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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

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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. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Polar Coordinates - Multiplication
z + z*
|z| = mod(z)
conjugate

2. Given (4-2i) the complex conjugate would be (4+2i)
Imaginary Unit
subtracting complex numbers
Complex Conjugate
Field

3. A number that can be expressed as a fraction p/q where q is not equal to 0.
i²
a + bi for some real a and b.
Rational Number
0 if and only if a = b = 0

4. No i
real
Polar Coordinates - sin?
Euler Formula
How to find any Power

5. Every complex number has the 'Standard Form':
Argand diagram
Polar Coordinates - Arg(z*)
Polar Coordinates - Multiplication
a + bi for some real a and b.

6. Numbers on a numberline
radicals
Argand diagram
(a + c) + ( b + d)i
integers

7. I^2 =
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Irrational Number
-1
Polar Coordinates - Division

8. Divide moduli and subtract arguments
standard form of complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Division

9. 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
i^2
We say that c+di and c-di are complex conjugates.
'i'
Polar Coordinates - Multiplication

10. The square root of -1.
The Complex Numbers
Imaginary Unit
(cos? +isin?)n
z - z*

11. E ^ (z2 ln z1)
Complex Number Formula
|z| = mod(z)
z1 ^ (z2)
|z-w|

12. 2ib
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division
z - z*
Polar Coordinates - Multiplication

13. Imaginary number

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14. For real a and b - a + bi =
Complex Number Formula
0 if and only if a = b = 0
Complex Division
four different numbers: i - -i - 1 - and -1.

15. 1
zz*
the complex numbers
i^2
conjugate

16. A + bi
Argand diagram
Complex Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
standard form of complex numbers

17. 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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18. Written as fractions - terminating + repeating decimals
Real and Imaginary Parts
rational
Imaginary Unit
Imaginary Numbers

19. Starts at 1 - does not include 0
Imaginary Unit
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cosh²y - sinh²y
natural

20. x / r
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Add & subtract
cosh²y - sinh²y
Polar Coordinates - cos?

21. A complex number and its conjugate
Complex numbers are points in the plane
conjugate pairs
|z| = mod(z)
(cos? +isin?)n

22. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
radicals
How to add and subtract complex numbers (2-3i)-(4+6i)
i^2
the complex numbers

23. The complex number z representing a+bi.
v(-1)
Affix
Complex Addition
transcendental

24. Where the curvature of the graph changes
Euler's Formula
point of inflection
De Moivre's Theorem
Polar Coordinates - sin?

25. (e^(iz) - e^(-iz)) / 2i
Complex Conjugate
point of inflection
Polar Coordinates - sin?
sin z

26. Not on the numberline
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication by i
non-integers
z1 / z2

27. To simplify the square root of a negative number
complex
natural
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - sin?

28. A subset within a field.
How to find any Power
Subfield
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex

29. 1
imaginary
Polar Coordinates - z
has a solution.
i^0

30. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
irrational
complex numbers
Field
How to find any Power

31. xpressions such as ``the complex number z'' - and ``the point z'' are now
Subfield
cos z
z1 / z2
interchangeable

32. R^2 = x
Field
z + z*
Square Root
Polar Coordinates - Multiplication

33. V(zz*) = v(a² + b²)
i^4
|z| = mod(z)
Real Numbers
e^(ln z)

34. Real and imaginary numbers
z1 / z2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
z1 ^ (z2)
complex numbers

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

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36. Root negative - has letter i
Argand diagram
subtracting complex numbers
imaginary
0 if and only if a = b = 0

37. When two complex numbers are subtracted from one another.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
standard form of complex numbers
Complex Subtraction
multiplying complex numbers

38. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Complex Addition
Roots of Unity
radicals
|z| = mod(z)

39. y / r
transcendental
z + z*
Polar Coordinates - sin?
cosh²y - sinh²y

40. x + iy = r(cos? + isin?) = re^(i?)
How to find any Power
Polar Coordinates - z
Polar Coordinates - Division
can't get out of the complex numbers by adding (or subtracting) or multiplying two

41. The field of all rational and irrational numbers.
Real Numbers
Polar Coordinates - Multiplication
Polar Coordinates - z?¹
How to find any Power

42. (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
a real number: (a + bi)(a - bi) = a² + b²
adding complex numbers
z1 / z2
How to multiply complex nubers(2+i)(2i-3)

43. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
multiplying complex numbers
Real Numbers
i²

44. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
the vector (a -b)
z - z*
Imaginary Numbers
The Complex Numbers

45. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Rational Number
0 if and only if a = b = 0
Absolute Value of a Complex Number
|z-w|

46. ½(e^(iz) + e^(-iz))
cosh²y - sinh²y
cos z
i^4
real

47. (a + bi) = (c + bi) =
Complex Exponentiation
Argand diagram
(a + c) + ( b + d)i
Imaginary Numbers

48. I = imaginary unit - i² = -1 or i = v-1
|z-w|
Polar Coordinates - Division
Imaginary Numbers
z1 / z2

49. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
How to solve (2i+3)/(9-i)
Polar Coordinates - cos?
the vector (a -b)

50. V(x² + y²) = |z|
Polar Coordinates - r
Complex Numbers: Multiply
Complex Subtraction
Complex Number Formula

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CLEP General Mathematics: Complex Numbers

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