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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. Real and imaginary numbers
complex numbers
cos iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z| = mod(z)

2. 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
a real number: (a + bi)(a - bi) = a² + b²
multiplying complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Numbers: Add & subtract

3. 5th. Rule of Complex Arithmetic
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
sin iy

4. A+bi
(cos? +isin?)n
Complex Number Formula
cos z
How to find any Power

5. 1
i²
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Every complex number has the 'Standard Form': a + bi for some real a and b.
Liouville's Theorem -

6. Every complex number has the 'Standard Form':
a + bi for some real a and b.
standard form of complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^4

7. 2ib
-1
Polar Coordinates - Division
ln z
z - z*

8. The field of all rational and irrational numbers.
Absolute Value of a Complex Number
has a solution.
real
Real Numbers

9. 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.
interchangeable
How to find any Power
Polar Coordinates - cos?
Complex Subtraction

10. Have radical
radicals
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary number
Polar Coordinates - z

11. E ^ (z2 ln z1)
imaginary
complex numbers
z1 ^ (z2)
|z| = mod(z)

12. Has exactly n roots by the fundamental theorem of algebra
Rules of Complex Arithmetic
Real and Imaginary Parts
Any polynomial O(xn) - (n > 0)
i^2 = -1

13. ½(e^(iz) + e^(-iz))
cos z
adding complex numbers
the vector (a -b)
Integers

14. 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
imaginary
i^1
We say that c+di and c-di are complex conjugates.
z + z*

15. We can also think of the point z= a+ ib as
the vector (a -b)
How to find any Power
Real and Imaginary Parts
multiply the numerator and the denominator by the complex conjugate of the denominator.

16. I
complex numbers
Complex Subtraction
(cos? +isin?)n
v(-1)

17. Derives z = a+bi
Polar Coordinates - Arg(z*)
Euler Formula
Affix
cosh²y - sinh²y

18. Written as fractions - terminating + repeating decimals
imaginary
cosh²y - sinh²y
rational
i^0

19. Like pi
e^(ln z)
Polar Coordinates - r
Argand diagram
transcendental

20. ½(e^(-y) +e^(y)) = cosh y
The Complex Numbers
cos iy
irrational
cosh²y - sinh²y

21. A subset within a field.
The Complex Numbers
Any polynomial O(xn) - (n > 0)
the distance from z to the origin in the complex plane
Subfield

22. 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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23. The modulus of the complex number z= a + ib now can be interpreted as
'i'
the distance from z to the origin in the complex plane
a + bi for some real a and b.
Absolute Value of a Complex Number

24. V(zz*) = v(a² + b²)
interchangeable
|z| = mod(z)
Integers
Complex Number

25. Given (4-2i) the complex conjugate would be (4+2i)
complex
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Conjugate
Polar Coordinates - Arg(z*)

26. V(x² + y²) = |z|
How to solve (2i+3)/(9-i)
i^0
the complex numbers
Polar Coordinates - r

27. Cos n? + i sin n? (for all n integers)
Subfield
i²
(cos? +isin?)n
For real a and b - a + bi = 0 if and only if a = b = 0

28. 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.
How to find any Power
Complex numbers are points in the plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Any polynomial O(xn) - (n > 0)

29. 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
z - z*
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Numbers

30. The square root of -1.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a real number: (a + bi)(a - bi) = a² + b²
Imaginary Unit
Roots of Unity

31. All numbers
conjugate
Complex Numbers: Multiply
Euler's Formula
complex

32. x / r
Euler's Formula
Subfield
Polar Coordinates - sin?
Polar Coordinates - cos?

33. I^2 =
zz*
-1
irrational
e^(ln z)

34. A number that cannot be expressed as a fraction for any integer.
integers
We say that c+di and c-di are complex conjugates.
a real number: (a + bi)(a - bi) = a² + b²
Irrational Number

35. R?¹(cos? - isin?)
i^1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z?¹
(a + c) + ( b + d)i

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

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37. z1z2* / |z2|²
point of inflection
Liouville's Theorem -
z1 / z2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

38. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Rational Number
Roots of Unity
Complex numbers are points in the plane

39. 2nd. Rule of Complex Arithmetic

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40. 4th. Rule of Complex Arithmetic
'i'
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Multiplication
Imaginary Numbers

41. 2a
z + z*
natural
the vector (a -b)
i^4

42. 1
i^0
Roots of Unity
Liouville's Theorem -
Complex numbers are points in the plane

43. xpressions such as ``the complex number z'' - and ``the point z'' are now
Real Numbers
interchangeable
Complex Subtraction
transcendental

44. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
i²
Roots of Unity
four different numbers: i - -i - 1 - and -1.

45. (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)
Polar Coordinates - Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two
zz*

46. Not on the numberline
For real a and b - a + bi = 0 if and only if a = b = 0
non-integers
i^0
Complex Exponentiation

47. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Rational Number
Argand diagram
The Complex Numbers
'i'

48. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Real Numbers
complex numbers
Absolute Value of a Complex Number
cos z

49. Imaginary number

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50. Root negative - has letter i
imaginary
Complex Division
Rational Number
a real number: (a + bi)(a - bi) = a² + b²