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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. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
integers
Any polynomial O(xn) - (n > 0)
v(-1)

2. 1
natural
Complex Number
i²
|z| = mod(z)

3. 4th. Rule of Complex Arithmetic
has a solution.
Complex Numbers: Multiply
Polar Coordinates - Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i

4. A + bi
zz*
imaginary
rational
standard form of complex numbers

5. (e^(-y) - e^(y)) / 2i = i sinh y
i^2 = -1
i²
How to solve (2i+3)/(9-i)
sin iy

6. 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.
Argand diagram
Liouville's Theorem -
How to find any Power
How to solve (2i+3)/(9-i)

7. Multiply moduli and add arguments
Complex Number
Affix
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication

8. When two complex numbers are multipiled together.
Polar Coordinates - r
Complex Multiplication
De Moivre's Theorem
complex numbers

9. Any number not rational
transcendental
Complex Conjugate
irrational
Subfield

10. The reals are just the
x-axis in the complex plane
Complex Subtraction
We say that c+di and c-di are complex conjugates.
radicals

11. Numbers on a numberline
integers
conjugate pairs
sin iy
subtracting complex numbers

12. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
How to solve (2i+3)/(9-i)
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - z
multiplying complex numbers

13. I
Rational Number
Real and Imaginary Parts
i^1
Imaginary number

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

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15. Has exactly n roots by the fundamental theorem of algebra
natural
Roots of Unity
Complex Numbers: Multiply
Any polynomial O(xn) - (n > 0)

16. (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)
How to solve (2i+3)/(9-i)
ln z
z1 ^ (z2)

17. Root negative - has letter i
imaginary
'i'
the vector (a -b)
subtracting complex numbers

18. Equivalent to an Imaginary Unit.
Real and Imaginary Parts
non-integers
Polar Coordinates - Division
Imaginary number

19. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Subfield
Complex Division
Roots of Unity
i²

20. 1
irrational
subtracting complex numbers
cosh²y - sinh²y
ln z

21. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Complex Conjugate
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane

22. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
Complex Addition
How to multiply complex nubers(2+i)(2i-3)
z1 / z2
adding complex numbers

23. (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
How to multiply complex nubers(2+i)(2i-3)
rational
transcendental
The Complex Numbers

24. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
the complex numbers
Integers
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to add and subtract complex numbers (2-3i)-(4+6i)

25. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - Division
Complex Numbers: Add & subtract
Absolute Value of a Complex Number
Complex Conjugate

26. 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 Multiplication
Subfield
We say that c+di and c-di are complex conjugates.
Imaginary Unit

27. V(x² + y²) = |z|
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Numbers: Multiply
interchangeable
Polar Coordinates - r

28. Starts at 1 - does not include 0
e^(ln z)
real
Affix
natural

29. All the powers of i can be written as
i²
e^(ln z)
Complex Numbers: Multiply
four different numbers: i - -i - 1 - and -1.

30. The complex number z representing a+bi.
natural
Polar Coordinates - Multiplication by i
How to add and subtract complex numbers (2-3i)-(4+6i)
Affix

31. I
v(-1)
Polar Coordinates - Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

32. R?¹(cos? - isin?)
point of inflection
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary Numbers
Polar Coordinates - z?¹

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

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34. 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
conjugate
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Conjugate

35. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
0 if and only if a = b = 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
radicals

36. 5th. Rule of Complex Arithmetic
interchangeable
complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Arg(z*)

37. R^2 = x
Square Root
Polar Coordinates - Multiplication
Polar Coordinates - Division
Euler's Formula

38. Divide moduli and subtract arguments
Polar Coordinates - Multiplication by i
Polar Coordinates - Division
multiply the numerator and the denominator by the complex conjugate of the denominator.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

39. V(zz*) = v(a² + b²)
|z| = mod(z)
conjugate
The Complex Numbers
a + bi for some real a and b.

40. A² + b² - real and non negative
has a solution.
For real a and b - a + bi = 0 if and only if a = b = 0
zz*
Imaginary number

41. Real and imaginary numbers
complex numbers
Euler's Formula
Imaginary Numbers
De Moivre's Theorem

42. x + iy = r(cos? + isin?) = re^(i?)
Every complex number has the 'Standard Form': a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.
Polar Coordinates - z
non-integers

43. No i
real
i^2
z1 ^ (z2)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

44. 1
Irrational Number
-1
i^2
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

45. To simplify the square root of a negative number
Complex Subtraction
Argand diagram
Complex Number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

46. A plot of complex numbers as points.
Polar Coordinates - Multiplication by i
complex
Argand diagram
Liouville's Theorem -

47. Not on the numberline
Complex Subtraction
The Complex Numbers
non-integers
'i'

48. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
standard form of complex numbers
The Complex Numbers
Complex Conjugate
ln z

49. Written as fractions - terminating + repeating decimals
rational
Rules of Complex Arithmetic
Polar Coordinates - z
multiplying complex numbers

50. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

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