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CLEP General Mathematics: Complex Numbers

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. A number that can be expressed as a fraction p/q where q is not equal to 0.
Euler's Formula
e^(ln z)
Rational Number
transcendental

2. A subset within a field.
Affix
Subfield
Polar Coordinates - Multiplication
Any polynomial O(xn) - (n > 0)

3. 1
Roots of Unity
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Multiplication
i^4

4. Imaginary number

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5. A number that cannot be expressed as a fraction for any integer.
x-axis in the complex plane
i²
Irrational Number
adding complex numbers

6. No i
Subfield
standard form of complex numbers
conjugate
real

7. The square root of -1.
i^0
Rational Number
Complex Number Formula
Imaginary Unit

8. 1st. Rule of Complex Arithmetic
complex numbers
i^2 = -1
x-axis in the complex plane
Complex Exponentiation

9. 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
Complex numbers are points in the plane
zz*
subtracting complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)

10. x / r
Euler's Formula
Polar Coordinates - z
Polar Coordinates - cos?
Polar Coordinates - z?¹

11. V(x² + y²) = |z|
Polar Coordinates - r
Euler's Formula
z1 / z2
Euler Formula

12. 1
Polar Coordinates - Division
multiplying complex numbers
i^0
Imaginary Unit

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

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14. To simplify a complex fraction
Absolute Value of a Complex Number
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
point of inflection

15. y / r
Complex Subtraction
Complex Exponentiation
x-axis in the complex plane
Polar Coordinates - sin?

16. A plot of complex numbers as points.
Complex Number
Euler Formula
cos z
Argand diagram

17. I
z + z*
i^1
Complex Exponentiation
conjugate pairs

18. z1z2* / |z2|²
z1 / z2
Square Root
(cos? +isin?)n
Integers

19. Have radical
Polar Coordinates - z?¹
Affix
radicals
We say that c+di and c-di are complex conjugates.

20. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
Complex Addition
conjugate pairs
Polar Coordinates - z

21. All numbers
complex
-1
'i'
For real a and b - a + bi = 0 if and only if a = b = 0

22. R?¹(cos? - isin?)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication
i^0
Polar Coordinates - z?¹

23. 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.
Complex Numbers: Multiply
Complex Conjugate
point of inflection
z1 ^ (z2)

24. 4th. Rule of Complex Arithmetic
non-integers
'i'
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex numbers are points in the plane

25. 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.
(cos? +isin?)n
ln z
How to find any Power
Polar Coordinates - Division

26. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
Complex Exponentiation
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Real and Imaginary Parts
natural

27. To simplify the square root of a negative number
i^4
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - z
Imaginary Unit

28. Equivalent to an Imaginary Unit.
Polar Coordinates - z
Imaginary number
(a + c) + ( b + d)i
integers

29. xpressions such as ``the complex number z'' - and ``the point z'' are now
Roots of Unity
Imaginary Unit
We say that c+di and c-di are complex conjugates.
interchangeable

30. ½(e^(-y) +e^(y)) = cosh y
cos iy
Complex Number
non-integers
How to add and subtract complex numbers (2-3i)-(4+6i)

31. Multiply moduli and add arguments
ln z
Polar Coordinates - Multiplication
interchangeable
Polar Coordinates - z?¹

32. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Numbers: Add & subtract
imaginary
multiplying complex numbers
ln z

33. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
multiply the numerator and the denominator by the complex conjugate of the denominator.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
adding complex numbers
ln z

34. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i^4
Integers
Absolute Value of a Complex Number
How to solve (2i+3)/(9-i)

35. Derives z = a+bi
Every complex number has the 'Standard Form': a + bi for some real a and b.
x-axis in the complex plane
Euler Formula
integers

36. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
Complex Division
the complex numbers
the vector (a -b)

37. ½(e^(iz) + e^(-iz))
'i'
four different numbers: i - -i - 1 - and -1.
cos z
Integers

38. 5th. Rule of Complex Arithmetic
Complex Division
real
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers

39. E ^ (z2 ln z1)
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Number
Polar Coordinates - Division
z1 ^ (z2)

40. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
irrational
Euler Formula
Real Numbers

41. 1
e^(ln z)
irrational
cosh²y - sinh²y
Complex Number

42. I^2 =
-1
cosh²y - sinh²y
Complex Number
Irrational Number

43. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
complex numbers
transcendental
interchangeable

44. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
Complex Conjugate
e^(ln z)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

45. 1
irrational
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^2
0 if and only if a = b = 0

46. A + bi
i²
standard form of complex numbers
ln z
interchangeable

47. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Multiplication by i
'i'
sin z
Every complex number has the 'Standard Form': a + bi for some real a and b.

48. Starts at 1 - does not include 0
Polar Coordinates - r
i^3
has a solution.
natural

49. When two complex numbers are added together.
Complex Addition
Polar Coordinates - Arg(z*)
i^2
integers

50. 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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^2 = -1
multiply the numerator and the denominator by the complex conjugate of the denominator.
the complex numbers

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CLEP General Mathematics: Complex Numbers

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