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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

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. (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)
|z-w|
irrational
Complex Division

2. Root negative - has letter i
imaginary
Rational Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Real Numbers

3. A+bi
i^0
Polar Coordinates - Multiplication by i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula

4. A number that cannot be expressed as a fraction for any integer.
Euler Formula
Irrational Number
subtracting complex numbers
imaginary

5. x + iy = r(cos? + isin?) = re^(i?)
cosh²y - sinh²y
Polar Coordinates - z
sin iy
imaginary

6. R^2 = x
Square Root
|z-w|
Complex Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i

7. When two complex numbers are subtracted from one another.
Complex Subtraction
Field
a + bi for some real a and b.
Liouville's Theorem -

8. 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
|z-w|
Real Numbers
imaginary

9. V(zz*) = v(a² + b²)
e^(ln z)
|z| = mod(z)
Polar Coordinates - z?¹
Polar Coordinates - Multiplication

10. 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 Division
ln z
-1
Complex Numbers: Multiply

11. To simplify a complex fraction
Polar Coordinates - Arg(z*)
multiply the numerator and the denominator by the complex conjugate of the denominator.
Irrational Number
Subfield

12. Starts at 1 - does not include 0
natural
Real and Imaginary Parts
z + z*
Complex Number Formula

13. xpressions such as ``the complex number z'' - and ``the point z'' are now
subtracting complex numbers
interchangeable
sin z
Roots of Unity

14. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
v(-1)
imaginary
Imaginary Numbers

15. 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
Polar Coordinates - z?¹
ln z
Rules of Complex Arithmetic
Complex Multiplication

16. 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.'
z + z*
Complex Subtraction
Polar Coordinates - z?¹
Complex Number

17. 3rd. Rule of Complex Arithmetic
v(-1)
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
De Moivre's Theorem

18. y / r
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - sin?
Euler Formula
Complex Number Formula

19. Where the curvature of the graph changes
Polar Coordinates - Multiplication
Polar Coordinates - Multiplication by i
point of inflection
Rules of Complex Arithmetic

20. When two complex numbers are added together.
i²
i^4
Complex Addition
a real number: (a + bi)(a - bi) = a² + b²

21. (a + bi) = (c + bi) =
The Complex Numbers
(a + c) + ( b + d)i
has a solution.
real

22. A² + b² - real and non negative
Subfield
zz*
Any polynomial O(xn) - (n > 0)
(cos? +isin?)n

23. For real a and b - a + bi =
0 if and only if a = b = 0
Complex Multiplication
i^4
De Moivre's Theorem

24. 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
Real Numbers
subtracting complex numbers
the complex numbers
Imaginary number

25. 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
-1
sin z
a + bi for some real a and b.
adding complex numbers

26. The complex number z representing a+bi.
radicals
Affix
0 if and only if a = b = 0
Euler's Formula

27. 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 distance from z to the origin in the complex plane
How to solve (2i+3)/(9-i)
Euler Formula

28. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Euler Formula
How to multiply complex nubers(2+i)(2i-3)
Complex Number Formula

29. Numbers on a numberline
Irrational Number
Liouville's Theorem -
integers
sin iy

30. 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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31. We can also think of the point z= a+ ib as
the vector (a -b)
complex
adding complex numbers
Polar Coordinates - z?¹

32. All the powers of i can be written as
complex
four different numbers: i - -i - 1 - and -1.
Imaginary Numbers
Integers

33. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
can't get out of the complex numbers by adding (or subtracting) or multiplying two
multiplying complex numbers
adding complex numbers
i^0

34. 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
z1 ^ (z2)
Complex Addition
i^1

35. Has exactly n roots by the fundamental theorem of algebra
Complex Subtraction
rational
Any polynomial O(xn) - (n > 0)
Euler's Formula

36. 1
Complex Numbers: Multiply
Integers
i^0
sin z

37. Real and imaginary numbers
complex numbers
De Moivre's Theorem
Complex Number
z + z*

38. (a + bi)(c + bi) =
0 if and only if a = b = 0
i^4
real
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

39. 1
Complex Numbers: Add & subtract
complex numbers
i²
i^4

40. Written as fractions - terminating + repeating decimals
transcendental
De Moivre's Theorem
rational
Complex Conjugate

41. I
four different numbers: i - -i - 1 - and -1.
z + z*
Euler Formula
i^1

42. Cos n? + i sin n? (for all n integers)
real
cos iy
integers
(cos? +isin?)n

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

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44. We see in this way that the distance between two points z and w in the complex plane is
non-integers
Complex Exponentiation
|z-w|
How to add and subtract complex numbers (2-3i)-(4+6i)

45. A subset within a field.
For real a and b - a + bi = 0 if and only if a = b = 0
Subfield
integers
cosh²y - sinh²y

46. (e^(-y) - e^(y)) / 2i = i sinh y
point of inflection
sin iy
Imaginary number
-1

47. Like pi
four different numbers: i - -i - 1 - and -1.
transcendental
Polar Coordinates - z?¹
standard form of complex numbers

48. ½(e^(iz) + e^(-iz))
four different numbers: i - -i - 1 - and -1.
cos z
Real Numbers
De Moivre's Theorem

49. 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
the complex numbers
the vector (a -b)
-1

50. All numbers
Polar Coordinates - Division
transcendental
complex
non-integers