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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. In this amazing number field every algebraic equation in z with complex coefficients
sin z
has a solution.
For real a and b - a + bi = 0 if and only if a = b = 0
a real number: (a + bi)(a - bi) = a² + b²

2. All the powers of i can be written as
interchangeable
How to find any Power
four different numbers: i - -i - 1 - and -1.
Field

3. Multiply moduli and add arguments
Polar Coordinates - Multiplication
i^1
How to add and subtract complex numbers (2-3i)-(4+6i)
How to multiply complex nubers(2+i)(2i-3)

4. ? = -tan?
Imaginary Unit
Complex Division
complex
Polar Coordinates - Arg(z*)

5. 2nd. Rule of Complex Arithmetic

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6. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Complex Division
integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

7. 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
Imaginary number
subtracting complex numbers
z1 / z2
Complex Addition

8. Every complex number has the 'Standard Form':
Complex Addition
a real number: (a + bi)(a - bi) = a² + b²
i^1
a + bi for some real a and b.

9. (e^(-y) - e^(y)) / 2i = i sinh y
Integers
the vector (a -b)
sin iy
irrational

10. Real and imaginary numbers
complex numbers
z1 / z2
For real a and b - a + bi = 0 if and only if a = b = 0
The Complex Numbers

11. 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
Complex Number
real

12. A² + b² - real and non negative
For real a and b - a + bi = 0 if and only if a = b = 0
zz*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Liouville's Theorem -

13. A number that can be expressed as a fraction p/q where q is not equal to 0.
sin z
Subfield
|z| = mod(z)
Rational Number

14. The field of all rational and irrational numbers.
Absolute Value of a Complex Number
Roots of Unity
Rules of Complex Arithmetic
Real Numbers

15. A plot of complex numbers as points.
complex numbers
Argand diagram
The Complex Numbers
subtracting complex numbers

16. 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
imaginary
'i'
Complex Exponentiation

17. 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
integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Conjugate

18. 1st. Rule of Complex Arithmetic
Real and Imaginary Parts
i^2 = -1
Polar Coordinates - sin?
Polar Coordinates - r

19. Starts at 1 - does not include 0
real
natural
Complex Exponentiation
The Complex Numbers

20. R^2 = x
Square Root
four different numbers: i - -i - 1 - and -1.
We say that c+di and c-di are complex conjugates.
Liouville's Theorem -

21. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
|z-w|
The Complex Numbers
i^4
cosh²y - sinh²y

22. 2ib
z - z*
Field
complex
i^3

23. When two complex numbers are divided.
Complex Division
|z| = mod(z)
the distance from z to the origin in the complex plane
Complex Addition

24. Written as fractions - terminating + repeating decimals
rational
(a + c) + ( b + d)i
'i'
Complex Number

25. 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
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*
Polar Coordinates - Division
Complex Numbers: Add & subtract

26. Divide moduli and subtract arguments
sin iy
Complex Addition
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Division

27. Rotates anticlockwise by p/2
complex
Polar Coordinates - z?¹
Polar Coordinates - Multiplication by i
Affix

28. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
i^2 = -1
Roots of Unity
Polar Coordinates - cos?
Complex Conjugate

29. ½(e^(-y) +e^(y)) = cosh y
Imaginary Numbers
cos iy
Roots of Unity
We say that c+di and c-di are complex conjugates.

30. 4th. Rule of Complex Arithmetic
sin z
-1
Complex Conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i

31. E ^ (z2 ln z1)
De Moivre's Theorem
a real number: (a + bi)(a - bi) = a² + b²
Integers
z1 ^ (z2)

32. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Complex Division
Integers
Real Numbers
Complex Number Formula

33. I
four different numbers: i - -i - 1 - and -1.
Complex Addition
adding complex numbers
v(-1)

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

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35. Like pi
transcendental
Polar Coordinates - cos?
Complex Subtraction
(cos? +isin?)n

36. 5th. Rule of Complex Arithmetic
Imaginary Unit
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
non-integers
Complex Conjugate

37. 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
Field
Polar Coordinates - z
'i'

38. Has exactly n roots by the fundamental theorem of algebra
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
sin z
Any polynomial O(xn) - (n > 0)
multiplying complex numbers

39. (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
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)
Complex Number

40. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
rational
(a + c) + ( b + d)i
multiplying complex numbers
standard form of complex numbers

41. Equivalent to an Imaginary Unit.
cosh²y - sinh²y
irrational
How to multiply complex nubers(2+i)(2i-3)
Imaginary number

42. A + bi
standard form of complex numbers
Complex Numbers: Add & subtract
multiplying complex numbers
i^4

43. E^(ln r) e^(i?) e^(2pin)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
e^(ln z)
Complex numbers are points in the plane
sin z

44. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
has a solution.
Complex Subtraction
standard form of complex numbers

45. V(x² + y²) = |z|
Polar Coordinates - cos?
a real number: (a + bi)(a - bi) = a² + b²
conjugate pairs
Polar Coordinates - r

46. y / r
ln z
Polar Coordinates - Multiplication by i
Polar Coordinates - sin?
'i'

47. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
(a + c) + ( b + d)i
complex
-1

48. We can also think of the point z= a+ ib as
the vector (a -b)
point of inflection
Imaginary Unit
can't get out of the complex numbers by adding (or subtracting) or multiplying two

49. z1z2* / |z2|²
Liouville's Theorem -
cos z
Real Numbers
z1 / z2

50. (a + bi)(c + bi) =
Polar Coordinates - sin?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z + z*
Every complex number has the 'Standard Form': a + bi for some real a and b.