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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. 3rd. Rule of Complex Arithmetic
Real Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Subfield
Complex Number Formula

2. All numbers
v(-1)
radicals
complex
has a solution.

3. R?¹(cos? - isin?)
Euler's Formula
Subfield
Polar Coordinates - z?¹
Complex Conjugate

4. Where the curvature of the graph changes
i^4
point of inflection
Polar Coordinates - Multiplication
How to solve (2i+3)/(9-i)

5. 2nd. Rule of Complex Arithmetic

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6. I
i^1
cosh²y - sinh²y
imaginary
|z-w|

7. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Square Root
subtracting complex numbers
Imaginary Unit

8. 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.
Euler Formula
-1
Complex numbers are points in the plane
x-axis in the complex plane

9. Rotates anticlockwise by p/2
Real and Imaginary Parts
Polar Coordinates - sin?
Polar Coordinates - Multiplication by i
cos iy

10. ? = -tan?
Complex Multiplication
integers
Polar Coordinates - Arg(z*)
i^1

11. We can also think of the point z= a+ ib as
(cos? +isin?)n
For real a and b - a + bi = 0 if and only if a = b = 0
the vector (a -b)
cos z

12. 1
i^0
i²
Complex Numbers: Multiply
Polar Coordinates - cos?

13. x + iy = r(cos? + isin?) = re^(i?)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - z
i^0
i²

14. 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
Imaginary Unit
multiplying complex numbers
the complex numbers
Roots of Unity

15. Equivalent to an Imaginary Unit.
i^3
De Moivre's Theorem
a real number: (a + bi)(a - bi) = a² + b²
Imaginary number

16. z1z2* / |z2|²
Subfield
complex numbers
Real and Imaginary Parts
z1 / z2

17. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
point of inflection
z1 / z2
Rational Number

18. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex numbers are points in the plane
standard form of complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

19. A number that cannot be expressed as a fraction for any integer.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Rules of Complex Arithmetic
How to add and subtract complex numbers (2-3i)-(4+6i)
Irrational Number

20. y / r
For real a and b - a + bi = 0 if and only if a = b = 0
|z| = mod(z)
Polar Coordinates - sin?
sin z

21. E^(ln r) e^(i?) e^(2pin)
Complex Exponentiation
i^4
e^(ln z)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

22. Written as fractions - terminating + repeating decimals
Integers
rational
Euler's Formula
Polar Coordinates - Arg(z*)

23. Like pi
transcendental
Complex Conjugate
Euler's Formula
The Complex Numbers

24. 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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25. (e^(-y) - e^(y)) / 2i = i sinh y
the vector (a -b)
How to solve (2i+3)/(9-i)
Euler's Formula
sin iy

26. E ^ (z2 ln z1)
z1 ^ (z2)
standard form of complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
imaginary

27. 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
cosh²y - sinh²y
complex
Polar Coordinates - Multiplication by i

28. 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.
Square Root
How to find any Power
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2

29. 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
Rules of Complex Arithmetic
v(-1)
Polar Coordinates - Arg(z*)
Absolute Value of a Complex Number

30. The reals are just the
Complex Numbers: Multiply
multiplying complex numbers
x-axis in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

31. I
Polar Coordinates - z?¹
Complex Numbers: Add & subtract
v(-1)
Polar Coordinates - r

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

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33. Given (4-2i) the complex conjugate would be (4+2i)
Integers
Polar Coordinates - cos?
a + bi for some real a and b.
Complex Conjugate

34. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
Field
'i'
the distance from z to the origin in the complex plane
How to solve (2i+3)/(9-i)

35. Real and imaginary numbers
multiplying complex numbers
complex numbers
Affix
How to find any Power

36. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
non-integers
the complex numbers
Absolute Value of a Complex Number
z1 ^ (z2)

37. A number that can be expressed as a fraction p/q where q is not equal to 0.
conjugate pairs
non-integers
Complex Number Formula
Rational Number

38. When two complex numbers are divided.
We say that c+di and c-di are complex conjugates.
Complex Division
the complex numbers
standard form of complex numbers

39. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
v(-1)
complex
(a + c) + ( b + d)i

40. 1st. Rule of Complex Arithmetic
i^2 = -1
Rational Number
The Complex Numbers
Complex Subtraction

41. All the powers of i can be written as
Absolute Value of a Complex Number
Complex Conjugate
(cos? +isin?)n
four different numbers: i - -i - 1 - and -1.

42. The field of all rational and irrational numbers.
Real Numbers
Rational Number
the complex numbers
-1

43. Root negative - has letter i
imaginary
the vector (a -b)
the complex numbers
conjugate pairs

44. Has exactly n roots by the fundamental theorem of algebra
How to solve (2i+3)/(9-i)
multiplying complex numbers
Any polynomial O(xn) - (n > 0)
integers

45. A² + b² - real and non negative
zz*
imaginary
Polar Coordinates - cos?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

46. Any number not rational
Subfield
e^(ln z)
(a + c) + ( b + d)i
irrational

47. Have radical
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Arg(z*)
radicals

48. A+bi
'i'
Rational Number
Complex Number Formula
interchangeable

49. 2ib
|z| = mod(z)
z - z*
Complex Numbers: Multiply
non-integers

50. When two complex numbers are subtracted from one another.
z + z*
Polar Coordinates - Multiplication
Complex Subtraction
Complex Conjugate