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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. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
multiplying complex numbers
irrational
The Complex Numbers
Polar Coordinates - z

2. 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
i^2 = -1
four different numbers: i - -i - 1 - and -1.
z1 ^ (z2)

3. 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.
z - z*
Euler Formula
Complex numbers are points in the plane
Any polynomial O(xn) - (n > 0)

4. Real and imaginary numbers
complex numbers
cos z
Euler's Formula
Euler Formula

5. The field of all rational and irrational numbers.
adding complex numbers
conjugate
Real Numbers
complex

6. (a + bi)(c + bi) =
i^0
irrational
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)

7. y / r
Complex Addition
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - sin?
Complex Exponentiation

8. Multiply moduli and add arguments
Polar Coordinates - Multiplication
can't get out of the complex numbers by adding (or subtracting) or multiplying two
adding complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i

9. 4th. Rule of Complex Arithmetic
rational
imaginary
conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i

10. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
adding complex numbers
imaginary

11. 2ib
complex numbers
z - z*
real
|z-w|

12. R?¹(cos? - isin?)
The Complex Numbers
Field
Polar Coordinates - z?¹
adding complex numbers

13. A+bi
i^2
i^4
the vector (a -b)
Complex Number Formula

14. Rotates anticlockwise by p/2
z1 ^ (z2)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
rational
Polar Coordinates - Multiplication by i

15. 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.

16. 1
'i'
i^2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos z

17. Every complex number has the 'Standard Form':
radicals
De Moivre's Theorem
cosh²y - sinh²y
a + bi for some real a and b.

18. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
z - z*
Complex Subtraction
four different numbers: i - -i - 1 - and -1.

19. We see in this way that the distance between two points z and w in the complex plane is
How to solve (2i+3)/(9-i)
|z-w|
Rational Number
Real and Imaginary Parts

20. 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.
Polar Coordinates - Division
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Integers
How to find any Power

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

22. Derives z = a+bi
Polar Coordinates - z?¹
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler Formula

23. When two complex numbers are multipiled together.
Complex Multiplication
e^(ln z)
sin z
How to solve (2i+3)/(9-i)

24. (e^(iz) - e^(-iz)) / 2i
sin z
Complex Division
has a solution.
irrational

25. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Rational Number
i^1
conjugate
ln z

26. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
v(-1)
How to find any Power
Complex Multiplication

27. Any number not rational
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - z?¹
Complex numbers are points in the plane
irrational

28. R^2 = x
interchangeable
the complex numbers
cos iy
Square Root

29. Imaginary number

30. xpressions such as ``the complex number z'' - and ``the point z'' are now
Imaginary number
How to find any Power
Rational Number
interchangeable

31. 1
the distance from z to the origin in the complex plane
irrational
cosh²y - sinh²y
conjugate pairs

32. 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.
irrational
Complex Numbers: Multiply
cos iy
natural

33. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Multiplication
Polar Coordinates - Multiplication

34. I
i^1
radicals
rational
How to solve (2i+3)/(9-i)

35. Divide moduli and subtract arguments
Polar Coordinates - Division
e^(ln z)
Complex Number Formula
How to solve (2i+3)/(9-i)

36. We can also think of the point z= a+ ib as
Affix
De Moivre's Theorem
Imaginary Numbers
the vector (a -b)

37. Not on the numberline
non-integers
Subfield
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Roots of Unity

38. A complex number and its conjugate
z - z*
conjugate pairs
Complex Addition
sin z

39. Has exactly n roots by the fundamental theorem of algebra
i^0
Complex Exponentiation
For real a and b - a + bi = 0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)

40. To simplify the square root of a negative number
radicals
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z - z*
Complex Numbers: Add & subtract

41. ½(e^(-y) +e^(y)) = cosh y
z1 / z2
Euler's Formula
cos iy
Rules of Complex Arithmetic

42. Given (4-2i) the complex conjugate would be (4+2i)
Affix
cos iy
Complex Conjugate
Polar Coordinates - Arg(z*)

43. 1st. Rule of Complex Arithmetic
i^2 = -1
De Moivre's Theorem
Complex numbers are points in the plane
i^4

44. All numbers
complex
Imaginary Unit
rational
Euler's Formula

45. (a + bi) = (c + bi) =
z - z*
0 if and only if a = b = 0
the distance from z to the origin in the complex plane
(a + c) + ( b + d)i

46. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Rational Number
transcendental
multiplying complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

47. I
Complex Exponentiation
(a + c) + ( b + d)i
Subfield
v(-1)

48. 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
complex
real
We say that c+di and c-di are complex conjugates.
conjugate

49. Cos n? + i sin n? (for all n integers)
i^0
(cos? +isin?)n
How to solve (2i+3)/(9-i)
i²

50. 5th. Rule of Complex Arithmetic
z + z*
Polar Coordinates - sin?
v(-1)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i