing a+bi.
Complex numbers are points in the plane
Imaginary number
four different numbers: i - -i - 1 - and -1.
Affix


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

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5. The modulus of the complex number z= a + ib now can be interpreted as
Roots of Unity
z - z*
Polar Coordinates - Division
the distance from z to the origin in the complex plane


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

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7. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - Division
x-axis in the complex plane
The Complex Numbers
standard form of complex numbers


8. Have radical
Complex Number Formula
Imaginary Numbers
radicals
Rules of Complex Arithmetic


9. 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.
Every complex number has the 'Standard Form': a + bi for some real a and b.
multiplying complex numbers
i^3
Complex Numbers: Multiply


10. y / r
cos z
sin iy
v(-1)
Polar Coordinates - sin?


11. In this amazing number field every algebraic equation in z with complex coefficients
cos z
Polar Coordinates - Division
has a solution.
irrational


12. All numbers
Complex Number Formula
Imaginary Numbers
complex
Roots of Unity


13. 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: Multiply
adding complex numbers
Polar Coordinates - sin?
Polar Coordinates - Division


14. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Polar Coordinates - cos?
natural
ln z
a + bi for some real a and b.


15. x / r
Euler's Formula
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - cos?
Euler Formula


16. (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
How to multiply complex nubers(2+i)(2i-3)
real
Imaginary number
Polar Coordinates - z


17. 1
i^2
Complex numbers are points in the plane
imaginary
irrational


18. I
Liouville's Theorem -
i^1
Real Numbers
has a solution.


19. (e^(-y) - e^(y)) / 2i = i sinh y
multiplying complex numbers
Roots of Unity
Polar Coordinates - z?¹
sin iy


20. 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 - r
the complex numbers
We say that c+di and c-di are complex conjugates.
cos iy


21. Imaginary number

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22. 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
Absolute Value of a Complex Number
Polar Coordinates - Multiplication by i
adding complex numbers


23. V(x² + y²) = |z|
natural
Polar Coordinates - r
ln z
How to find any Power


24. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
i^4
conjugate pairs
point of inflection


25. Has exactly n roots by the fundamental theorem of algebra
z1 / z2
z + z*
Complex numbers are points in the plane
Any polynomial O(xn) - (n > 0)


26. 3
(cos? +isin?)n
i^2 = -1
subtracting complex numbers
i^3


27. Like pi
The Complex Numbers
'i'
i^1
transcendental


28. 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
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
transcendental
multiplying complex numbers
Rules of Complex Arithmetic


29. A subset within a field.
interchangeable
Euler's Formula
|z| = mod(z)
Subfield


30. Rotates anticlockwise by p/2
Rational Number
Integers
Polar Coordinates - Multiplication by i
a + bi for some real a and b.


31. When two complex numbers are added together.
z1 / z2
(a + c) + ( b + d)i
Polar Coordinates - Division
Complex Addition


32. 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
Complex Numbers: Multiply
Polar Coordinates - Arg(z*)
the complex numbers
four different numbers: i - -i - 1 - and -1.


33. 1st. Rule of Complex Arithmetic
i^2 = -1
How to find any Power
ln z
conjugate pairs


34. The square root of -1.
Imaginary Unit
Euler's Formula
Complex numbers are points in the plane
Complex Addition


35. 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
can't get out of the complex numbers by adding (or subtracting) or multiplying two
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z1 ^ (z2)


36. A complex number and its conjugate
sin z
conjugate pairs
Complex Subtraction
complex


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

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38. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
standard form of complex numbers
Rational Number
Complex Numbers: Add & subtract


39. 3rd. Rule of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cos z
For real a and b - a + bi = 0 if and only if a = b = 0
the distance from z to the origin in the complex plane


40. 1
The Complex Numbers
i²
Subfield
non-integers


41. Not on the numberline
adding complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
non-integers
radicals


42. Any number not rational
Integers
z + z*
How to multiply complex nubers(2+i)(2i-3)
irrational


43. All the powers of i can be written as
Imaginary Numbers
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication
z + z*


44. Divide moduli and subtract arguments
Polar Coordinates - Division
multiply the numerator and the denominator by the complex conjugate of the denominator.
non-integers
cosh²y - sinh²y


45. A+bi
Complex Number Formula
has a solution.
0 if and only if a = b = 0
Complex Number


46. E^(ln r) e^(i?) e^(2pin)
i^2
e^(ln z)
Absolute Value of a Complex Number
|z| = mod(z)


47. V(zz*) = v(a² + b²)
|z| = mod(z)
De Moivre's Theorem
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)


48. I
z - z*
v(-1)
irrational
Rules of Complex Arithmetic


49. We can also think of the point z= a+ ib as
point of inflection
the vector (a -b)
interchangeable
Polar Coordinates - z?¹


50. Derives z = a+bi
Euler Formula
cos z
Real Numbers
Polar Coordinates - Multiplication