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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. Has exactly n roots by the fundamental theorem of algebra
Absolute Value of a Complex Number
Complex Multiplication
Any polynomial O(xn) - (n > 0)
Liouville's Theorem -

2. 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
Polar Coordinates - Multiplication by i
We say that c+di and c-di are complex conjugates.
Imaginary number
z + z*

3. 1st. Rule of Complex Arithmetic
i^2 = -1
Complex Numbers: Multiply
z1 / z2
four different numbers: i - -i - 1 - and -1.

4. 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
Square Root
complex numbers
Absolute Value of a Complex Number

5. 1
cosh²y - sinh²y
Square Root
Real and Imaginary Parts
i^0

6. Numbers on a numberline
Rational Number
integers
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Subtraction

7. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
Integers
the distance from z to the origin in the complex plane

8. No i
i^3
zz*
real
Complex Conjugate

9. 4th. Rule of Complex Arithmetic
Polar Coordinates - cos?
Complex Subtraction
transcendental
(a + bi) = (c + bi) = (a + c) + ( b + d)i

10. I^2 =
Complex Numbers: Multiply
-1
0 if and only if a = b = 0
Rules of Complex Arithmetic

11. (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
adding complex numbers
(a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)
Euler's Formula

12. 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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13. Not on the numberline
non-integers
Argand diagram
Complex Conjugate
i^4

14. Cos n? + i sin n? (for all n integers)
a real number: (a + bi)(a - bi) = a² + b²
e^(ln z)
point of inflection
(cos? +isin?)n

15. V(zz*) = v(a² + b²)
How to solve (2i+3)/(9-i)
complex numbers
real
|z| = mod(z)

16. In this amazing number field every algebraic equation in z with complex coefficients
the distance from z to the origin in the complex plane
Rational Number
has a solution.
Real and Imaginary Parts

17. ½(e^(-y) +e^(y)) = cosh y
|z-w|
For real a and b - a + bi = 0 if and only if a = b = 0
x-axis in the complex plane
cos iy

18. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
conjugate
Complex Number

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

20. The square root of -1.
Imaginary Unit
Affix
i²
Imaginary number

21. 1
i^2
Imaginary Numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Number

22. Derives z = a+bi
Euler Formula
Rational Number
(a + c) + ( b + d)i
subtracting complex numbers

23. y / r
z - z*
Polar Coordinates - sin?
(cos? +isin?)n
adding complex numbers

24. 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
Complex Numbers: Add & subtract
conjugate
imaginary
Field

25. 2nd. Rule of Complex Arithmetic

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26. A plot of complex numbers as points.
Argand diagram
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
the vector (a -b)

27. When two complex numbers are multipiled together.
Complex Multiplication
interchangeable
Polar Coordinates - Arg(z*)
Complex Number Formula

28. We can also think of the point z= a+ ib as
the vector (a -b)
Polar Coordinates - Arg(z*)
Rules of Complex Arithmetic
Absolute Value of a Complex Number

29. A² + b² - real and non negative
How to find any Power
integers
zz*
conjugate pairs

30. 3
has a solution.
complex numbers
i^3
Roots of Unity

31. Divide moduli and subtract arguments
Polar Coordinates - Division
'i'
a + bi for some real a and b.
De Moivre's Theorem

32. The complex number z representing a+bi.
imaginary
The Complex Numbers
Affix
interchangeable

33. When two complex numbers are subtracted from one another.
Polar Coordinates - cos?
Polar Coordinates - z?¹
Complex Exponentiation
Complex Subtraction

34. (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)
Absolute Value of a Complex Number
z1 ^ (z2)
Complex Number

35. A number that can be expressed as a fraction p/q where q is not equal to 0.
Liouville's Theorem -
Subfield
Rational Number
imaginary

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

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

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38. When two complex numbers are divided.
Complex Division
Polar Coordinates - Multiplication
x-axis in the complex plane
cos iy

39. xpressions such as ``the complex number z'' - and ``the point z'' are now
(cos? +isin?)n
Complex numbers are points in the plane
interchangeable
rational

40. ½(e^(iz) + e^(-iz))
cos z
How to solve (2i+3)/(9-i)
Integers
Rules of Complex Arithmetic

41. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
i²
a + bi for some real a and b.
How to solve (2i+3)/(9-i)
multiplying complex numbers

42. Written as fractions - terminating + repeating decimals
i^2 = -1
rational
x-axis in the complex plane
complex numbers

43. R^2 = x
Square Root
Any polynomial O(xn) - (n > 0)
We say that c+di and c-di are complex conjugates.
multiplying complex numbers

44. The reals are just the
standard form of complex numbers
Rational Number
irrational
x-axis in the complex plane

45. A complex number and its conjugate
Liouville's Theorem -
conjugate pairs
adding complex numbers
z1 / z2

46. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Real Numbers
i^4
Complex Multiplication
conjugate

47. 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
subtracting complex numbers
(cos? +isin?)n
For real a and b - a + bi = 0 if and only if a = b = 0
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

48. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Subtraction

49. For real a and b - a + bi =
We say that c+di and c-di are complex conjugates.
Polar Coordinates - sin?
0 if and only if a = b = 0
Integers

50. Any number not rational
How to multiply complex nubers(2+i)(2i-3)
cos z
i^4
irrational