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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.
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. Written as fractions - terminating + repeating decimals
Complex Addition
rational
We say that c+di and c-di are complex conjugates.
Absolute Value of a Complex Number

2. (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
rational
sin iy
How to solve (2i+3)/(9-i)
Subfield

3. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
a + bi for some real a and b.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
For real a and b - a + bi = 0 if and only if a = b = 0
Absolute Value of a Complex Number

4. 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
Subfield
Imaginary number
complex

5. (e^(-y) - e^(y)) / 2i = i sinh y
zz*
Rules of Complex Arithmetic
cos z
sin iy

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. 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
v(-1)
adding complex numbers
Complex numbers are points in the plane
Rules of Complex Arithmetic

8. Rotates anticlockwise by p/2
cosh²y - sinh²y
z + z*
Polar Coordinates - Multiplication by i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

9. We can also think of the point z= a+ ib as
the vector (a -b)
the distance from z to the origin in the complex plane
Irrational Number
Rational Number

10. x / r
imaginary
point of inflection
Polar Coordinates - cos?
subtracting complex numbers

11. All the powers of i can be written as
Imaginary Numbers
v(-1)
four different numbers: i - -i - 1 - and -1.
multiplying complex numbers

12. For real a and b - a + bi =
the complex numbers
0 if and only if a = b = 0
Complex Division
has a solution.

13. V(x² + y²) = |z|
Real and Imaginary Parts
Polar Coordinates - r
Real Numbers
multiplying complex numbers

14. When two complex numbers are subtracted from one another.
Polar Coordinates - sin?
Complex Subtraction
a real number: (a + bi)(a - bi) = a² + b²
standard form of complex numbers

15. 1
cosh²y - sinh²y
Complex Subtraction
Rules of Complex Arithmetic
multiplying complex numbers

16. The field of all rational and irrational numbers.
Irrational Number
a real number: (a + bi)(a - bi) = a² + b²
conjugate pairs
Real Numbers

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

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18. y / r
z + z*
ln z
Polar Coordinates - sin?
Complex Multiplication

19. Root negative - has letter i
subtracting complex numbers
imaginary
Integers
Euler's Formula

20. (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
multiplying complex numbers
subtracting complex numbers
How to multiply complex nubers(2+i)(2i-3)
conjugate

21. xpressions such as ``the complex number z'' - and ``the point z'' are now
integers
interchangeable
How to add and subtract complex numbers (2-3i)-(4+6i)
non-integers

22. z1z2* / |z2|²
z1 / z2
multiply the numerator and the denominator by the complex conjugate of the denominator.
transcendental
rational

23. 4th. Rule of Complex Arithmetic
z - z*
(a + bi) = (c + bi) = (a + c) + ( b + d)i
'i'
(cos? +isin?)n

24. 2nd. Rule of Complex Arithmetic

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25. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - Multiplication by i
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)
rational

26. A subset within a field.
0 if and only if a = b = 0
Subfield
real
Imaginary Numbers

27. The modulus of the complex number z= a + ib now can be interpreted as
Polar Coordinates - z
radicals
four different numbers: i - -i - 1 - and -1.
the distance from z to the origin in the complex plane

28. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - Multiplication by i
Rational Number
a + bi for some real a and b.
Complex Numbers: Add & subtract

29. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Subfield
Complex numbers are points in the plane
adding complex numbers

30. A complex number may be taken to the power of another complex number.
Rules of Complex Arithmetic
Complex Exponentiation
Complex Number Formula
a real number: (a + bi)(a - bi) = a² + b²

31. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
a + bi for some real a and b.
i^0
How to find any Power
(a + bi) = (c + bi) = (a + c) + ( b + d)i

32. Not on the numberline
Complex Number Formula
non-integers
rational
Polar Coordinates - Multiplication by i

33. 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
subtracting complex numbers
Subfield
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)

34. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex numbers are points in the plane
Subfield
ln z
z1 ^ (z2)

35. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Arg(z*)
i^4
rational

36. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - sin?
We say that c+di and c-di are complex conjugates.
Imaginary Numbers
Liouville's Theorem -

37. All numbers
complex
Absolute Value of a Complex Number
a real number: (a + bi)(a - bi) = a² + b²
|z-w|

38. A+bi
Complex Number
Complex Number Formula
cos iy
Polar Coordinates - Division

39. Equivalent to an Imaginary Unit.
Imaginary number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a + bi for some real a and b.
has a solution.

40. The reals are just the
Every complex number has the 'Standard Form': a + bi for some real a and b.
De Moivre's Theorem
x-axis in the complex plane
Polar Coordinates - z

41. Where the curvature of the graph changes
Real Numbers
x-axis in the complex plane
z1 / z2
point of inflection

42. 3
i^3
point of inflection
ln z
i^2 = -1

43. 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
non-integers
Complex Subtraction
z + z*

44. 1
Complex Conjugate
Complex Numbers: Multiply
i^0
a real number: (a + bi)(a - bi) = a² + b²

45. Starts at 1 - does not include 0
real
sin z
natural
adding complex numbers

46. 1
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Numbers
sin z
i^2

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

48. A plot of complex numbers as points.
cos z
Affix
Field
Argand diagram

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

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50. 2a
z + z*
Polar Coordinates - cos?
standard form of complex numbers
the vector (a -b)