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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. 1st. Rule of Complex Arithmetic
cos z
i^2 = -1
'i'
a + bi for some real a and b.

2. (e^(-y) - e^(y)) / 2i = i sinh y
sin z
Polar Coordinates - Multiplication by i
sin iy
i^3

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.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Absolute Value of a Complex Number
the distance from z to the origin in the complex plane

4. A complex number and its conjugate
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z + z*
rational
conjugate pairs

5. A² + b² - real and non negative
Complex Numbers: Multiply
rational
z + z*
zz*

6. When two complex numbers are divided.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
v(-1)
the distance from z to the origin in the complex plane

7. Like pi
|z| = mod(z)
transcendental
How to multiply complex nubers(2+i)(2i-3)
How to add and subtract complex numbers (2-3i)-(4+6i)

8. 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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9. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Rational Number
v(-1)
standard form of complex numbers

10. The complex number z representing a+bi.
How to find any Power
rational
Affix
Complex numbers are points in the plane

11. Root negative - has letter i
Polar Coordinates - sin?
0 if and only if a = b = 0
imaginary
conjugate pairs

12. When two complex numbers are added together.
Absolute Value of a Complex Number
Complex Addition
Polar Coordinates - Multiplication
transcendental

13. Starts at 1 - does not include 0
Polar Coordinates - r
For real a and b - a + bi = 0 if and only if a = b = 0
natural
Complex Exponentiation

14. I
Liouville's Theorem -
conjugate pairs
i^1
Complex Number

15. E ^ (z2 ln z1)
i^2 = -1
a real number: (a + bi)(a - bi) = a² + b²
(cos? +isin?)n
z1 ^ (z2)

16. ½(e^(-y) +e^(y)) = cosh y
cos iy
Real Numbers
Euler Formula
(a + c) + ( b + d)i

17. 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 + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler Formula
We say that c+di and c-di are complex conjugates.
Complex Numbers: Add & subtract

18. 5th. Rule of Complex Arithmetic
Euler's Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Liouville's Theorem -
Real Numbers

19. Cos n? + i sin n? (for all n integers)
Euler Formula
Complex Number Formula
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
(cos? +isin?)n

20. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Roots of Unity
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - r
conjugate

21. ? = -tan?
Absolute Value of a Complex Number
Polar Coordinates - sin?
Polar Coordinates - Arg(z*)
v(-1)

22. I
0 if and only if a = b = 0
v(-1)
conjugate
Integers

23. Numbers on a numberline
How to add and subtract complex numbers (2-3i)-(4+6i)
real
irrational
integers

24. To simplify a complex fraction
Complex Conjugate
cos z
sin z
multiply the numerator and the denominator by the complex conjugate of the denominator.

25. 3
cos z
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^3
a + bi for some real a and b.

26. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
e^(ln z)
Polar Coordinates - z?¹
Field
Imaginary Numbers

27. ½(e^(iz) + e^(-iz))
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
adding complex numbers
cos z
i^2 = -1

28. (a + bi) = (c + bi) =
|z| = mod(z)
How to find any Power
(a + c) + ( b + d)i
Imaginary number

29. I^2 =
non-integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
-1
Roots of Unity

30. The field of all rational and irrational numbers.
Polar Coordinates - sin?
Real Numbers
(cos? +isin?)n
the complex numbers

31. Imaginary number

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32. The modulus of the complex number z= a + ib now can be interpreted as
a real number: (a + bi)(a - bi) = a² + b²
adding complex numbers
the distance from z to the origin in the complex plane
Polar Coordinates - Arg(z*)

33. 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 Numbers: Multiply
adding complex numbers
sin z
Polar Coordinates - Arg(z*)

34. 1
Polar Coordinates - Division
i²
a real number: (a + bi)(a - bi) = a² + b²
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

35. Rotates anticlockwise by p/2
adding complex numbers
Polar Coordinates - Multiplication by i
Complex Subtraction
the distance from z to the origin in the complex plane

36. Equivalent to an Imaginary Unit.
Imaginary number
rational
Polar Coordinates - Division
Complex Multiplication

37. 1
non-integers
i^4
conjugate
rational

38. The reals are just the
Integers
Rules of Complex Arithmetic
Euler Formula
x-axis in the complex plane

39. A complex number may be taken to the power of another complex number.
De Moivre's Theorem
the complex numbers
Complex Exponentiation
Euler's Formula

40. We can also think of the point z= a+ ib as
adding complex numbers
zz*
the vector (a -b)
sin iy

41. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - z?¹
Polar Coordinates - cos?

42. A number that can be expressed as a fraction p/q where q is not equal to 0.
irrational
integers
Real Numbers
Rational Number

43. 3rd. Rule of Complex Arithmetic
Polar Coordinates - sin?
radicals
Field
For real a and b - a + bi = 0 if and only if a = b = 0

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

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45. E^(ln r) e^(i?) e^(2pin)
Subfield
sin iy
e^(ln z)
subtracting complex numbers

46. 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
a real number: (a + bi)(a - bi) = a² + b²
standard form of complex numbers
Complex Number
subtracting complex numbers

47. 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.
the complex numbers
Complex numbers are points in the plane
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.

48. Not on the numberline
non-integers
cos z
i^2 = -1
Irrational Number

49. Divide moduli and subtract arguments
Polar Coordinates - Division
Absolute Value of a Complex Number
Complex numbers are points in the plane
sin iy

50. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
the complex numbers
rational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i