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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. 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.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Addition
How to find any Power
Real Numbers

2. The complex number z representing a+bi.
complex numbers
0 if and only if a = b = 0
Affix
can't get out of the complex numbers by adding (or subtracting) or multiplying two

3. I^2 =
-1
Polar Coordinates - Multiplication
Subfield
Complex Conjugate

4. Root negative - has letter i
imaginary
Imaginary Unit
Polar Coordinates - Multiplication by i
multiplying complex numbers

5. 4th. Rule of Complex Arithmetic
rational
De Moivre's Theorem
radicals
(a + bi) = (c + bi) = (a + c) + ( b + d)i

6. A subset within a field.
rational
Subfield
Polar Coordinates - Multiplication
(a + bi) = (c + bi) = (a + c) + ( b + d)i

7. Multiply moduli and add arguments
Liouville's Theorem -
Complex Numbers: Multiply
Polar Coordinates - Multiplication
Every complex number has the 'Standard Form': a + bi for some real a and b.

8. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
z1 / z2
Absolute Value of a Complex Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
The Complex Numbers

9. We can also think of the point z= a+ ib as
Complex Exponentiation
the vector (a -b)
|z-w|
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

10. We see in this way that the distance between two points z and w in the complex plane is
|z| = mod(z)
conjugate pairs
|z-w|
(cos? +isin?)n

11. Any number not rational
cosh²y - sinh²y
point of inflection
irrational
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

12. x + iy = r(cos? + isin?) = re^(i?)
conjugate pairs
Polar Coordinates - z
transcendental
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

13. 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
multiply the numerator and the denominator by the complex conjugate of the denominator.
(cos? +isin?)n
How to multiply complex nubers(2+i)(2i-3)

14. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
zz*
How to add and subtract complex numbers (2-3i)-(4+6i)
the distance from z to the origin in the complex plane
standard form of complex numbers

15. E^(ln r) e^(i?) e^(2pin)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
e^(ln z)
i^2 = -1
radicals

16. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
How to find any Power
sin iy
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. When two complex numbers are multipiled together.
Complex Conjugate
Polar Coordinates - Multiplication by i
Complex Multiplication
z - z*

18. A complex number and its conjugate
conjugate pairs
imaginary
Absolute Value of a Complex Number
integers

19. R?¹(cos? - isin?)
Complex Exponentiation
x-axis in the complex plane
Polar Coordinates - z?¹
can't get out of the complex numbers by adding (or subtracting) or multiplying two

20. 1
We say that c+di and c-di are complex conjugates.
x-axis in the complex plane
Polar Coordinates - r
i^0

21. x / r
v(-1)
Polar Coordinates - cos?
Real and Imaginary Parts
How to add and subtract complex numbers (2-3i)-(4+6i)

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

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23. Every complex number has the 'Standard Form':
a + bi for some real a and b.
Imaginary Numbers
sin iy
can't get out of the complex numbers by adding (or subtracting) or multiplying two

24. 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.'
Imaginary number
Complex Number
i^1
sin z

25. Divide moduli and subtract arguments
multiply the numerator and the denominator by the complex conjugate of the denominator.
|z-w|
Polar Coordinates - Division
a + bi for some real a and b.

26. When two complex numbers are added together.
Complex Addition
i^0
standard form of complex numbers
z1 ^ (z2)

27. 2ib
z - z*
Complex Number
Euler's Formula
conjugate pairs

28. A number that cannot be expressed as a fraction for any integer.
sin z
Irrational Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z

29. A complex number may be taken to the power of another complex number.
Complex Exponentiation
irrational
z - z*
i^3

30. Have radical
zz*
point of inflection
radicals
imaginary

31. 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-w|
subtracting complex numbers
Complex numbers are points in the plane
Real and Imaginary Parts

32. I
v(-1)
Euler Formula
conjugate pairs
Polar Coordinates - Arg(z*)

33. Derives z = a+bi
ln z
Euler Formula
0 if and only if a = b = 0
subtracting complex numbers

34. Cos n? + i sin n? (for all n integers)
How to multiply complex nubers(2+i)(2i-3)
(cos? +isin?)n
sin z
Rules of Complex Arithmetic

35. 1
z1 ^ (z2)
Euler Formula
i^2 = -1
i²

36. When two complex numbers are divided.
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Number Formula

37. No i
i²
real
Any polynomial O(xn) - (n > 0)
z + z*

38. To simplify the square root of a negative number
i^2
the complex numbers
adding complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

39. Given (4-2i) the complex conjugate would be (4+2i)
z + z*
Complex Conjugate
point of inflection
conjugate

40. To simplify a complex fraction
How to find any Power
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + c) + ( b + d)i
Complex Division

41. Real and imaginary numbers
complex numbers
Euler Formula
Complex Multiplication
Polar Coordinates - r

42. 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
Complex Subtraction
the complex numbers
i^3
Real and Imaginary Parts

43. 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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44. A+bi
complex numbers
Complex Number Formula
z - z*
z1 ^ (z2)

45. 2a
(a + c) + ( b + d)i
the complex numbers
x-axis in the complex plane
z + z*

46. 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
the vector (a -b)
Polar Coordinates - Multiplication by i
x-axis in the complex plane
Rules of Complex Arithmetic

47. A² + b² - real and non negative
radicals
real
zz*
How to find any Power

48. 1
e^(ln z)
Euler's Formula
Complex Addition
i^2

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

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50. A number that can be expressed as a fraction p/q where q is not equal to 0.
Polar Coordinates - z?¹
cosh²y - sinh²y
Rational Number
natural